Sacred Geometry Foundation

1. Welcome: now in this course, we're going to be exploring sacred geometry. But you might be wondering what is sacred geometry anyway? And this is actually a very difficult question to answer. It's akin to asking, What is the meaning of the universe or what is the meaning of life? These air giant questions. But let me take a stab at it. Sacred geometry is the study of the qualitative aspects of geometry, and this is different than the quantitative aspects that you may have learned in school. You see, geometry can be used to explore the inner world as well as the outer. And in that regard, it's really amazing and unique. Geometry is the universal language when you think about it. If there was an alien intelligence, how would we communicate with it? We couldn't use language. They wouldn't know our languages. We couldn't really use mathematical symbols because symbols are abstract in particular to the culture that they're in. We have to use geometry because it's a universal language, a universal means of expression. And as such, geometry is very close to the idea of truth with a Capital T, and this is really refreshing in this age, of post truth to reconnect with what Israel and what is true, it gets into the platonic idea of the transcendental of where you find truth. You also find beauty in goodness, and so the study of sacred geometry can be an inner voyage of discovery, but it can also be used as a lens to explore the universe outside. So it's really amazing in that regard, and it can also unlock whole vistas of creativity for you. The more you work with geometry, the more it transforms your consciousness and the more it activates you. So in this course, to get there to unlock those doors, we have to work step by step. And in the first chapter, we're going to build up some essential skills, like bisecting a segment or bisecting an angle, drawing a parallel or perpendicular. These are things you may have learned in school, but we have to revisit them now and to build up these building blocks of how we can work with geometry. And then you'll learn how to divide segments and how to construct regular polygons. And all of these skills are entertaining and interesting ways of working with geometry. And it's something you have to experience. It's not so much intellectual exercise is something that you have to do for yourself. So in this course you can do it with a straight edge in a compass on paper. Or you can do it using the free app. Either way, you'll be able to explore geometry. But the essential thing is not just to watch me do it, but to do it for yourself. Also, that's where it will actually transform you. And once we build up these basic skills, we can actually get into the sacredness and experiencing the sacredness of geometry when you start to see how the golden ratio is connected with squaring the circle in the 345 triangle and the proportions of the moon and the Earth, and how all of that is encoded in the Great Pyramid and it goes on and on. So hopefully this course will act as a entry point for you into exploring not only in our world but also the outer world and exploring the mysteries of physics and how fine artists have encoded these geometries in their works and how city's air often designed using specific geometries. So all of this is available to you Should you go down this road of exploring sacred geometry? So what are we waiting for? Let's dive in. 2. Traditional vs digital drawing: there's certain advantages of the digital medium over the traditional one. So if I make a drawing like this in the digital medium, everything is more accurate. I can draw it much more quickly. I can change colors right away. I could Inc these by hand, but it would take a lot of set up to put the red ink in a special attachment on my compass , and it would be much more deliberate in a more lengthy process. But it might be more satisfying to do it by hand. I find that the digital medium is more mental, more intellectual, and then the ah, the physical medium is more about body awareness, about breathing, about lighting. Each one has its beauties. In this course, I'm going to capture everything off of the digital app because it's much more practical. My body and hands aren't in the way. When I'm drawing, I'm using an iPad with an apple pencil, and I really love the apple pencil. It's a wonderful input device. It's like I've gone for decades using a mouse, and I've just rediscovered the joys of the pencil. So if you do, you have an iPad that supports an apple pencil. I think it's a really worthwhile investment. If you don't that's fine. You can. If you have an iPhone, you can use this app with your fingers. That's totally fine. The only disadvantage of using your fingers is that it's less accurate. You can't quite see what you're doing because your finger is covering up the place where you're drawing, but it still will work, and in this course you can draw by hand or you can do it in the APP And this. This app, by the way, is the best that I've found, and I've tried pretty much every app that's used for drawing. And the developers of this app have really streamlined it as much as possible. And they keep the experience very close to the traditional experience of drawing by hand. And I really appreciate the fact that we don't have to enter any numbers in this app. It's not really a cad program, so you shouldn't be looking for those kind of features like layers and in putting by numbers and so on. It's just not here. We're just trying to replicate the experience of drawing by hand. And that said, there are certain time saving features in the APP. So, for example, I'm just gonna pan over here with my fingers, just dragging two fingers across the screen and I'll just drawing a line segment here. So if I wanted to bisect that line, that is to divide it into two equal segments. Traditionally, I would have to draw a circle and put the compass point on one end point of the line and draw that circle out. Let's say to the other end, and then I could do the same thing over here by putting the compass point at this end point and dragging, dragging it over to the other end point. And then I could put my pencil right at this intersection point. Make a point. What I would do by hand is I would position the pencil point there and then turn the pencil to kind of drill down and macon indentation. But here in the app, I can just use this intersection tool in just tap there, and it puts a point in very precisely. And then I can drawing a line segment here, connecting these two dots, and then I can place a point at their intersection and I can see that the line has been bisected. Now, I'm just gonna undo by tapping on this arrow here several times. And so if I wanted to buy sect that using a tool in the op, I could do that by using this button right here in That works by dragging from one point to the other. And there it is. I've bisected the line just like that. So it's kind of cheating in a way, because we don't have to go through all the steps that we would in the traditional medium. And I was unfortunate enough to be old enough to have started my architectural career as a manual draftsman. So I got all that training of how to do it by hand, and I feel enriched by that. So if you've never done things by hand, I really highly recommend trying that out. You're gonna learn some new qualities of drawing that you can only get by that manual method in the app. I have another kind of cheat tool right here next to this one, and I can drag from one point to the other. It automatically puts in the perpendicular. So again it's faster and easier to use the app, but we miss out a bit on the physicality of the experience. So I'll leave it up to you to decide which way you're going to take this course, whether you're going to do it by hand with a compass in a straight edge, or you if you're going to use this free app to do your drawing. 3. Manual drafting tips: in most of this course, I'm going to be drawing with an app, and so I'm gonna be taking screen captured videos of the APP. And there's a lot of advantages to that method because when I'm drawing by hand, my hands get in the way and you can't quite see what I'm doing yet with the app. There's no such obstacle. But I want to make one video using the traditional instruments just to give you some tips on using instruments like these, and also to give you my personal take on the experience I really love drawing by hand. It's something that really you should experience for yourself. There's something magical about using your whole body, about hand eye coordination, about the whole experience that is much more fulfilling honestly than drawing in a nap. The APP is much more convenient. It's more accurate, has a lot of advantages for sure, but there's something about this time tested method of drawing with the compass in the straight edge that is really transformative. So I'd like to give you some tips if you're gonna proceed in this manner of using the straight edge just drawing a line. For example, So, first of all, all you need is a straight edge. You don't need a ruler that has divisions on unnecessarily, although these days it's hard to find a straight edge that doesn't have marks on it. So the ruler you can use and that's fine. But just be aware that we're not going to need to measure anything in this course because we're using. We're looking at sacred geometry in a qualitative aspect rather than a quantitative one. If you're taking a cad course or drafting course than measurement would surely be part of it. But in secret geometry, what we want to do is just follow the geometry itself, and it will guide the way. All of the decisions that we have to make are guided by the geometry itself and not measurement. And so for that reason, the traditional instrument is just a straight edge. So if you're going to draw a line, what you want to do is line it up and then hold down the ruler before you draw your line. You want to be aware that you're not gonna want to draw a line like this when you draw it. You don't wanna change the angle that you're holding the pencil at. You want to make sure that's always constant, and so before you draw your line, you want you know you want to think about that, and then as you draw it, you want to rotate the pencil tip. What that does is it keeps the line it keeps. The pencil tip evenly worn down keeps it sharp. Having a sharp pencil is really important, so you can use a mechanical pencil like I have or you can. You have a traditional pencil, but if you do have a traditional pencil, make sure that you keep it sharp. Also, if you're using a compass, there are two kinds of tips that come with compass is there's the There's the chisel point , which is what I prefer because you can sharpen it using a piece of sandpaper, and that's really critical in keeping your circles crisp. The other type that often comes with less expensive compass is is the conical point. But if you have that, you need a special sharpener to keep it sharp, and you really need to pay attention to that because accuracy is really very much related to the thinness and accuracy of your lines. Okay, So if I want to draw a circle, the first thing I'm gonna do on this line is located point for the center. Let's say it's gonna be right here. I'm gonna hold down the pencil tip right there and I'm gonna press into the paper in turn, and this actually drills down into the structure of the paper and makes an indentation. And then when I come over here with the compass, I can actually drag the tip of the compass on the paper itself. And I can feel when that drops into the hole, and that will give me a tactile verification that I'm in the right place and then I'm going to draw the circle. And then if I want to draw another circle to create the Jessica Discus form, I'm going to position the point of my pencil here very carefully, drill down and make a hole there. When I do that, I can also label these A and B, and then I'll come back here and again, feel the point dip in there. And then when you make the circle, a nice thing that will happen is when you come around and go through A. You'll actually feel the lead drop into that hole, and that will give you confirmation that you're in the right place. So those were just some tips I have for using a compass and using a straight edge, trying to be as accurate as possible. 4. Organizing sketches with tags: welcome to Euclidean sketches, and I'd like to start by showing you how toe navigate through the interface here and open up different drawings and how to tag them. So the APP ships with a number of samples that you see here. It also has some art samples. I'll click art samples tagged here on the left. You can see them a different selection of sketches here say I want to go into a sketch of this tap on path Aguero's tree, and that takes me right in there. If I want to go back, I can click the icon here in the upper right and then click the grid icon to go back to this grid of sketches. You'll notice that I have a lot of other tags here because I've I've been using the APP for some time. I have, for example, traditional constructions and has a lot of different sketches that I've made over time. It has polygons aesthetic subdivision in these different tags that I have. Just help me keep things organized. If you want to create your own tag, you can do that just by tapping create tag and give it a name. I'll just call this test, and so now I have a tag. Right now, there's nothing tagged as test. So let's fix that good of samples and then click select in the upper right. And I'm going to tap on a couple of different sketches here that I'd like to tag as test and then I'll go to the tag icon at the top and tag that as test noticed that it has test and samples both. I'll say, done and then I'll go to test. You can see that these now show up in this list, and they also show up under samples because they're tagged with two different tags. This does not duplicate the drawings. It just is a different way of viewing the same information. So tags air just for organisational reasons. In fact, if I delete this tag now, if I come up here and click on this Lipsitz button right here at the top, I could rename the tag or delete it, say delete tag. Are you sure you want to delete it? Yes, that does not delete those drawings. If I go back to samples, you'll see that those drawings air still in here. There are few other tax that are worth mentioning. If you click on all at the top, it shows you all of the sketches that you've ever made. If you had a tag lis, it'll show any sketches that don't have any tags. If you go to downloads, you'll see sketches that have been shared with you by someone using air drop. This is not downloading off the Internet. It's downloading and a peer to peer kind of way. And we could do that if we went to a specific drawing. Let's say I'm going to select this one here and then I'll share that. And then if I had a person next to me who had airdrop on, they would show up at the top of the screen and I could share this drawing with them, and then the drawing would show up on their screen under downloads. There's also a trash category, so these have been deleted. So if I select particular drawing here and then I can delete that forever right here, or I could restore it back to where it was originally. In this case, I will delete it because I know I have a copy of that so that's pretty much all you need to know about tags and how to navigate through the different sketches that you'll be creating in this course. 5. Sketch management tips: in this video, we're going to create a new sketch and will tag it and will also delete it so that you have the full complement of file management skills under your belt. Let's begin by tapping on new sketch at the upper left, and this starts us off with a clean slate on the bottom. We see the segment tool is active, and that works by dragging from one place to another. It's very straightforward. Let's just draw one line segment and then optionally. You can give this a name in the upper left tap on that red pencil and then tap on title and you can give this a name. I'll type line, has the name and then dismiss that info window. And then let's go over to the upper right menu icon and save this and you do that by topping the red disc, get icon and then reopen that menu and go back to the dashboard and go toe tag all and you'll see that there's the line drawing that we just made so I can tag this by selecting it here and then going to the tag icon, and I'll tag that as one of my sketches and I'll say, Done So now if I go to the my sketches tag shows up in this list, let's say I want to get rid of that drawing now. After all, it's just a line. I don't really want it. I'll select that again, and this time I'll click on the trash icon and that moves the drawing into the trash category so you can go to the trash and you can see it in there. You can leave it there, or you can delete it permanently if you want, by selecting it again and deleting it forever. Observe that the line drawing still seems to be in this icon here, just under the word new sketch. That's kind of a temporary buffer, and that will exist until you create a new sketch so I can actually access that if I top on that. But it's unsaved, it's just in the buffer, and I see that because the icon of the pencil on the upper left is red. That means it's unsafe to I could save this again and then go back to the dashboard, go to all and there it is. So there's actually a safeguard system built in to the op. You just have this safeguard as long as you don't create a new sketch. If I create a new sketch now and then let's a draw a circle and then go back to the dashboard. Well, now the buffer just has that circle sketch in it, so that could be convenient if you're just going back and forth. But you should always. If you make a drawing that you want to keep, you should always make sure to save it. I'll save this now. No that to save it. You don't actually need a name, but it saved now, and it shows up in my grid. So titles are entirely optional, and you can easily change titles later on down the road. So I think that's pretty much everything you need to know about file management in the op. It's very straightforward. It's pretty forgiving, and I think you'll find it easy to use 6. Bisecting line segments: I'm going to buy sector line in this video, and I'll start by using the traditional construction technique, and then later I'll show you some quick ways of doing it in the APP. Let's begin by drawing in a line segment of arbitrary length. Now I'd like to line this up with the screen. So I used the hand tool and then used two fingers on the screen to pan or zoom. And also, if you rotate your two fingers relative to one another, you can rotate that on the screen. What I'm going to do is use the top edge of the screen as a guide to help me line this up and make it horizontal. Then I can bring that back in the center of the screen and maybe zoom out somewhat like that. Note that the bottom of the screen doesn't really work as a guide because it fades out down there. If you want to make a line vertical, you could use the left or the right edges of the screen. Either way, it's fine, so this is simply analogous to drawing a line on a piece of paper and rotating the paper. Just a good way to get started. Now, I'd like to label the end points of the line so that we have a common vocabulary, and you can do that by choosing one of the different tool sets at the bottom of the screen . Right now, we're in the drawing tool set. There are different tools that's here's a decoration tool set the color and the label to a sense. I want to use the label tool set, and it's icon is Letter A and then let her Alfa. The Latin alphabet is useful for labelling points, and the Greek alphabet is more commonly used to label angles. So I'll stick with the Latin alphabet and then I'll just click the left endpoint in the right endpoint. Have it automatically label these with sequential letters in the alphabet. Incidentally, if you wanted to change a letter, you can just drag left and right on this control down here. Let's say I wanted that to be point G. I can then just tap on the right end point. It changes the label just like that. You can also have numerical sub scripts if you like. So let's say we wanted this to be G sub one. I could then tap on the item here on the right, and that opens up a different control with numbers in it. So now it's G one. If I tap on the right, it should say G one now. So this is a very flexible way of labeling things. I want that to say Be, though. So I'm gonna go back here, make that's a dash, then tap on the left. You make this Sebi, and then if I top on the end point on the right, should change it back to letter B. So now we have line a B, and my goal is to bisect that in the middle. I'll go to the drawing tool set here, and I'll drawing a circle from A to B and then another circle from B to A. This is called the Vesa Capisce CAS form, which means vessel of the fish. It's also known as a man, Darla or almond. It's Ah, very common foreman, sacred geometry, and it's the basis of many geometric constructions. Each one of the circles has its CenterPoint on the circumference of the opposite circle, so it's a symbol of duality, of equality and It's the also known as the womb of creation, where a lot of geometric forms spring forth out of this very simple and fundamental relationship. In this video, we just want to bisect the line, and we can do that by drawing in a vertical. But first we need to figure out where we're drawing that from In those two circles intersect each other at the top in the bottom, and we need to identify those points first. There's a tool here on the right of the toolbar, which is Intersection No, just tap on the top and the bottom of the vesicles, and that identifies those two points accurately. Now I'll go to the label tool, and I'll just label that cnd by tapping on those two points again. And then we'll go back to my drawing tool set. Draw a line from Point C to D when then put in an intersection point or the two lines cross , and then I label that this point E. Now we'd like to determine if a equals E B. It looks like they do, but we can't really be sure unless we check it out and that's here under decorations. Now They're different ways to decorate your drawing, and you can do it by measuring lengths of lines, angles, angles and degrees or right angles. I'm concerned with the lengths of lines, and the way that this works is you drag from 0.80 and then you drag from Point A to B and will only show up as a decoration if in fact it is truly equal and you can see that it iss . This can be toggled off it on, and that's proof that leave bisected the line. So this is how you would do it traditionally, although you might not draw the whole circles, you might just draw little arc segments to locate points cnd. This would be the technique that you had used to bisect any line segment. If you're using the APP, you don't need to go through all of these steps necessarily. For example, I'm going to zoom out and drawing a new line segment below of some arbitrary length. I'd like to label that points F N G. Now I'd like to buy SEC that, and there's a specialized tool for that purpose right here, and it works by dragging from Point F to G and it puts in the by section point right there . So I'm gonna label that and then I'll verify that F H equals H G. And it does. There's a similar tool adjacent. This puts in the perpendicular by sector so I can drag from F to G. Not only does it bisect the line, but it also puts in the perpendicular. In this case, I don't really need that. I'm going to undo by tapping the arrow in the lower right hand corner of the screen to go back a step. So now you see how you can bisect the line Eser either. Using the traditional method, we're using the specialized tools in the APP. 7. Controlling drawing element appearance: this drawing isn't as aesthetically pleasing as it could be because it's all grey. I'm going to zoom into this vessel capisce, CAS form here and show you how you can colorize the drawing that's available here in this third tool set from the bottom. It looks like an artist's palette of paint, and we have a whole list of different colors that you can scroll through on the bottom. Here. The color that is default is on the extreme right edge of the palate. It's this one right here. I'm going to change the colors here. I'm going to scroll over to read and choose Bright red. And then I'm going to click on this, uh, interface element here, and I'm going to choose a thick line, and then I'm going to tap right on the line, a B and that automatically applies that style to the line. Now I'm going to de select the red color and change this style to this dash style here in the middle, and then I'm going to tap on the line CD, and that doesn't colorize it. It just changes the line type to a dashed line type. If I wanted to colorize it. I could select a color, let's say black and then tap on the line. It will colorize it without affecting the line type. You can toggle these modes on and off by tapping on them. So if I if I toggle off black and toggle on dashed, I can then tap on the two circles. If I wanted them to be affected in that way, or I could come in here and reset that back to a thin line and make that black, see how that looks, that's that has a nice look to it, very precise looking. You can also decorate the points and thats done here. In this interface, the points have different styles you can choose from here. I'm going to go with the top style, which is the thickest point, and I'm going to use a bright red dot on points A E and B because this drawing is really about bisecting a line and those are the most significant points so we can leave it like that. I think the drawing is much more readable, with a little bit of customization here in terms of color, line type and point styles 8. Bisecting angles: in this video. We're going to bisect some angles, and we'll do that with lines. They're different line tools at the bottom of the screen. So far, we've explored the segment tool, and that makes a line segment that has start and end points. There's also the line tool, and this draws a line that doesn't have start or end points. In fact, if you zoom out, you'll never get to the end of the line. It goes off infinitely. The Ray Tool is sort of a hybrid of the segment, and the line in that it starts at a point, but it goes off in one direction infinitely. All this undo and just draw in one line segment here, and I want that to be relatively horizontal. So I'm going to line that up top of the screen, and that's good. I'll draw another line here that crosses the other line at some arbitrary angle in a label , This point a there to bisect the angle. A. We could do it on either side. Let's say we try to bisect in the upper right, so what I'm going to do is start by drawing a circle from a out here at some arbitrary size . And then I'll label these two points here in here. Note that the label tool automatically adds the intersection points for you. There's no need to do that as a separate step. Now I'm going to draw in another circle from B again at some arbitrary size. Now draw the same sized circle at Point C. This is something that's very easy to do with the compass, because you just don't adjust the compass width and you draw another circle in the APP. We have to go to an additional step to make this happen. We have to first put in a label here at this point, and then we will use this specialized tool right here. What this tool does. Is it copies of length of a segment into another location? So I'm going to drag from Point B to D, left the pencil or lift your finger and then drag from Point C. And as I'm dragging from Point C, I can place that parallel to the other segment or anywhere along this circle for convenience. I think I'll drop it right here. Then I'll draw another circle from seeing over. So circles BNC have the same radi i A label this point right here and draw a line connecting a and E that line by sex, the others. To make this more clear, let's decorate. I'm going to change the line type to a small dash, and I'd like to dash out the circles and then to make the lines more prominent, I'll choose a color, turn off the dash line type and then tap on the lines. Now, just to verify that we have indeed bisected the lines. Let's go here she was the angle, and I'm going to drag from B to a and then again from 80. And that establishes ankle B A. And then I'll do again from E T. A and then a to C. There's your proof. These two angles are, in fact, equal. So now let's do this in a more efficient way. That's optimized for the app. So just now get over here, drawing a new line down here somewhere and drawing another line at some arbitrary angle, this time a little steeper. I label that as point F in this case. Then I'll use this specialized tool right here, which is for bisecting angles, and I'm going to drag from some arbitrary point on this line down to F lift the pencil and then drank from F over to some arbitrary point along this line that will essentially put in the by sector just so that we can verify this. I'm going to add some points to the lines here that will come in here and measure from G two F and then after h that establishes angle G F h. And then again h two f an f toe I and you can see that this indeed has bisected the lines as well. So you have two different methods one traditional, one digital for bisecting angles. 9. Perpendicular through point not on line: in this video, we're going to draw perpendicular lines again, using traditional and digital techniques that's began by drawing in a line. And then I'll decorate that as a red line. And I'd like to place a point at some arbitrary location on screen like that little label that as point a now my goal is to draw perpendicular line through Point A. It's perpendicular to the red line. To do so. I'm going to draw a line segment from a and I'm not going to try toe eyeball it or estimate its location here. That would not work. Instead, I'm going to draw this line. It's some arbitrary angle like that, and then I'll label. This is Point B. Now I'd like to bisect line a B. I'll do that by drawing in a circle from A to B and another circle from B to a, then under our line, connecting these two points on the vessel Capisce Kiss form, which I label as C nd that also puts in the by section point at Point E draw a circle from Point E. T. A. That's label this point where the circle intersects with the red line right here as F, then draw in a new line segment from A to F, and it turns out that F is perpendicular to B F. Let's verify that first I'm going to decorate it, make it red. Then I'm going to check it out and use this perpendicular tool and click on a F. And then BF, which it's labeling internally as line s one. Because it was the first line that I drew. And because it's highlighted in blue, the decoration is complete. We can rest assured that they are indeed perpendicular. So that is the method that you use with the traditional compass and straightedge. I'm just going to pan over here and put in an arbitrary point, appear somewhere which are label as point G. Now I'd like to draw perpendicular to the red line, using the digital tools in this app. There's a perpendicular tool right here, and the way that this works is a bit counterintuitive because you don't start at G. You start on the perpendicular, you start on the line that you want to draw perpendicular from, and you dragged the pencil or your finger along the red line. And as I move the pointer. You can see that I can draw that perpendicular Lee to the line anywhere I want. So the secret is just to drop it on point g. And there you have it. I'll just decorate that. I label this point and also verify that these two lines are indeed perpendicular. And they are so there you have it. Two different methods for creating perpendicular. 10. Perpendicular through point on line: in this video, I'll show you how to construct a perpendicular line that goes through a point on the line. So for this we need a different procedure. I'll draw a circle from Point A on the line at some arbitrary size, and then I'll label this point B and the opposite point C. Then I'll draw another circle from B. This needs to be big enough so that it goes beyond point A a label, this point de Now, if you're using a compass, you would draw another circle from Point C with exactly the same radius as the previous, and you'd have the circle immediately if we're using the APP. We have to use this tool to copy the distance from B to D, and then we'll start from C. We'll drop that in right here. Optionally we could construct the full circle, which is what you would do with a compass. I label this s Point E. No, I'll draw a line connecting in a and that will be perpendicular to the red line. I'll make this line red and just verify that it's perpendicular. Indeed, it iss a pan over here and place a new point on the original line, which I'll label as F. And then if we're using the digital technique, we'll use the perpendicular tool and then drag along the red line until you reach F and then lift your pointer and that's all there is to it. I'll just make that read and again verify that these two lines are perpendicular and indeed they are. So there you have it different ways of drawing perpendicular lines through a point on the original line. 11. Drawing parallel lines: in this video, we're going to draw a parallel line through the point. A. I'll begin by making a point on the red line at some arbitrary location. I label that be, and then draw a circle from A to B and another one from Beat A. This creates another point over here on the right, which are label see. Create another circle from seed A, and this creates a Vesa Capisce CAS shape, which you could instantly use to bisect segment BC if desired. That's not my goal, so I'll just continue on. I'll place another point over here and label that d I'll draw another circle from D to see . So now we have three circles along the line with all the same size, a located point right here and then draw a line connecting point A. To this new point. E. It turns out that this line is parallel to the original line. I'll just make that read, and then to prove that it's parallel, we'll have to draw in a diagonal line somewhere. I can draw that between a and B or between E and C. It doesn't really matter. I'll choose it over here and then, just for sake of clarity, I'll create some points on that line and another one right here and then a label. These points and then measure angles. So I'm going to measure this angle from 80 and then eat F that sets up angle A e f. And then all the measure of the similar angle down below G to see and then c T. And because these two similar laying angles are equal, that proves that these lines are indeed parallel. If I want to create a parallel line using this specialized parallel tool, it's very easy. All you have to do is drag the pointer from eggs in existing line and place it anywhere you want. So if I had a point, let's say right here I could then label that, and then I could make a parallel line that connects to that point. So it's very easy to do using the parallel line tool. But now you know how to construct parallels using the digital tool as well as the traditional method 12. Two and three-point circles: in this video. Let's think about circles and points. When you draw a circle, it emanates from a center point, and it goes to some radius that value that you set. Now, if I want to draw a circle that goes through two points, I'll put two points on the screen here, and I want a circle to go through those two points. To make that happen. I need to bisect the line implied by those two points. I can do that with this tool, and that locates the center point. Now I can draw a circle from the center out to either one of the points, and you can rest assured the circle goes through both points because the center is in the middle of those two points. By definition, it's bisected. Now. What happens if I want to draw a circle through three points, put three points arbitrarily on the screen. Now that's a much harder challenge. The solution is to intersect two of the sides perpendicular by sectors. I'll do that here. I'll draw in the perpendicular by sector between those two points and these two that will locate a center point which I label right here as point A. Now, if I draw a circle from point A, you can have it go through any one of these three points and you can rest assured that it will go through. All three will be precisely tangent to those three points, so you can use this method. Whenever you have a triangle of any sides, you want to have a circle circumscribe it. What you need to do is figure out where the center of the circle is by drawing in two perpendicular by sectors. That would be this point here. Point B. Now I can draw a circle from be out to one of the ends of the triangle, and we have accomplished the goal. There's even a specialized tool for this purpose. If I draw on three points, I can use this tool right here. Just tap on the three points and the up, fingers it out for you. But now you know how it's done. The traditional and the digital methods 13. Circle and triangle relationships: Okay, so we have a triangle, and I'd like to put a circle inside of it that is technically called inscribing a circle. To do that, if you're doing it manually, you would need to bisect at least two of the angles. I'm going to use thespian Shal ized, bisect tool here for that purpose. As I've already shown you in a previous video how to bisect angles geometrically. So I'm going to drag from a to B and then from B to C to put in the by sector. I'll do that again by dragging from B to C and then see the A So we have to buy sectors the point where they cross. I'm going to label d, and that will be the center of a circle that I can draw here. And I want I can drag that out and snap it perpendicular Lee to the lines. That point is now tangent to the line. That means that the circle passes through that line in precisely one point. I label that point E we can optionally find additional points of intersection with this tool on the right edge of the toolbar. I could just tap here and there, and it finds those points of intersection. It's not arbitrary. It's not just kind of close to that. It's precisely there. And if we zoom in here, you can see that is the one point where the circle in the line come together. Incidentally, if you draw a line that goes to a circle close to the edge, that will be called a chord because the line will actually pass through the circle in two points. I'm going to label these points F and G, and while I'm at it, I'll also change the circle to a darker red color. So now I have successfully inscribed a circle within the triangle. Now I'd like to draw a new triangle. All choose a green color for that. And while I'm in this color mode, you can actually draw additional lines, but only lines. You can't draw circles or any other types of objects. So I drew in another arbitrary triangle, and I'll go back to my toolbox by clicking on the compass symbol. Let's say I want to do the same thing again. Well, I can use a specialized tool right here for that purpose and the way it works is you just tap the three end points of the triangle and that adds a point in that point is the center of a circle that is inscribed. I'll make that a dark green color. So what are these other tools under here Now? There's a little icon here in the upper left hand corner, which indicates that this is actually a pop up menu. So the next one here, this one? If you look at the icon, it shows that it divides the sides equally. Let's try that. I'm gonna tap these three end points of the triangle again, and that puts in a new point. I'm gonna label these points H and I So point I is useful if I drawing lines. So I'm gonna drawing lines from I to the end, the endpoints of each corner, they're of the triangle. And then I'm going to label these points here. I'm gonna label everything just by tapping. I can measure a distance here, so m two J should be equal to Jada end. In fact, it is. If you go all the way around the triangle, you'll find that that's true all the way around. So that's what that point is all about. It divides the sides equally. There's some other options here. This one, for example, that will locate the center point of a circle. That circumscribe is the triangle. So if I just tap M O and N, that will place a new point in there, which I will label as Point P. Now, if I draw a circle from Point P, you can see that it's circumscribing the triangle. I'll give that a color as well. We'll make that light green. Let's see, there's an additional tool here that we should look at, and this is the case where lines are perpendicular to the edge. So if I try that, I'll tap M Oh, an end. And that puts in a new point, which I will label que. And now, if I draw a line from Q I M. That line will be perpendicular to O n and similarly for drawing a line connecting que to end cute. Oh, one would expect these lines are perpendicular, but let's just check that out. I'll tap on this line in this line and you can see that they are perverting perpendicular and again this one and this one are perpendicular and finally this one and this one are also perpendicular. So you can now appreciate what these tools do. They find these specific geometric conditions as circles relate to triangles. 14. Lines tangent to circles: in this video, I'm going to examine circles and lines that air tangent to the circles. So, for example, let's say I want to draw a line from Point B over here to the circle, and I want that line to be attached to the circle at just one point of tendency. What you don't want to do is try toe, eyeball it and get it kind of close. It'll be accord where it will miss the circle. You'll never get it exact. What you need to know is how this is constructed geometrically. The solution is to find the midpoint between points A and B all use this tool right here that puts in the mid point, and then I will label that as Point C. I can then draw a circle from sea to a and then also label these two points where the circles cross. Those are the points of tendency, so I'll have to do is drawing the line from B to D and from B, T. E. And those lines are precisely tangent to the circle at those points. We can do this a little bit more easily, and I'll just show you that down below. I'm gonna make a parallel line here which I will call the red and also drawing a circle over here, which is the circle. I'm trying to get something tangent to And I'll make that blue also label that as point F and put a point over here somewhere along the line in a label that point G. So if I want to draw a tangent line now, I can use this specialized tool right here in the way that it works is you can tap on the point first and then the circle and it will put in the line. Yeah, you can tap on the circle first, and then the point either way is fine and it will draw those lines in tangent to the circle optionally if you want, you can find the points of intersection with this tool and those the points of tendency right there, which I will able as h and I 15. Theorem of Thales: in this video, I'd like to point out a geometric curiosity that has to do with circles, and it's actually the oldest Durham of all. It's called the Theorem of Valleys, and I think it's the only the're, um, I'm going to cover in this course, and it's very simple. I'll start by drawing a line, and then I'd like to find the center of that line So you know how to buy sector line, and I'm going to label that with the points A and B and C will go in the middle. I'll draw a circle from sea to a and then I'm going to place an arbitrary point somewhere on the circumference of the circle, and I label that as Point D. Now I just need to connect eight D and D to be it. Turns out, the theorem of Valley says that this angle a D B will always be a right angle. Let's just check that out. It looks like it's true. If I go to the hand tool and move d around, you can see that that relationship always remains true. No matter where I move D. It's kind of an interesting and curious fact 16. Trisecting an angle: in this video, I'm going to show you how to try sect and angle. This is actually considered to be impossible, but it is possible if you use a technique called noises, sometimes called verging or the marked ruler technique. Let's start with two lines crossing like this to form an angle. I'd like to try Sect that angle, my first drawing a circle from Point A at some arbitrary size like that. Then I'll go ahead and label these two points B and C, and I'll draw another circle from C T A and then draw a line from a over to this circle over here. And I want to make sure that as I bring this over, the circle highlights because when I lift my pointer, the end point of that line will be connected to the circle, and I can use the hand tool to move that end point. If I move that around, you'll see that it remains on the circumference of the circle. No matter what I do, this is analogous to taking a ruler and pivoting it at point a and moving it around, and you can imagine different lines that you could draw so if you're doing this by hand, you don't actually draw in this line yet. You're just going to have the potential to draw that line in. But for now, we have to set this up in the APP like this. I'm going to label this point over here as point D and I'm also going to draw in a cross piece that connects being see like that. I need to use a special tool here in the APP, which is this one right here. And this allows me to transpose a distance. If you were doing this by hand, you just open your compass from point A to B. In fact, that's the position the compass should still be in from drawing the circles from before. But what I'm gonna do is drag from a to B to set the distance, and then I'm going to drag from D over here, and I'm going to place this point right on the line and I'm going to zoom in there and you can see that that point is on the diagonal line. It's not on what I'm calling the cross piece. I'm going to label this point point e you know the amount a little bit just to emphasize this condition. I'm going to attach a decoration to this that shows that these two segments are equal. So I'm going to drag from a to B and then I'll drag from E two d. They should be equal. That is the crux of this technique. If you're using a straightedge and compass what you do, she opened the compass to this A b distance and you hold it against the ruler. And then you you pivot the ruler at point A and as you pivot that you can see that one end of the compass has to be on the circle in the other end. Point E has to be at this magic point where pointy crosses that line. It's at this point right here where that were, pointy crosses the line that you've try selected the angle. So just to complete this, I'll go to the toolbox Here, use my by section tool and drag from B to a and then from a t. E. So I've bisected angle B A E. And just to demonstrate this, I'll drag this again. And so the bisected angle remains active, but as you can see it, when I drag this over and e crosses that line, it's at this very angle right there that we've successfully try selected the angle. I'll just complete this by drawing in these colors here on these lines to show that these are the lines that have been these air, the angles that have been try sect ID Angle B A C has been divided now into three equal angles. 17. Doubling the cube: in this video, we're going to solve an ancient problem of doubling the Cube as the myth goes. In ancient Greece, there was a plague, and the oracle was consulted. The Oracle said that the only way to stop the plague was to double the volume of their alter, and the altar was a cube. Let's say that the altar had an edge length of one that would give it a volume of one because one cubed is one. If we want to double the volume of the altar, it will have a volume of two. So what would the edge length be then? It turns out that the edge length has to be the cube root of two, because that is the only number that, when cubed, results in two. And it turns out that it's impossible to geometrically find the cube root of two. However, Isaac Newton was the one to figure out that you could do it with a noises technique, which I'm going to show you. So it starts with the vets, a capisce CAS form that you see here, then draw a line connecting the top of the vesicles with the left side. Incidentally, I've colorized segment A, B and Red because we're going to assume that that is the unit length of the old altar. We need to find the new length, which is the cube root of to. So I need to now put in a point down here which I'll label s Point D And then I'll draw another line connecting D and B, and I'm just going to style this a little bit differently. I'm going to change this to a dashed line, and I'll make it black so that these lines are just a suggestion of where those relationships extend to. And now I'll draw a new line this time our segment from sea over here to some arbitrary point along the horizontal line. And I'll give that a new label E. Now I need to transpose the unit length a B onto this diagonal line. To do that with a compass, you would just move the compass over to E and mark, where it we're intersects with the dog in a line here. What we need to do is use this tool and drag from a to B and then drag from E over in place . The point right there. I'm going to label that as point F. And just to be more clear, I'm going to change the color of this diagonal line to Blue. Actually, I don't want it to be dashed. I'll make it a thick line and make it blue. And then I'm going to turn that off and change the color to Red. And now draw a new line segment from FTE just so that I can change the color of that. And this gives us a visual reminder that a B equals E f. And just to emphasize that I'll make a decoration that proves this a B is equal to E. F. Well, the solution that we're looking for can be solved if we use the hand tool here and move this point e along the horizontal line. Intel Point F crosses that dash line right there that determines the cube root of two Segment CF is the new altar edge link that we need to avoid the plague. So there you have it, a solution to an ancient problem with the noises technique 18. Squaring the circle: in this video, I'm going to attempt to square the circle and I say attempt because this is literally impossible. It was proven in the 19th century that you can't exactly square the circle. What does it mean to square the circle or circle the square? It means to bring the circle in the square into a kind of equality so that they have the same circumference and perimeter. And this is impossible because of the transcendental nature of pie. You see, pie can never really be rationalized and brought down to Earth. The circle represents heaven or the infinite. It's the most perfect of forms, the most symmetrical. The square is the most terrestrial, the most mundane of forms. It's how we divide land. It's how we make grids and create buildings. It's the most. It's the closest to humankind, shall I say. And so squaring the circle is really an activity that brings the mundane together with the transcendent. And as such, it's part of the human spirit to attempt to square the circle, and I'd like to show you an approximation that is 99.9% accurate. Let's begin by drawing a line horizontally and then draw a circle on that line, a label, the center of the Circle Point A and then I'll go ahead and draw another circle and another one and a label These points B and C. Now I'll draw 1/4 circle from Be out to the edge of the construction and a label. These points D and E. I'd also like to put in a perpendicular by sector of the segment D E, and I'll do that using the perpendicular tool here, and I'll drop that in at point B. I'll label the top and bottom crossings as F and G. And now I'll make a circle from G on the bottom, and I'll have that go up here so that it's perpendicular to the edge. There. That makes a tangent point right there, which are label as Point H. And there's a symmetrical tangent point on the other side, which is point I. This large circle also crosses the original horizontal at J and K. This is really all we need to square the circle. At this point, I'm going to draw a large circle from B out to J. This is the circle that we've been looking for I'm going to cull. Arise that as a thick lined red circle and now for the square, the square actually fits right around the smaller circle. So to construct that, I'm going to draw a parallel line from the horizontal appear and drop it at F and another one at G and then going the other way. I'll drop drag a parallel out from the central line and drop that at D and it e. This forms the square that we need. I'll go to the artist's palette here and select a different color, maybe blue, and then I'll go ahead and draw in the square and there you have it. We've squared the circle with 99.9% accuracy That's so accurate. The margin of error is within the line thickness. 19. Method for dividing into arbitrary number of segments: in this video, I'll show you a technique for dividing any segment into an arbitrary number of equal pieces . The advantage of this technique, of course, is its arbitrariness. You condemn vied it into any number of segments that you want. The disadvantage is its complexity. Once you understand the idea, though, the procedure is fairly straightforward. So I have a segment a B here that I'd like to divide into, Let's say, five equal parts. We begin by drawing a line from a at some arbitrary angle and then draw a circle from a at some arbitrary size A label this point. C. Add additional circles according to the number of divisions that you want, so I'll draw a circle from sea to a and then repeat that making equal sized circles. Now I'm going to label these points D, E, F and G. So now we have five equal segments going up the line. We need to transpose this information on to our segment A B, and we're going to do that by drawing in a parallelogram. But first let me style thes objects so that I can draw some emphasis away from the circles . I'll choose the dotted line type, and I'll just tap on each one of these circles so they're not so prominent. And then I'll turn off line type and select black as the color and tap on the diagonal line to make it a little bit bolder. I'll draw a parallel line and place that at B. We'll also make that black. Then I'll draw a line connecting G and B. That's one side of my parallelogram. I need a corresponding line at A. I could draw a parallel, or I can use this tool to transpose a distance. I'll drag from G to be and then from a and I can see right here that it connects. And just to prove to you that this is a parallelogram, I'm going to label this as H and then measure it. Let's say the distance from A to G should equal a distance from H two B, and it does. And then a H should equal G eight g b. And it does okay, so we have essentially a parallelogram here. Because of that, we should be able to copy the circles on the upper edge down to the lower edge, so I'm going to use this tool and measure from A to C and then position it at H in place. That right there now I can draw circles as I did before, and you don't actually need a circle that be because we have all the information we need. Now I'll draw lines Well, First, I'll label these and then I'll draw lines connecting these dots. And it's where those lines crossed The original segment, a B that we have are needed divisions, so I'm going to label these points. So let's verify that we've divided that red line into five equal parts. I'll measure the distance from A to M. That should be the same as MN, and it is and then m and should equal and no and N o should equal Opie an o. P. Should equal PB. So that's proof that we've divided it correctly. 20. Dividing with the Seed of Life: And in this video I'll show you a technique that you can use to divide segments into odd numbers. And this was pioneered by Hannah Jazzy, and I learned about it from the book Drawing Geometry, which is in the bibliography. I'm going to draw a horizontal line to start out, and then I began by drawing a circle right in the middle and a label that okay, B and C. And then I'll draw another circle from B to A and then from C T A. And then I'll label these points up above and down below. And then I'll draw circles from D and E from F and G. And we have this pattern, which is called the seat of Life in sacred geometry. And it's from this pattern that we're going to make our divisions just to emphasize that I'm going to change this to a thick red line from Segment B. C. So that is the segment that I want to divide. I'm going to save this at this point. Let's say we want a divide segment BC into three parts. You can do so by I'm gonna change this to Black in a draw a line from F all the way up to the top point here, which I don't have labeled and then down to G. That actually divides that red line into three parts. And we can verify that by labeling this and then by measuring segment B H should be equal to segment H I. And it is, and then h I is equal to I C. And that's true. So I've successfully divided that into three parts. I'll go up to the menu, appear and then reset. Now let's say I want to divide that into five parts. It can do that again by drawing in some black lines. At first, I'll start by connecting B and E, C and D and then f up to this apex point and then back down to G. This actually gives it gives us enough information to divide this into five parts. What we need to do next is bisect this distance from B over to this point and again from C over to this point. And if I label this and then measuring, you'll see that we've done it right. All of these air equal lets reset again. Now let's try seven parts to do that. I'm going to draw a black line again, and I'm going to start out the same way CD A. I'm gonna do the same below, and then I'll do a diagonal and then up like this and down. So this divides the line and two points, and that should be enough to divide this into seven parts. And we can transpose this distance here from here to there and just keep doing that, and this will divide it as well. Now I've lost the thick red line there, so I'm just going to replace that and I'll erase this central point again. I can prove it to you by labelling it and measuring it. I'm just going to jump from there to way over here. Let's say to show that those are equal and I'll do the end as well. There, in fact, all equal. I won't belabor the point. Let's come up here and reset one more time, and now I'd like to divide it into nine parts. I'll set Black is the color and I'll start kind of the same way, or I do these crosses above and below, and then I'm going to do these big crosses all the way across the figure, like so and then in here, we need to go like that and then across here and then finally up here like this. These points actually divide the line into three parts already. Then I just need to copy these over and on the other side as well. And again, I'll make this red and erase the central point A. And if you count that, you'll see that there are nine parts, and I'll just test this real quickly. I don't place a few labels in and then go ahead and measure a few random ones. That's a T to you. Is that the same? Is X Y indeed the deaths? So as you can see, this actually does. Divide the line into nine equal parts, so you can use this seat of life pattern to divide line segments into specific odd numbers of segments. 357 and nine 21. Dividing with the Starcut: in this video, I'm going to describe a new method for dividing lines into discrete numbers of equal segments. And I'm using what is called the Star Cut diagram, also known as the Sand Wreckers diagram. And I learned about this from an excellent book called Patterns of Eternity, which is in the bibliography. So this diagram is based on a square, and the edges are all connected at the mid points, and I'll show you in the next chapter how to construct this diagram. But for now, I'd like to concentrate on its uses. So already it's divided the red line into two segments because the diagram itself divides Theo Edge at the midpoint. Now let's see how we can use this diagram to divide this segment a B into three parts. I can do so by drawing in some lines, so I'm going to go over here, draw lines connecting these points, and that actually divides segment A B into three parts. And if you'd like to see that, I can go ahead and label this and then measure it and you can see those two segments or equal, as are these segments. I'll just reset. Let's say we want to divide it into four parts. Well, that's actually trivial because you can buy Sect the segment here, if you like. The diagram also gives us a way to create that by drawing this over like that, and that actually divides the red line in four parts. I'll reset. What about five parts? Draw a line down here up here and do that the same on both sides. So that actually divides the red line in 1/5. And then what I can do is just transfer that measurement over, and that actually divides it into five equal pieces. So again reset. What about six parts that possible? Let's see. How about from the corner up to here and then that divides the line already into these two pieces? And then I can transfer that measurement over, and you can see that it divides it into six pieces. Reset. Now seven is always a tricky number because you can't construct a seven sided polygon, only approximations or possible. But we can actually use this diagram to exactly divide the line into seven pieces, and thats done by dragging a line from this point over to this one and doing that on both sides. So those air sevenths and I'll just transfer this over and you can see that it transfers perfectly. Reset. What about eight parts? That's challenging. It's really not so bad. Go up like this and there's two. Transfer that over and there we have it. A parts. What about nine nine starts in the corner goes up through the middle, and it also starts here and goes down to where that line intersex. So this is kind of a secondary intersection point, and that determines these points. And then I can transfer this distance over and you can see that it's equal nine parts. How about 10? 10 is actually somewhat easier. 10 goes from here, up to there like that, and then you just carry this across. There's 10 11 works like this, and then I have to carry this all the way across. There's 11. Finally, I can do 12. I'm going to go from upper corner down here, and then there's a secondary from this point to the middle. We do that the same on both sides like that, and then we have these two points here. Let's see if that carries through. So there you have it. We've divided the line from 2 to 12 segments using the simple diagram. It's really a wonder. 22. Equilateral triangles: in this chapter, we're going to be constructing regular polygons. The term regular means the polygons have all equal sides. If I want to make a triangle that's easy. I can just draw three lines. But if I want to draw a regular triangle, that's more challenging because all three sides have to be equal. So I'm going to reset. To do that, I'm going to draw a circle and then I'm going to draw a line from the center of the circle over and that creates points A, B and C, and then I'll draw another circle from B to A. So now I have the vesicles discus form, and I can construct a regular equal lateral triangle in here. That triangle is determined by its edge length A B, so you can use this method if you want to create unequal lateral triangle. If you know the edge length, however, what if you don't know that and you want to create a regular triangle inside of circle, I'll show you another method for that. Well, just back up a little bit and then I will draw on additional circle over here, which I will able de and then I can draw a triangle in here, and I can draw a matching triangle over here. So that's a method just using three circles you can use to inscribe a regular triangle, also known as an equal lateral triangle inside a circle. 23. Squares: In this video, I'll demonstrate how you can inscribe and circumscribe a square. Let's begin by drawing a horizontal line and then place a circle on that line. Let's label the points A, B and C and then put in a perpendicular at B. I label these points D and E. So already with a 90 degree angle, I have enough to create a square. I can just draw that in. But what if he wanted the square to be rotated 45 degrees with respect to the original line ? Well, I'll just go back a few steps to do that aching, bisect the angle DBC and the angle below angle CBE. We can just connect these dots and we have a square rotated Now What if we wanted a square that was circumscribed about the original circle? Let me just go back to this point, and I'm going to change the color of the original circle to black for clarity. I'll draw a new circle from A to B and another one from C to B and then d two b and E to be so we have four circles surrounding the original circle, then drawing your square 24. Constructing the Starcut from a square: Once you circumscribe a square about the circle, you have all the points needed to generate the star cut diagram. I'd like to de emphasize the circles, so I'm going to change the color of the central circle to Gray, and I'll also change the line weight to the dotted line type. And I'm not going to choose a very pale grey here. And I'll make all of these circles fade out like that and also get rid of these lines as well. So now I just really have a square that we're paying attention to change the color to black , and I'll change this to a regular line. Now I can simply draw in this V shape and then do the same below and then do the same on the left and the right. That's all there is to it. 25. Pentagon and pentagram (Western): In this video, I'll draw a Pentagon and its related pentagram, and I'll use the most common method that you're likely to find in a Western textbook. Off geometry. This method is not my favorite because it relies on length by section and angle by section , and I think there are more elegant ways to do this, which I'll show you in other videos. But this technique is common, and it's worth learning anyway. I'm going to draw a perpendicular line to this horizontal through B and A label the point up here de and the one below E. And then I will bisect the segment a B to create a new point here. If and then I will bisect the angle D F B. And that creates a new point up here g and then create a parallel line to the horizontal goes through G. And this gives us two points that are actually on the Pentagon H and I. Now I'll draw a red line connecting D, N H and D and I. So those are two edges of our Pentagon, and now we just need to carry it through. So use this tool and set the compass distance to D. H. And then put the point of the compass on H and place that there and then do that again and again. And if you do it right, this will come and precisely connect at point I Now I can change the colors of these lines , and we have our Pentagon. We can add a pentagram very easily simply by connecting these dots in this pattern. 26. Pentagon and pentagram (Eastern): ID like to show you another method for constructing a regular pentagram or Pentagon, and this one comes from 19th century Japan. I have, ah, horizontal line, and I'll start by drawing a circle on it. And then I'll go ahead and label the points, draw a circle from C to B at a new label and then add another circle from D to see. So now we have three equally sized circles. I'll draw a circle encompassing all of them, centered at sea with the Radius a C. I need to put in a perpendicular to the horizontal line running through C. I'll label its points of intersection. Next, draw a line from F to be aniline from after D. This creates new points that's shown draw a new circle from EFTA, I and J and another one from F to G and H. This creates all the points we need for the Pentagon or Pentagram. A label these No, we just need to draw in the Pentagon. Actually, I'll choose bright red for that with a thick line, and I'll put a pentagram inside of that. And just to prove that this is a regular form where all the edges are equal. I'll just go ahead and measure those e k should equal km and then k m should equal mn and then and NL are equal en El and Elah e are equal. And finally, Ellie is equal to e que So I've proven that we have a regular pentagram and Pentagon. 27. Pentagon and pentagram by edge length: in this video, I'll show you how to construct a Pentagon based on an edge length. Here we have a segment A B, and I'd like to make a Pentagon so that a B is one of the signs. I'll begin by drawing a circle from A to B and then another one from Beat A. I label the top in the bottom points and then draw a line between them forming a right angle down the middle. I label this central point as Queen E, also a label the outer points and then draw a circle from E two F or G. Circumscribe the vessel Capisce case Next label the bottom point and the top point and then draw a line a line here from H two a and from H two B. This creates significant points at the top of these two circles, J and K drawing circles from J to A and K to be so Now we have four equally sized circles which all decorate here in black. So you can see this a little bit more clearly. We actually have an important point up here where these two black circles come together, which I'm going to label as point l and this is very close to point I. So don't be deceived about which point is part of the Pentagon. It's l, which is the point we need. So I'm going to zoom out here and then let's draw this in in red. It's a thick line, and I'm going to go from the top point l to do that. Maybe I need to zoom in. So I'm going from L to J and also from L. Tu Que And then, of course, Jade A and K two b completes the former. I'll just draw this down here to make it thicker. Also, I'll put the pentagram inside of that just by connecting the significant points. And finally, I'll just prove that this is a regular Pentagon or pentagram. By going, LJ should be equal to J. A. It is J. A. Should be equal to a B A. B should be equal to B K and working our way around. We're showing all of these equalities, and now we've completed the cycle. So there you have it, a Pentagon or pentagram based on an edge length 28. Pentagon by neusis: in this video, I'll show you how you can construct a Pentagon by noises, also known as the marked ruler technique or verging. Let's start by drawing a circle on the line and another one of the same radius to form the vessel. Capisce, CAS. Form a label. These two central points as a and B, and I'll also draw a line through the center, forming a right angle with the horizontal. I'd like to draw in some arbitrary line from a up to this vertical line somewhere. I label that new point C. Now, if you're doing this by hand, you're not going to actually draw that line in. Instead, what you do is you hold your ruler on point A and you pivot that ruler around a and we can simulate that here with the hand tool in the app. So by moving point c up and down on the line, you can see this is sort of like pivoting the ruler around point A to figure out where this is going to go. We need to transfer the radius of the circle to the line. So we're going to do that by using this tool and by dragging from a to B, and then I'll drag from C over to this point on the line and a label that as Point D if you're doing this by hand, what you do is you hold the compass up to the ruler with the same radius set and you just pivot the ruler and you take a look at where Point de is. That's the opening of the compass. And when that crosses the circle right there, you're in the right position. I'm gonna zoom in there and take a closer look, see if I can get it as accurate as possible. So I'm going to move point see up and down until that point just crosses the line. Dad. A point of intersection here and then when I move Point C. It's a little bit easier to see when these two dots coincide. So that's the point that's needed to construct the Pentagon. So now we just need to draw a circle from C to D and then label these points of intersection and finally just connect the points and you have a Pentagon 29. Inscribed and edge length hexagons: creating hexagons is very simple because hexagons are actually built into the structure of reality. If you will, I'm just going to draw a circle on this line and another identical circle next to it. And in a additional once we have three equally side circles. Each one has its center on the other ones edge. I can go ahead and draw in the hexagon directly. That results in a hexagon, which is inscribed within the central circle. But what if we wanted to make a hexagon based on a given edge length? I'll just go back to the point where we have just two circles, and then I'll add additional circles, all with the same radius forming this seat of life pattern. And that is based on this edge length, which is the width of the original Jessica Pisco's form. And then I could just connect the dots, and we have a hex gone there, so hexagons are really part of structure. You'll find them in snowflakes and honeycombs, perhaps because it gives you the strongest form for the minimum amount of material 30. Pentagon and hexagon sharing and edge: in this video, I'm going to reproduce a drawing first made about 500 years ago by the master Renaissance artist Albrecht Durer. This drawing Mary is the hexagon, and the Pentagon forms together along a shared edge. So let's begin by drawing a horizontal line and then three circles like this. As you know, that's enough already to create a hexagon. I'll just ink that in and blue. Now I'd like to continue on, and I'll draw additional circles from the top left corner and the top right corner back to the center. This gives us a new point at the top, and I can draw a line from that point down through the center. And then I'm going to locate a new point right here where that vertical line goes through the original central circle. Then I'll draw lines from the left of the hexagon through that point and the right edge of the hexagon through that point, and this creates new points of intersection at the tops of these two circles. Well, then proceed to draw two additional circles from these points back to the hexagon that creates a new point of intersection at the top. Now we have enough to draw in the Pentagon. I find this form very significant, and I've written a lot about it in my blawg and in my films. 31. Septagon and septagram approximation: In this video, I'll show you how to construct an approximation of a seven sided regular polygon, sometimes called a hep to gone or septa gone according to the Greek or Latin prefixes. I prefer the Latin prefix, except not only because it reminds me of the CEPT in the game of Thrones, but also because it's easy to remember with September no October, November December, these air, all Latin prefixes, meaning 789 and 10. Of course, these were shifted two months out of whack because of Julius and Augustus Caesar inserting July and August into the calendar. And the advantage of this method is that it can be constructed within the vessel capisce SKUs form. So I'm going to draw in two equal sized circles, make a perpendicular line on the left and the right there so that they go through the centers of the circles and then make another perpendicular going the other way at the bottom, forming a square. And now I want to draw diagonal lines across here in order to locate this CenterPoint, from which I will draw a circle and I'll snap that perpendicular to the edge. So now we have a smaller circle inside this square. And this is the circle in which we will inscribe the SEPTA gone for clarity. I'm going to change the color of that circle to black. And I'd also like to de emphasize some of these other elements that we're just getting us to this point that we are now. Now zoom in on that circle. Okay? Now I need a couple of other lines, namely from the top left corner of the square, down to this point where the two circles of the vessel capisce, cas intersect and then a symmetrical line back up to the other side like that. This gives us points that we need, which I'll label. I label the top of the circle as a and then we have be and seeing here for these diagonals that go down to the Vesa capisce CAS circles. So now we actually have enough to construct the SEPTA gone. Believe it or not, all we have to do is carry these around. So we'll use this tool in the app, and I'll drag from a to B. If you're doing this by hand, you would open your compass from A to B and then you would put the point of the compass that be, and then swing this around and mark circle where it crosses and then repeat the process. I'll go carry that around this point, and I'll just label these points as well D and E. And then I could continue this around. I'll show you what happens. I'll go from D T E and then e over. Be careful not to snap to those large circles and then continue around, and this last blind is going to show us how much error we have. So I'm going to zoom in there, and as I draw this out, you'll see that it's not quite on that point. It's very, very close, but it's not exact. So because of that, I'm going to back up and I want to spread the error out not right there. But I'm going to back up a couple of steps, and instead of leaving it up there at point C, what I'm going to do is measure from a to C and then carry this around on the right side of the circle, and in this way I could get the error to all be down here on the bottom segment, and I can just draw in a segment manually down there. And the advantage of this method is that this segment remains horizontal. Although it's not exactly the right size, it's very, very close. Now I'm going to color the's segments in in a thick, dark blue line. I'll just draw in the SEPTA gone and finally I'd like to draw in a septa Graham inside of that and that can be done. Let me just continue labeling this. That could be done by drawing in lines connecting E and A and then A and G and then G and B be in F, F and D DNC and finally C and E. 32. Octagons and octagram star: Now I'm going to draw Octagon ins both inscribed and circumscribed. I'll begin by drawing a circle on the line and then draw another circle from the left quadrant over to the right and then do that again symmetrically from the right to the left . This creates of escape. Pyszka us draw a line through the vertical center to create the cross access. I'm going to color the original circle black for clarity. Now create additional circles from the top of this central circle to the bottom and again from the bottom to the top. Now we have cross quarter points at 45 degree increments. Draw an X through the center like this. Now I'm going to switch to read, and I'll use a thick line wheat draw in the octagon, which is inscribed in this black circle. Next, I'd like to draw an octagon that circumscribed about that circle. So to do that, we need to draw some additional construction circles. I'm going to start at the top quadrant of the original circle and draw a circle down to the center and then do the same thing on the bottom and on the left and on the right draw another circle from the center out and snap it to this intersection point. Then we can draw in a square. Actually, I'm gonna draw that in another color. I'll use black, but with a thin or regular line. Wait, I'll make it thin and then draw in a line from that point over to the corresponding point on the right, and then draw another one down and over and back up to create a square that circumscribe Sethi circle. Then draw another square, starting up here and going down to this point and then work your way around to create a square at a 45 degree angle. I'm going to zoom in there now and choose a thick line. Wait, and this time I'll choose a darker red color. Now I can draw in the circumscribed octagon by carefully tracing over these points, and you have to be a little bit careful because there are so many different intersection points here. This is where the apple pencil comes in handy. If you're using your finger, you might have to zoom in a little closer on each one of those points in order to connect it correctly. Finally, I'm going to draw in a symbol called the Octa Graham star in the inscribed Octagon. And one way to do this is to start drawing from a vertex and then skip to There's 12 I'm going to skip those and go over here now skip to and go down here that creates this pattern . 33. Enneagons and their stars: in this video, I'm going to draw an any of gone, sometimes called a Nana Gone. I like the term any of gone because it's all Greek, whereas Nana Gone is sort of a hybrid of English and Greek. So either way, we're drawing a nine sided regular polygon, and this is impossible to do it exactly. So again, we have the situation a lot like the SEPTA gone, where approximations are, the best we can hope for, and I'm going to show you. The most accurate approximation that I've come across comes from a book drawing geometry, which is in the bibliography. I'll start by drawing of Essick Episcopals form by drawing two circles and then I'll put in the cross axis. Now I'm going to label these points and then bisect the distance from A to B and again from B to C. This gives us some new points, which all able F N g. And then I'm going to play circles at F and G like that. So they're fitting within the vesicles discus form. Now I'll draw a line segments radiating out of the top of the vesicles in connecting to these points that creates some new points in here h and I, which are significant. I'm going to draw a circle from D to H and I like that. And that creates two new points J and K on that last circle. And I'm going to draw circles from A to J and C two K. Now I'm going to draw a three point circle connecting Hey, J. Okay. And this circle is actually the one that we're interested in. So I'm going to change its color two black And this black circle is what I'm going to inscribe any of gone within. Now I'm going to erase the original Jessica circles because they're kind of in the way now and I'll zoom in. And now I can label the top and then these two points these air, all edges of the any of gone, we just need to find two edges near the bottom. So I'm going to set the compass distance from a T m and then put that point down below, do the same thing on the right, and then label these points now to show you how much error there is. Visually, I'm going to set the compass at that same distance this time m o. And then, as I draw it over here, you'll see that it misses Point P. It looks like about the thickness of a point. So that's how inaccurate it ISS. So what I'm going to do is switch to my color palette and select, read the thick line. Wait and then I'll just start at the top and draw in the any of gone. And then within that I will draw the any a gram and their different versions available. I'm going to make the sharpest version, and you do that by starting at the top and then skipping three points and then continuing on clockwise, skipping three points and continuing on in. That way we have the n e a gram. If you're interested, you can make the wider version of the star. I'll change the color and just make it a regular line. Wait and I'll start at the top, and this time I'm going to skip one point, and this creates the wider version of the star. It kind of has a dynamic effect. You can also skipped two points, but that results in an equilateral triangle. We can add two additional equal lateral triangles, and this will fill out the full complement of stars that are available within the any of gone. 34. Enneagon by neusis: Now I'm going to draw on any of gone by noises, and the advantage of this technique is that you can be as accurate as your hand eye coordination allows. I'll start but with a vertical line, and I'll draw of Asako Pyszka us. And I'll just label these circles A and B also label these additional points, and then I'll draw a segment connecting B and E if I'm. If you're using the APP, you can draw a segment from A to some arbitrary point on this lower circle. If you're doing this by hand, don't draw that in yet. This is the line that we're going to be moving. I label this point f and then I want to measure the distance from A to B. So if you're doing this by hand, you don't have to change the spread of the compass is, and then put the point of the compass at F and mark the diagonal line. At that point, which I label has G. Now I'm going to go to the hand tool and move point F along the circle. And as I do that, the key point here is to get G on the line like that. I'm going to zoom in and try to get that as accurate as I can. Move that back and forth and then lift the pointer right at the moment when G crosses the line, I think that's about as accurate as I'm able to make. If you're doing this by hand, you hold the straight edge on point A and pivot it around. Point a and you hold the compass is or dividers against the straight edge set to this A B distance and you keep one point of the compass on on the circle. In the other point will be point G. And as you pivot that straight edge, you want to line everything up. So that G is right where the point of the compass passes the line be. So now I'm going to decorate this and just show you that a B equals F G. This creates a new point. Where are diagonal line crosses spee upper circle at H in that point is on the any of gone . So now I need to carry that around, so I'm going to measure from H T and then carry that point up to the top like that and then do that again, and I'll just go ahead and label these points. And then I'm going to measure that distance I j. And then put the point at sea. Measure that over, do that again and then label these points. Finally, we need point on the bottom, no measure from H E and then carry that over here and a label that as well. If you'd like to visually see how much error there is, you can measure this distance and then put the compass over here. And ideally, that will line up perfectly with Point DE. If there's a gap, it's because you didn't perfectly lineup Point G earlier on. So now I can go ahead and ink in by any of gone by connecting all the dots around the upper circle. 35. Tiling patterns: in this chapter, we're going to explore automatically generated polygons, and this is something that is really cheating if you're a purist who uses a straight edge in a compass. But if you're using the app, why not take advantage of all of its functionality? So the automatic polygon tool is here, and if you hold it down, you'll see there's additional polygons that you could make of these specific forms. But the last one here is the most flexible because it allows you to make polygons of any number of sides in the way that it works is you drag on screen to establish the edge length and then lift the pointer. It has a line bisecting down the middle, and what you do then is you put the pointer on the midpoint and you drag either above or below. And it says, three. If you keep dragging, it will say four and then five, and so on, all the way up to as high as you want to go. So after you've just laboriously learned how to create all of the different regular polygons from 3 to 9, this might be a bit frustrating, because now you see how easy it is to do it without any construction. But hopefully you've learned a lot about geometry in the previous chapter, and you're excited to take advantage of this tool in the APP because it allows us to explore geometry at a higher level. Of course, you can construct these shapes manually one at a time, and that's great practice. But when you want to start exploring things like tiling patterns, that could be very tedious. And so an app like this can be handy in terms of allowing you to think about things on another level. So, for example, I can draw in triangles and squares very easily here, and I can just explorer possibilities more readily. In this way I can develop ah ceramic tiling pattern just by laying down these polygons and seeing where it takes me off. Just pan over here and try a different form. What if I were to start with a hexagon and then put I don't know, squares. Maybe on these edges? Where does that take me? It looks like I can put a equal lateral triangle between those, and in this way you can just discover these patterns that are just waiting for you in two dimensional space. I'll try another one over here. What if I were to start with an Octagon? What could I do with that? I'll try putting squares on these cross quarter points. And then I could put another octagon over there and another one here, and you might be able to recognize this pattern already. This is a very common tile pattern that you'll find in kitchens and bathrooms everywhere. I'll try another one this time. Maybe I'll start with a larger figure like a 12 cider. And then, with that, I could put I have a lot of different choices of what I could put on its edges. Let's see. What if I tried just a square down here? Then I could put another square over here, and it looks like then I could put a hex ago in there. And so you get the idea. You can just go ahead and explore regular polygons very rapidly using this tool 36. Quality vs quantity in form: I'm going to start this video by drawing a Vesa Capisce SKUs form and then using the polygon tool inside of it to generate a equal lateral triangle. Now I'm going to color the circles red and then of a back and create additional polygons. So using the same edge, which are label A and B. So now I'm going to drag from a to B and then position the pointer at the midpoint and drag upward until I can create a square and then repeat that and make the Pentagon hexagon except a gone octagon. Any of gone? As we do this, we're exploring each one of these regular, probably guns that we learned how to construct manually. But if you just keep going, you'll find that as you go more and more outward, the shapes seem to lose their character until they start looking like approximations of circles. And it doesn't really matter how Maney edges that one has, because it just is a big, faceted circle. Essentially so. At a certain point, we lose track as humans of the distinctness of the geometry, and it all blends together, and it just becomes something that we call many there are many sides doesn't matter how many but its needs. Lower number forms three through nine that seemed to each have a different quality to them . And that's what we like to explore in sacred geometry are the qualities of the numbers, not the quantity. 37. Recursion of form: armed with our automatic polygon tool, we can explore a fascinating subject called Rikers in. So, for example, I'll draw an equilateral triangle and then I'll bisect two of its edges and then draw a new equal lateral triangle within that and then again bisect two edges and then repeat and do it again and again. And as you can appreciate, this process will continue forever. In fact, you can only zoom in so far in this app. Now I've reached the maximum magnification, so I could only really go so far in the APP. But in reality, you could have this internally subdivide forever. That's just the triangle. What does it look like? If we explore this idea with a square again, I will bisect two adjacent edges and then create a new square and again. And if you consider how much time this would take you to do by hand, you can really appreciate power of this automatic feature. It allows us to explore subjects that we otherwise this wouldn't have the patience to do by hand. And so these forms seem to go in and go somewhere. They're rather intriguing. I'll also look explorer, The Pentagon see how that looks and it follows the same rules. You just bisect two adjacent to edges and then make a new Pentagon connecting those edges. It doesn't matter which a Jason edges you bisect works the same every time, and it's entirely accurate here. And so you can appreciate these patterns now in a way that you probably wouldn't take the time to do if you were doing it by hand. You can also subdivide a circle. Interestingly enough, I'll draw a circle and then draw a line through its centre and then bisect. And then I can erase the line now that I have these points, because now I can draw circles that are mutually tangent and I can bisect again and again. And as you can appreciate, this would go on forever. So we have yin yang pattern that has a Ryker shin into itself, sort of like the infinity within. So this is a great way of exploring Ryker Shin using the automatic polygon tool 38. Triangulating the circle: in an earlier chapter, I took you through an approximate solution to the age old problem of squaring the circle. In this video, I'd like to introduce you to a modern problem of triangulating the circle, which really is analogous to squaring the circle. Except we're using an equilateral triangle this time. So we're looking for unequal lateral triangle, whose perimeter equals a circle that it centered on. My solution has a 25 sided polygon, so I used the automatic polygon tool and drag out a short edge near the top of the screen. Create a 25 sided polygon, and then you may need to navigate center that draw a line from the center through the bottom Vertex, and then draw a circle from the center to the bottom used the Intersect tool to intersect the polygon at the top with the circle. At these two points, we don't really need the polygon anymore, so I'm going to delete it segments and his top inside the circle near the edge. To get rid of these segments, I'm going to zoom in here at the top and delete the segment, leaving those two points Now I'm going to color the circle red. That's the circle whose circumference we're trying to match with unequal lateral triangle. Next draw a perpendicular line through the center point. I'm going to label the Circle A and then I'll label these points as such. Now I'm going to draw a circle from B to A and from C T. A. This creates new points at the top, and then I'll draw a line from B to G and C two h and then make a parallel line to that that goes through D in a parallel line that goes to E. This creates a new point at the top, which is the apex of our equilateral triangle. I've marked it, Point I. So now we want the equilateral triangles points to stick through the circle, the same amount on each side. So I'll use this tool to measure the distance from I two D and then transpose that distance down here and then do the same thing again on the right. This creates new points, which I've labeled and also create a parallel line goes through K and M. Now all I need to do is draw in the segments in red to show you equal lateral triangle. So I've drawn this in a cad program, and I was really amazed to discover that the perimeter of the equal lateral triangle equals the circumference of the circle to 99.9999% accuracy. 39. Golden ratios from the vesica piscis: this chapter is about the golden ratio, which is sometimes called the Golden Section, or the divine proportion. And in this video, I'd like to show you how you can derive this geometrically from the vessel Capisce Kiss form. So let's start by drawing a circle on this horizontal line and then another circle from the intersection Point on the line back to the center, a label, these points of Intersection A through D. Now I'll draw another circle from a to see, and then I'll draw a line segment from the bottom of the vesicles, or almond shape up to this intersection Point, which are label as E N F. And that diagonal line cuts the horizontal line at G. And so this point G is the point which cuts the segment BC precisely at the golden ratio, or golden section. I'm going to color that in just for emphasis, and then I'll take a screen capture so that I can draw on this. So to understand the golden ratio algebraic Lee, that is in terms of ratios. I'll just describe that to you. To the hole is to the larger as the larger is to the smaller the whole specie that is, to the larger in exactly the same proportion as that larger is to the smaller. So both of these ratios are equal. And you might ask, What are they equal to? Well, they're equal to you. The golden ratio, which is symbolized by the Greek letter Phi and thats approximately equal to 1.618 I say approximately because this is an irrational number that keeps going on forever. So there are more digits if you'd like to know them and it goes on forever, so I'll show you. As far as I've memorized, I'll go back. And so far we made this diagram. But it's a symmetrical. I'll draw another circle from D to be. And then I'll draw in a segment from E Up to this point, which are label as H. That creates a point of intersection called I, and that also creates a golden ratio. Except now the larger and the smaller segments are reversed. So now BC is toe. I see, as I see is to be I so for any given segment that you have, you can divide it at two points to unique points, which form this golden ratio or golden proportion Golden section. We can also do this in another way. If we draw a circle from see to a and from B to the, that creates two additional circles, each with the same radi I as the other larger circles. Then I can draw in a segment from E right up through the middle, and I'd like to label that point Jay in this point K, where the line intersects with the original vesicles circles, which I'm going to decorate in black, you know, also decorate this new golden ratio, which we now have here. So this golden ratio, the whole J E, is two k e as K is to J K. That forms a golden ratio. You can also draw a line down below Joe label and color, and that forms another golden ratio. But this time the whole is K. L. So Kael is two k e as K is Teoh. So now we have these two different ways of seeing how the golden ratio emerges directly out of this foundation geometry 40. Golden ratios from an equilateral triangle: the golden ratio isn't only construct herbal from a Vasa capisce, CAS. In fact, it's interrelated, with lots of different kinds of geometries. So in this video, I'd like to show you how it emerges from unequal lateral triangle to construct that triangle. I'm going to draw a circle on this vertical line and two additional circles like that. Now I can draw in the equilateral triangle by connecting these intersection points to emphasize that I'm going to change the color of the triangle lines to black and also color in the circum circle surrounding that in black. Next, let's bisect each one of the edges. Another way of thinking of that is I've identified the mid points, and now I will connect those mid points with lines. You can think of this as a kind of Riker Shin within where we have these segments, which I could sketch in in black. So we have the try and go within, and now we actually have all we need to construct six golden ratios. I'm going to draw those in in red. Also, I'll use a thicker line and a thicker dot, so the first golden ratio I'd like to draw your attention to is right here. I shall label a B and C so segment A B is cut at sea in this forms of golden ratio. So a B is two C B A. C B is to a C. Also, we have the same type of ratio over here. So seedy is two CB as CB is to be d. So we have a corresponding golden ratio on the right and you guessed it. We have golden ratio is going this way two of them, and this way, bringing the total up to six different golden ratios that emerged directly out of the relationship that a circle has with its inscribed equal lateral triangle. 41. Golden ratios from a square: in this video, I'm going to explore the relationship between a square and the golden ratio. The key to unlocking the golden ratio from the square is too bisect. One of the edges I'll bisect CD, and that creates this new point E. Now I'm going to draw a circle, centred it E and the radius is going to go up here to a or B. That actually is enough to encode the golden ratio. We just need to know how to unlock it. The way you do that is by extending CD. We have a new point over here F. This actually creates a golden ratio right here so that the whole segment CF is two. CD as CD is two DF. We can take this further and create a golden rectangle by making a parallel line to be D through F and a parallel line to CF through B. That creates a new point over here G and that also encodes a new golden ratio right up here . This is a golden rectangle, so rectangle a G cf is a golden rectangle which has the proportions of one to fi. It turns out that this rectangle B G D. F is also a golden rectangle has the same proportion, although it's smaller and turned to 90 degrees and similarly over here we have another golden rectangle that weaken sketch in. So you see the square in codes three golden rectangles right off the bat. Actually, four if you think about HB I. D. A. G cf and then the other two smaller golden rectangles on the sides, and you can look at this diagram and realize that it's only half of the story. So if I draw in another square and extend the lines, we have additional golden rectangles down here and here. And, of course, two more going across the other way. So the square is very intimately connected with the golden ratio through this circle that has a radius like that, going from the mid point of one of the edges of the original square to the opposite corner . 42. Golden ratios from a triangle and square: now that you've already seen how to derive the golden ratio from the triangle and the square ID like to show you how you can find it from the relationship between these two forms. So I'm going to use the automatic polygon tool and draw in an equilateral triangle, and then I'll draw a square on one of its edges like that. I'll use the hand tool and just straighten this out. Now I'll draw a line along the lower edge of the triangle to extend that and then a circle from the top of the triangle to the opposite corner of the square. This actually encodes two different golden ratios. And just for sake of symmetry, I'm going to draw a square on the other side as well. Now I'm going to label these points along the line. That's where we find the golden ratio. So the segment A C is cut at B at the golden ratio, and similarly, the segment B D is cut at sea, which warms another golden ratio. So the golden ratio isn't this abstract or rare proportion. It's actually encoded in many different types of simple geometries, such as triangles, squares, circles and, as you'll learn in the next video pentagrams 43. Pentagrams and the golden ratio: the pentagram is actually built on the golden ratio. I'm going to draw a Pentagon, and I'll just straighten that out and then I'll inscribe a pentagram within it. This edge is in the golden ratio. With respect to this edge, I'll just undo a couple of times. In fact, this segment is cut at the golden ratio by this point, and similarly on the other side. So each edge of the pentagram defines two different golden ratios. So that's a total of 10 golden ratios just from the edges course. We can also inscribe a pentagram within, and it would coat encode another 10 different golden ratios, such as this one and the one on the other side. You can even draw circles connecting these dots, and that defines new points, which can be used to create additional pentagrams. All this to erase the circles now and so these additional pentagrams march outward along the arms. So, for example, I could draw in these circles and then drawing another pentagram over here. And then I'll erase these circles and you get the idea that we could make ever smaller pentagrams reaching out along that arm. And of course, each pentagram is another temple to the golden ratio. In a sense, the Pentagon and the pentagrams within are just replete with golden ratios. More than any other form, the pentagram is a study in the golden ratio. 44. Golden ratios from adjacent forms: you can find the golden ratio from two adjacent squares by drawing a diagonal through both of them and then drawing a circle whose radius is equal to the edge length of one of the squares from one corner of that diagonal and then from the other corner of the diagonal make a tangent circle. This creates points a B and see and see actually cuts that edge a B at the golden ratio. So now we have a golden ratio right here. You can also find the golden ratio from two circles. I'm just gonna erase this circle up above. So if we're looking at two circles, we can find that by drawing a perpendicular through one of the circles there and then drawing a line from the center of one of the circles through the lower quadrant of the adjacent circle. This creates points D, E and F, and these also form a golden ratio. Comparing the same shape to itself is often a pathway to finding this mysterious and yet ever present ratio 45. Golden ratios in art: in this video, I'd like to take you through, Ah, quick tour of the history of Western art, starting with Leonardo da Vinci and going forward through time. So in this image, here we have Le Bel Farah near around 14 90 and I've over laid this geometry on the canvas , and I've anchored it right there at the top of the railing and the top of the canvas itself . And what's amazing is that the the convergence point of the golden spiral converges right on that piece of jewelry on the woman's forehead. This Farah near which was a popular adornment in those times and so also noticed that her eyes are also framed by the vertical divisions within that golden rectangle. Let's move forward to 14 92 the year that Columbus discovered America. Here we have the most iconic image, perhaps in the entire world, the Vitruvian man. So Leonardo was depicting the ideal proportions of man based on the writings of the ancient Roman architect of a true VI ous, and he's showing that the human body is framed by the square and the circle. But what is really amazing is if you analyze this with the golden rectangles. You can see that the golden ratios shown in the Yellow Arrows point two significant features within the body. For example, the golden ratio on the square points to the center of the circle. The golden ratio on the circle points to the heart center, which is the central Shaqra in the energetic system of the body. So we have the we have the physical and the spiritual centers identified with golden ratios . Here's my analysis of the Mona Lisa showing this method of constructing the golden rectangle from the square, and the square has shown with a circle inscribed in it, and you can see that the center of that points right at her. I and the Ark, which is what is used to create the golden rectangle, perfectly frames her head. Her body is within this square, down below her head up above. Here's another image by Leonardo da Vinci and 15 15 and it's really amazing how precise this is. The point of his finger points exactly at that point, where we have golden ratios both vertically and horizontally, the golden rectangle is framed on three edges of the canvas. In this image, we have John the Baptist, sometimes called Bacchus, pointing to as above so below or perhaps in the Christian mode, on Earth as it is in heaven. And it's fitting very well in with the steam of the golden spiral, which is something that is self similar that is the same. Above and below. This is Michelangelo's creation of Adam Fresco in the Sistine Chapel. Ah, whole Siris of authors, which I included on this slide, discovered this proportion by measuring the ceiling. And they note, they noted that the point where the fingers come together is precisely at the golden ratio . I'm just illustrating here with my proportion, er, to call that out. Graphically, this is perhaps the second most popular artwork in the world after the Mona Lisa, and you have to ask yourself, Why is that so popular? Perhaps it has something to do with the hidden golden ratio proportions of these images. Here's Vermeer's girl with a pearl earring, and you can see when the golden rectangle is framed, the top and the bottom of the canvas that these vertical divisions seem to point out her eyes, and the horizontal division of the golden rectangle goes right through one of her eyes. Also by Vermeer, we have the geographer and the eye of the geographer and the eye of the compass are precisely indicated by golden ratios above and below. Moving forward in history. We're looking at the Death of Socrates by Jacques Louis David but the end of the 18th century, and you can see with the golden rectangle framed at the top in the bottom of the work that the golden ratio runs right through all of the figures heads, That's the horizon line and the diagonal of the golden rectangle. Above is what controls the angle of Socrates finger as he's pointing to truth right before he dies. That points to the convergence point of that golden spiral. Here we have another painting about a death, the death of Caesar, this time the end of the 18th century, and the golden ratios point right to the center of Caesar's head, where his third I would be also one of his killers arm His arm is following that arc of the golden ratio. The figure on the extreme right seems to be framed by in his elbow touching edge of that golden rectangle. It's very resonant with this proportion. So I believe that this artist started with this geometry and then proceeded to flesh out the artwork From there. Here we have the Grande Odalisque by Young. In the beginning of the 19th century, we have this golden rectangle framed on three edges of the canvas and the ark being a mimic of the women's back, and the curve of her back is very much like a narc. Also, her face and the details there of her mouth and chin, framed very nicely by the divisions within the golden rectangle. Here we have Salvador Dali's corpus hyper Cuba's in the mid 20th and I've shown golden ratio divisions in the gold color here, both vertically and horizontally, both sides of them. And you can see from this grid that the entire artwork was planned using this golden ratio proportion. So in the end, we have to recognize that geometry is very much a part of our aesthetic appreciation of artwork. It's something that we might not understand consciously, but some of the most important works in Western history were structured using this proportion on the canvas, and I believe the reason that we find them so attractive is because that proportion resonates with something in us with something in our visual subsystem in our brains or perhaps even deeper, perhaps even in our d n A. There is something that resonates with the golden ratio. 46. Golden ratios in science: in this video, I'd like to take a look at the golden ratio through science through a different lens. And here we have the study of metrology or measurement, and my proportion, er is open here to show that the kilometer is actually a golden ratio with respect to the mile. So here we have the world's two most popular systems. We have the system in the United States. The mile, or sometimes called the Imperial System and the kilometer part of the metric system are related in this golden ratio way, and it's 99.5% accurate. So it's not exact, but it nevertheless is very close to this relationship. Here we have a graphic showing the Earth and the moon together and the measurements. So it's really amazing. I think that the polar diameter of the earth is 10,000 kilometers times the square root if I and that's 99.97% accurate. Also, when you look at the combined polar diameters of Earth and moon together, that's 10,000 kilometers times fi again with the same astonishingly high degree of accuracy . Now there's nothing in the metric system that would lead us to this conclusion. The metric system was initially defined by a meridian running through Paris that went around the entire earth, went through Paris through the North Pole through the South Pole and then back through Paris again. And that was defined to be 40,000 kilometers. Exactly since that time, the way the metric system has shifted, but essentially they're the same meters. But there's nothing in that definition that would encode the golden ratio, as you see here. But nevertheless it does, and the reason that it does is because the Earth and the moon have this proportional relationship with the golden ratio. So if you take the size of the earth as one than the combined sizes of Earth and moon together is the square root. If I and that can be combined into what is called a Kepler triangle, where the high pot news is equal to the golden ratio. I looked at the planet Saturn when this beautiful image was coming back from NASA and I hold. I held up my golden ratio calipers to it, and I was really blown away to see that the planet itself is in a golden ratio. With respect to its rings. And there's nothing in physics that says this must be so. And nevertheless, it iss so quite a beautiful geometry underlying the proportions of perhaps the most beautiful planet after the Earth. In our system, Here's the visible spectrum of light down below, and you can see that the three primary colors Aaron a golden ratio relationship with respect to one another. This was discovered by Clay Taylor, and I've illustrated it here with my proportion. Er, here's the DNA molecule. This is a three D model of the DNA molecule coiled up much as possible. It shows that the major and minor grooves Aaron a golden relationship with one another, and simultaneously and independently of that. The pitch is in the golden relationship with the diameter, and the Axial View shows a molecule with 10 part symmetry. And then we have. If you asked me to design a molecule based on the golden ratio, there's no way I could ever probably anyone could have come up with a better molecule that encodes the golden ratio in many different simultaneous ways. And here's a really big picture idea. This this is a sent. This is a log rhythmic scale on the left. It shows the visible universe at the top. That's everything that we can see with any telescope all the way down to the human scale and then all the way down to the smallest thing of all the proton. So the proton is really the smallest thing that can be measured in that book ends one into the scale. What's really amazing is the human scale happens to be right at the golden ratio with respect to all things. The human scale is 10 to the zeroth power or approximately one meter. And remember, this is a scale log, rhythmic scale or orders of magnitude scale. So any creatures that are approximately one meter in size would fit the bill. And so it's, I think it's really amazing that are experience of the universe happens to be at this golden ratio point, the golden ratio. It's just written throughout this entire mathematical universe 47. Golden ratios in design: in this video, I'd like to point out where we can find the golden ratio in things that people have made. Oftentimes, they're things of great beauty or things that we see a lot in our daily lives. So here we have a thing of great beauty. The Stradivarius file in and you see it is really a masterwork in encoding the golden ratio . Here's the piano keyboard, and it's keys are actually proportioned using the golden ratio. This was discovered by a participant in one of my secret geometry workshops. Harry Webley. He had one of these proportion er's. He went out and started measuring things, and he discovered a couple of really amazing things, this being one of them and this being the other. These LP's and 45 have labels, which are at a golden ratio with respect to the vinyl and this. This is something that I don't think anyone has noticed before, but it's something that is perhaps part of the aesthetic beauty and love of these old records. You have a golden rectangle, probably on your person right now in the form of a credit card, And most credit cards aren't aren't perfect Golden rectangles there. It's more like a 98% correlation. And because it's not exact, it makes me think that whoever designed the proportion of the credit card just designed a rectangle that was aesthetically pleasing to them. And that happens to be the golden rectangle, although they probably weren't aware of it. Here we have examples of the golden ratio in many different car logos. These air different analyses that I've done that point out, that many of these logos are obviously designed using the golden ratio. And I think these air cases where the designers are aware of the power of the golden ratio and they intentionally encode them in their logos because they know that that will be something that will just resonate with people. 48. Tangent circles: this chapter is about the moon and the earth, and in particular, we're going to find the proportion of the size of the moon as compared to the Earth in multiple diagrams. So let's start with a line segment, which I've already labelled A and B. I'm going to bisect that line segment to find the midpoint, which I'll label see, and then I'll just go ahead and draw a circle out from the center to the end, then draw perpendicular through C and label the top Point D and then bisect CD. To find this point, E put a circle centred at E with a radius of E. D. Next, draw another circle from A and have it go perpendicular to that circle so that there's a point point of Tangent C there and do the same thing on the other side. Oh, then identify that point of the topics point f and finally draw a circle centred at F with the radius F. D. So now if that's the Earth, then this would be the moon. Now there's nothing in physics which says that our planet and its natural satellite should have such pure geometric proportions, which can be identified with a simple diagram, but nevertheless that is the case. It's really quite miraculous. And this is part of what makes sacred geometry sacred. It's this geometry. This very simple foundational geometry describes our planet, our bodies, even our DNA. 49. Square and pentagram: the true proportions of moon to Earth can be expressed in the relationship between a square and a Pentagon. I'm going to start by drawing a square and then a Pentagon that shares one edge. The automatic polygon tool conveniently gives us the center points of these figures. If you're drafting this by hand, you'd have to construct those points using different methods. But since I have those points, I can easily draw a circle from the center of the Pentagon to the center of the square, and I'll draw another circle. It's concentric with this circle that goes out to the edge of the Pentagon like that. Miraculously, this is the earth, and this is the moon. This is slightly less accurate than the previous method shown in the previous video, but this is even more fundamental. It's this relationship between the square and the Pentagon, which truly encodes the proportion of moon and earth 50. Sacred geometry of the pentagram: in this video, I'd like to share with you the sacred geometry connecting the pentagram, squaring the circle, the true proportions of moon and Earth and the Great Pyramid. All in one diagram. I began by drawing a horizontal line, and I need to start this construction by drawing a pentagram. I'm going to use what I call the Eastern method, which I covered in an earlier chapter. That method, what we do is we bisect these two lines here and placed circles as you see me doing so that they're mutually tangent. And now I'm going to construct a perpendicular through the center. And I'm going to place a circle at the bottom quadrant and make it tangent to the circles and another concentric circle, which is tangent on the other end. This creates significant points which are labelled a and B. Now I'm going to construct the pentagram and I'll use a dark red color for that and connect these points of intersection. Now I'm going to color this circle, actually in red to represent the Earth. And then I can actually erase some of these construction circles as they're no longer needed. Now I'd like to explore squaring the circle. To do that, I'll draw a circle from the center out, two points A and B, and this will be the circle that I'm trying to square. No color that in orange squaring that is easy. All we have to do, his drag parallel lines out from the axes until they are tangent to the earth circle. And then I can connect the dots to form that square whose perimeter is approximately equal to the circumference of the circle and then going to erase thes construction lines around here. But I will leave that vertical one going through the center for now, to represent the moon in its proportion with the earth. All we need to do is draw a circle like so, and then I'll color that in blue, so that represents the proportion of moon to earth. Now we finally need to correlate these three elements, the pentagram squaring the circle proportions of moon and earth. We need to correlate that with the Great Pyramid, and that can be done using the tangent tool. I'll tap on a and then the red circle to drawing a line that's tangent, and I'll do the same thing on the other side, tapping on B and the red circle. I also need to drawing a line on the base of the square to represent the base of the Great Pyramid. And then just to identify this, I'll draw in a triangle that represents the cross section of the Great Pyramid of Giza. And then I'll erase these lines. So that's a triangle, all right. But why does it represent the Great Pyramid? It's because the angle, it's very exact. I'll measure that by tapping on a the lower left corner and then the lower left corner of the square. And that forms an angle, which is 51.8 degrees approximately. That's another way of saying 51 degrees 51 minutes. And in the 19th century, casing stones were discovered at the base of the Great Pyramid and they were intact. They were measured very accurately to be 51.8 degrees. So here we have a representation of the Great Pyramid, who's very form in codes squaring the circle, the pentagram and the true proportions of moon and earth 51. Starcut technique: Now I'd like to show you how the star cut pattern can encode the proportions of moon and earth. I began by drawing a square with the automatic polygon tool and I'm just going to square that up with the hand tool. And now I'd like to bisect each one of the sides in order to construct the star cut pattern . And then I'll draw in lines in black to form that star cut. And there it is. The secret to finding this is by drawing a line from this point which I'm going to label a up to BNC. So if I draw a line from A to B and from A to C, that will do it, this creates points up above DNE and those points form the edge of a new square. So drag out a new square up here. I can then sketch in some segments now that go from a up to this point up to DNE and then I can erase thes lines which extend beyond I'll also decorate this so that these lines were not so prominent. They've given us the size of that square above. Now I can sketch in a circle from the center of the large square and the smaller square to construct circles which represent the proportions of the moon and the earth. It's the most accurate approximation that I'm aware of. 52. Golden rectangle: in this video, I'd like to show you how we can derive squaring the circle, the moon earth proportions and the great Pyramid slope, all from a golden rectangle. This shows you how these forms are all interconnected with one another through geometry. So where you have one, you have all the others. I'll start by constructing the golden rectangle. And perhaps the most expedient way to do that is simply to draw a square, which I'll do with the automatic polygon tool. And then I'm going to bisect this right edge and draw a circle from the mid point over to the opposite corner points of this square. Draw a line along the right edge, and this creates a point up above, which is actually one of the corners of the golden rectangle that we're aiming to construct . Next, Use the parallel tool and drag a parallel line off the square and have it go through that point in another parallel in the other direction like that. This creates another point up here, which is yet another corner of the golden rectangle. So I'd like to color that in in gold with a thick line, and I'll just trace over this golden rectangle also label it's corner points and erase all of the construction geometry, which is no longer needed. I'll also erase these points by tapping on them. So we're left just with a golden rectangle, and this will be the genesis of all the other geometry. So I'm going to draw another square along the edge, BD. And that unfortunately, lost our labels. So I'm going to go back and relabel that. And then I couldn't label the other points on the square in the center of the square as well. Now it's bisect the segment cf to create a new point here, which is called H and then draw a circle centered at H with a radius HC. Next draw perpendicular line to segment CD that goes through BD and continues to the edge of the circle like so, let's label that point. I draw lines, see I and I f. And because these lines are part of a semi circle by the theorem of families, we know that these former right angle also I'll draw rather all bisect segment I f to create a new point here called J. Now I'm going to connect J and G with a line in that Linus parallel to BD. I'm going to transfer a measurement from this by using this tool. And if you're doing this by hand, of course, you would simply position the compass, point it f and open it to J in the APP I have to drag from F to J. And then I'll drag from G over here to wear this circle intersects with D. H. That creates a new point, which is called K, and I'm going to decorate that with a linear decoration to show that FJ equals G. K. So now I can draw a circle centred at G with a radius G K. And it turns out that this circle has the same length approximately as the square B e f de . I guess I'll trace over this edge to make it red. So now we've squared the circle. Next, I'd like to derive the moon and earth proportions, and that's easily done because wherever you square the circle, you automatically have the true proportions of moon and earth. And that's shown here by drawing a circle centred at G that fits inside the square. And another one that centered up at the top of the circle there and connects back to the square. What color? These in green To show the earth in the moon, respectively. Now, I'd like to show the great pyramid angle, and we actually already have that here. I'll just measure it and I'll tap on G. Okay. And each. And it shows you that that angle is approximately 51.8 degrees. It turns out that is the slope of the Great Pyramid. So what I'd like to do next is connect the dots G and H. Sorry, G and K and then K to this point. Here were the circle and square come together and then we'll go back to G. And that forms a kind of great pyramid in the diagram. So now you can appreciate how the golden rectangle actually encodes all of this geometry. We have the great pyramid squaring the circle and the proportions of moon and earth all in one sacred geometry diagram 53. Mean orbits: in this video, I'd like to show some of the patterns revealed in a little book of coincidence by John Martino, which is also part of the Quadrevion Bind Up in both books are in the bibliography, so these geometric patterns govern the mean orbits of the planets. Planets orbit in elliptical orbits, but if you take the mean orbit or average distance that the planet has from the sun, that would define a circle. And so re plotting these orbits as circles representing their mean distances gives us these geometric figures with astonishingly high accuracy. So, for example, here is a 15 pointed star, and there's a red circle around the outside and one on the inside. The one on the inside represents the earth's orbit, and the one on the outside is Saturn's orbit. In addition, in this diagram, the circles also represent the sizes of the planets in both are 99.8% accurate, both the mean orbits and the sizes. It's really astonishing. Here's another one for Venus and Earth, so Venus is the inner orbit. Of course, an earth is the outer one, and their orbits are governed by this square between the circles. As you can see, just by glancing at these icons, we have these other geometric relationships that govern the orbits of the planets. 54. 3-4-5 triangles from square: in this video, I would like to show you how the 345 triangle can be constructed without having to measure anything in the 345 Triangle is the first Pythagorean triple that is the first triangle that has entered your sides that it fulfilled the Pythagorean theorem, such that three squared plus four squared equals five squared. It turns out that this triangle is very important in sacred geometry, and you'll see that in the subsequent videos in this chapter to start with, we need to figure out how to construct it, and we'll do that by drawing a square. So I used the automatic polygon tool to do that and then bisect the top edge and the right edge. You can actually bisect whichever edges you want. I'm doing those two adjacent edges, and now I'm going to draw a couple of lines here like that, and those lines are perpendicular to one another. And I'll just show you that here and now I'll draw a segment connecting these two edges, and that actually forms a 345 triangle, which are color in red. So there's my 1st 345 within this square. You can also create additional 345 triangles if you draw perpendicular line from the lower edge and snap that to the corner there and then draw a line down to the corner of the square. This form is a smaller 345 triangle with a different orientation, so I'm going to trace over that. But it has the same proportion with either of these triangles. You can find additional 34 fives by drawing perpendicular lines. So, for example, I could draw perpendicular and attach it there, and this actually forms to 345 triangles within the red one. They both share the same proportion and they get progressively smaller. You can keep going in this way, subdividing the triangles to be ever smaller versions that are all shared the same proportion. Similarly, you can do that over here on the Blue Triangle by drawing in perpendicular. In this way, the square is absolutely replete with 345 triangles 55. Euclid's Proposition 47: Now I'm going to demonstrate something called Proposition 47 and this actually dates back to Euclid about 2000 years ago. It turns out Euclid Elements is probably the most translated, published and studied of all books, aside from the Bible in the West, and this had a huge impact on people up until the 19th century, when it kind of fell out of favor. The elements are an analytical course in geometry, and it teaches you not only how to do geometrical proofs but also how to think rationally. And as such. It was really a treasured learning experience for thousands of years. Bear with me, I'm going to erase a bunch of these elements here, which are no longer needed. I just want to have this one, 345 triangle, and now I'm going to construct a square on each edge because the Pythagorean theorem says that the sum of the squares of the two edges is equal to the square of the high pot knees, and we can show that visually here by subdividing each one of the square into a certain number of great elements. So on the bottom edge, that's a on edge which measures three units long so I could show that by dividing that edge into three equal parts. And what better way to do that than the star cut? So I'm going to bisect each edge and then draw in the star cut pattern. And the reason I'm doing that is so that I can use that pattern to help me divide this now into three equal parts. And I'm doing that by drawing in these lines. And now that the lines Aaron, I can use the intersection tool to drop in points here around the square. And then I'll race thes lines and the star cut pattern, and I'm left with a grid. I can then connect the dots and form a three by three grid. Let's do something similar on the square over here on the left. Oh, bisect the edges and then draw in the star cut pattern. It's funny we get a red line there, I'm going to change the color of that. That wasn't my intention, and I'm going to continue drawing in lines. Okay, now to get the star cut here for a four part division, the secret is to draw lines from the lower to the right and then to mirror that and then do that on the other side as well. Now I can go around and put intersection points all the way around and then erase these lines as they extend beyond and then erase the star cut pattern itself. Now I can connect the dots, inform a four by four grid. Looks like there's a few other dots in there that I could add to the grid just to fill it out. Now, over here on the high pot news, I'm going to buy second the edges and draw in the star cut pattern and then think about five part division works like this on both sides and then in both directions and then going the other way as well. Now I work my way around putting intersection points around the perimeter of the square and then erase all these lines and then the star cut pattern itself and then connect the dots. Looks like I have to erase certain points. Still, I will go back to a race and tap on these points to get rid of them. One by one should be left with a grid of five by five, and you can optionally add intersection points on the grid if you are concerned with the aesthetics of this diagram, I've heard it said that this diagram would be a wonderful way to communicate with an alien intelligence because you could show him this. And even though you don't share a common language, we all share geometry. There's a universal language of math that transcends language, and that would be a way to establish communication. I find that fascinating. But now we've successfully created Euclides Proposition 47 which is itself a proof of the Pythagorean theorem, because nine units on the bottom plus 16 units on the left equals the 25 units on the high pot news. 56. Earth's tilt angle: Euclid is Proposition 47 which is what we see here encodes the Earth's tilt angle. And what is that? Well, that's the angle that the Earth's axis has with the ecliptic and that is responsible for creating the seasons on this planet. And it's really amazing that this diagram encodes that so we can find that by drawing a couple of diagonal lines here that go from the corner of the right triangle up to the corner of the high pot News Square, and then another line from the lower right corner of the triangle over to the corner of the side there. And it turns out that these lines are at a right angle to one another, and we can measure the angle here. I'm going to tap on a few points here to measure an angle, so that angle is 23.4 degrees, which is the Earth's OBL equipe, or tilt angle. Complement to that angle is given. Here I'll tap on a few other points. It's 66.6 degrees because 23.4 plus 66.6 is 90 and I'd like to show you a graphic that I've prepared that illustrates this. So here we have this angle, given a 66.6 degrees. That's the angle between the Earth's North Pole, which is the axis of rotation of our planet, which forms the days and the nights and the ecliptic, which is the the orbital path that the Earth takes as it travels around the sun and forming the year. So it's pretty amazing that this simple, sacred geometry encodes this relationship. 57. Connecting 3-4-5 triangle to squaring the circle: Now, I'd like to show you how the 345 triangle and this so called Proposition 47 connects with the rest of sacred geometry. To illustrate this. I'm going to start by bisecting this one square right here to find the midpoint of that edge. And I'm also going to draw an X inside of this square. This will give me a couple points. I'm going to label A and B, and I'll also label this point up here. See? So now I'm going to draw a circle centred at a goes up to see and I'm going to zoom out. Now I'll extend the segment. See A So it got passes through the circle on the bottom in a label that point D. Next draw a square with edge length CD to the left. A label. The center of that square is E. Now I'm going to draw a circle centred at E that passes through B and another circle centered at B that goes through a Finally, I'll draw a circle from E to A and then I'm going to use the hand tool and just rotate this around, and it turns out that this circle has the same length approximately as this square. I'm going to color that in red here. So this is squaring the circle. And if that's true, then that would represent the Earth and that would represent the moon. And if you like, we can also add the Great Pyramid into this mix by drawing a line from B out here to these points of tangent C of the circle in the square. And I can measure that angle and show you that it is indeed the great Pyramid Slope angle. 58. What is the flower of life?: no course on sacred geometry would be complete without a discussion of the flower of life. And the flower of life pattern is actually ancient. It dates back at least to the reign of Asher Bon Appe, all the Assyrian King in the sixth century B. C. E. And it's shown up now and again in different cultures in the world, including China. And here we have DaVinci exploring this pattern in his codex Atlantic US. Here's the pattern again at the osario on in ancient Egypt. Here's the cause Mahdi pattern in Westminster Abbey, and it really has a beautiful, sacred geometries all over the place. But in particular there's a detail that's on Westminster Abbey's Twitter page. It shows the flower of life pattern up close. In the next video. I'll show you how you can create this pattern. 59. How to draw the flower of life: Now I'd like to draw the flower of life and that starts with a circle. So the flower of life is something of a spiritual metaphor. Let's say that you're an infinite consciousness creating a universe and space doesn't exist yet. This is your entry point into space and time. The point at the center of the circle is your origin point. It's like the center of the Big Bang and then this boundary isas faras the universe exists . It's like the whole universe exists within that circle. And this circle is really just a shadow of a higher dimensional reality. You can imagine the circle is a cross section of a sphere. This is really a hyper sphere. And what we're looking at here is the simplest possible shadow of that, and we're representing. That is a circle. Nevertheless, in this simplified model, we can explore the creation of the universe. So we have the universe now in the circle and being this infinite creative intelligence. Maybe you're not satisfied with this simplicity of just existing within this hyper sphere. So what you do one day is you move your the locus of your consciousness to the boundary of what is known that is some point on the edge or circumference of that boundary. And then you survey another boundary using the same radius value. And this creates duality. So now we have all things that have polarity and from this vessel capisce, CAS form. We have this explosion of creativity that comes out of here. And we've seen that in this course that a lot of these forms that we draw start with the vessel Capisce, cas. At this point forward, we're not going to make any arbitrary decisions. It's all gonna flow from this geometry where the most interesting points in the universe it that at this moment I would say it's right here, above and below. And so eventually we're going to go to those places and create a new boundary. And that creates more of this pattern. And what I'm going to do is continue the pattern around in the middle there, more or less, we have this characteristic floral pattern that I'm going to color in red and you see that circle bounds thes six pedals. And so that's what we identify as the flower of life pattern. But it fills space, so we can keep going with this pattern in cycles that go around this. I'm just gonna do this a few more times and there we have another cycle. I'll do another one. Starting in. What you want to do is look for these intersection points on the out outer edge of your of your drawing in every time you make a circle, it's always with the same radius. There's nothing arbitrary about that. We're just letting the pattern itself guide us in the movement. Then what I'd like to do is color circles that are tangent to the original. So we have that one is tangent, as is all of these here that I'm coloring. So now we have six around one, and this is another important stage in the process. And as you can imagine, we're gonna just do additional cycles. There's the complete flower of life, and what we would do is we would trim off all of the circles that go around that blue circle. We would erase all of that if you're doing this by hand. But of course, we can't trim or modify this in this app, and so we have to be satisfied at this point. And so this represents the complete flower of life diagram 60. Drawing projection grid for a dodecahedron: all the regular Polly Hydra are construct herbal within the flower of life grid, for example, the tetrahedron, the Cube, octahedron, Acosta, Hedren and Dodecahedron. Now, these air three dimensional forms and so we can represent them in a two dimensional way using this grid and what I'd like to do is draw the dodecahedron for you. And to do that, we need to construct these vanishing lines. And to do that in a zoom in to that blue circle as much as I can and draw a line segment. And what I'm gonna do is maybe I'll make that in a slightly different color or line thickness. I think what I'll do is draw it with a dashed line type and maybe a medium grey. So it stands out. Now I'm going to draw a line from the center of the top red circle down here to this red circle and then do the same thing on the other side and then draw lines that connect to these other Red Circle centers like that. So now we have these four radiating rays, if you will, coming out of the top circle connecting to those circles down below. Now I'm gonna use a hand tool and rotate the drawing and do the same thing again, starting at the top and coming down here now I don't want to do it that way when undo a draw, using the colors and the line type that I set up before someone to draw it like this. And you want to be really careful that you don't snap to the wrong point here. It's really easy to do because there's a lot of geometry on the screen and then I'm going to rotate again and then continue drawing now. Noticed that I don't have to draw the ray on the left. It's already there from before, so I can. It can draw additional lines. Whatever is needed to complete this projection grid. Now rotate again. Draw again. Again. I don't have to draw that first gray, and now I'll rotate again. It looks like the Rays on the left and the right are already there. But I have to put the rays that go down like this, and similarly, I think we have just one more 0.2 dio and that's this one here. So I'm going to draw these construction lines down now I think I made a mistake on undo. I need to be really careful that I start that line from the centre at the top and not off and zoom in there and to show you that now that the trick for the requirement is that you have to you can't navigate while you're drawing. So what might be helpful? It's deism, inasmuch as you can, where you can see the start and end points simultaneously and then draw it out. Now again, I see it. That's wrong. Maybe if I draw the other way from here up to here, I'm gonna have more luck. There we go. And then again, the last line that we need it starts here in the ends up above. So you want to take a close look at what you've drawn and you may need to erase lines like I can see over here. I made a mistake all the rates, this line here because actually, it's not that when it's this one here. Okay, let's fix that. So I'll draw a new line that goes from the center down below, up to the center of above. I think that is correct. So you want to take a really close look at what you've drawn, you make sure that the lines that you intend to connect are connected at the center points of the red circles. In the next video, we'll use this projection grid to draw in the actual dodecahedron. 61. Dodecahedron within the flower of life: Now that we have the lines of projection sketched in in the dotted line type, I'd like to sketch in the dodecahedron, and for that I use a black color with a thick line type, and I'll start by drawing a line from the center of the entire pattern up to this intersection point above. And then I'm going to zoom in there and draw in this segment and then this one. Now I made a mistake. I have to go right from here to there so that the line goes from the projection grid to the other intersection on the projection grade, and I'm going to mirror that on the other side. And then I'm going to connect these points like this, and that creates a Pentagon in perspective. Now I'm going to don't draw in the next antagonist face, which starts in the upper left and goes over here. And then it's going to continue on. And this is hard, sometimes hard to visualize, But it's going to continue on over here, and then it will continue back into the center. No, I think I missed one just slightly. There. You see how these points don't connect? I need to erase this line in that point and re sketch it in like that. I want to do the same thing on the other side so I can use what I've already drawn as a guide. So I'm mirroring this over in my mind in drawing this over like that, that makes sense. And then again, like that, and into the core. Now we have the top of the dodecahedron complete now for the bottom part. Okay, Now the bottom part is going to start here and go over to that point, going to draw a line connecting those dots. And then I'm going to draw lines going up like that. And then I'm going to draw a focus on the left side. I'll draw a line in like that and then similarly above like that. Then I can connect these dots and then do the same thing on the other side. We'll connect these dots and then these and again here, and I think we have it now. There is a dodecahedron inside the flower of life 62. Metatron's hypercube: in this video, I'm going to draw Megatons Cube from the flower of life. Megatons Cube is a two dimensional shadow of a four dimensional cube, sometimes called a hyper cube or a tesseract to get started. With this, I'm going to choose a black color and the thick line wait, and this shadow looks like a cube within a cube. On first approximation, I'm going to show you what that looks like here by sketching in a cube in connecting the centers of the red circles on the outer perimeter of the pattern. And then I'm going to draw in the Cube like that, and then the back faces of the Cube would go like that. So that's the Outer Cube and now for the inner cube. So there's a cube within a cue ball, right? But there's more to Megatons Cube than just that, being a four dimensional structure, their interconnections that we have to also represent, and we can start by sketching in an equal lateral triangle in the core, and then another one below. And then we're going to do another one for the larger cube and another one going the other way. So now we have these nested triangles With this Ryker shin going on, I am going to look at the top circle and create radiating rays coming out of it a lot like we did when we drew the Dodecahedron. So I want to connect the top circle with the circles on either side. Then I want to do the same thing below. And then I want to do the same thing on the left and the right, and then the lower left and then the lower right? Hopefully I didn't miss anything. I think I may maybe made a mistake over here. I noticed this doesn't connect properly, so I'm going to erase that and that point, and that then I need to recreate that I will draw that in. So that's the complete set of lines that defines Megatons Cube. But if you look this up on the Internet, most likely what you'll find is this pattern, plus a number of circles that air highlighted and we already have those circles here, but just for emphasis, what I'm going to do its color them in. So I need to zoom in here and just color these in and black, and it's not all of the circles is just 13 circles. It's these circles that I'm coloring in now, so we're skipping certain circles like that. So what we see in black is Metta Theron's Cube, the two dimensional shadow of Ah, higher dimensional object. What we've just completed is Megatons Cube as a two dimensional shadow, but we're really dealing with a four dimensional object. Here is what megatons cabe looks like in three dimensional projection. The inside becomes the outside, which becomes the inside again. So inside and outside are all part of the same thing. And it's so difficult for us to really visualize this because four dimensions air literally beyond us. And so we don't have an intuition for that. If Megatron is, Cube is the two dimensional shadow than this simulation represents a four dimensional cube . It's rotating, so it represents a dynamic, three dimensional shadow, which is probably about as close as we can get to really visualizing this form. But it really has a lot of power, even in the two dimensional shadow form. And I've done sacred geometry workshops where we create a large scale mandala of megatons Cube, and I can really tell you that it's extremely powerful. There's something about resonating with higher dimensions that really is an invitation for personal transformation. 63. Next steps: I just wanted to say thank you for taking this course. I hope you really enjoyed it, and I hope you keep drawing. If you'd like to keep exploring sacred geometry, the best advice I can give you is just to relate my own personal experience, and perhaps that will inspire you to look into different areas on your own. For me, I discovered sacred geometry in the late nineties in the context of crop circles of all things. And here's a website that you can use to explore the phenomenon. I think it's important to set aside the question of who made the crop circles, whether it was aliens or the military or the Illuminati or just some people trying to make a hoax or an art project. In the end, it doesn't matter. What does matter is that geometry, and the more that you work with geometry, the more it can transform you. This website has an extensive archive that has aerial photography going all the way back to 1994. You can click on any one of these years and explore the most interesting circles from those years, and so I recommend checking this out and taking a look at these different geometries. There might be some that inspiring you to create your own artwork or just inspire you to replicate the drawing. See if you can draw it yourself. That's certainly how I started, and it led to me exploring geometry in detail. It eventually led to me pioneering some techniques which I wrote about in a couple of books I wrote Enhancing Cad drawings with photo shop in 2005. A few years later, I followed that up with enhancing architectural drawings and models with Photoshopped. In both of these books show how you can take a vector drawing in auto cad and convert it into Photoshop, where you can add texture and pattern and shadow and really make these drawings stand out. And these techniques could be used in architecture, engineering and construction, and they can also be used to create digital artwork, the likes of which you see in my galleries. Here. I also host sacred geometry workshops where we create group projects which are really very powerful, and later on at the end of the workshop, we have to sweep those designs away, which can be a very powerful and transformative experience. Exploring geometry led me to get deeper into physics and fine art, and it caused me to look at city designs. This is Beijing, the Forbidden City. It caused me to look at architecture from all different periods to explore the human body and how it relates. I also created a Siri's called Secrets in Plain Sight in 2010 and now this has been seen by millions of people. And it's led to a lot of interesting conversations online with people who've been turned onto geometries that you can find all around you. I also have a art gallery here relating to the secrets in plain sight, and there's some overlap that you'll find with my Sacred geometry academy. But again, there are some images that are only available here in this gallery, so you can browse through here and see how sacred geometry has inspired me over the years to create different artworks and explorations. So I just can't emphasize enough how powerful geometry can be. If you just stick with it and keep drawing, you can really transform you so that the point is to keep working with geometry. However you feel inspired to do that, whether it's just drawing with a pencil and a straight edge, or decorating your drawings as watercolors or acrylics, or creating physical structures with sea shells and stones or colored sands however you want. The point is to let the geometry unlock different features in your consciousness. It's really a mysterious process how that works, but it's generally very positive and inspires a lot of creativity, so keep drawing.