Transcripts
1. Introduction and Project: Optical illusions
have fascinated humans for centuries
in profound ways. Their study dates back
to ancient Greece, where philosophers
like Aristotle observed how visual
perception could be deceived. In fine art,
impossible objects are designed to defy logic and
three dimensional space. In this course, I will
teach you how to draw a cute hexagram inspired
by the Penrose triangle, using a ruler and a compass
and how to decorate it, emphasizing its three D effect.
2. Constructing the Grid: Okay, let's create this fun
impossible object pattern. I'm using a 20 by 20 square
piece of watercolor paper. You can use any size similar
to this A force fine. You can use card depending on how you'd
like to decorate it. I think I'm going to
want to paint it. To find halfway
through the page, it's actually 20.2, so I'm
going to make a mark halfway. My halfway is 10.1, and I'm going to do that
on the other side as well. 10.1. Then I'm going to draw a horizontal line
for the two marks. I'm not going to go quite to
the edge and mark halfway, which again is 10.1 because
my paper is square. I needed to find the middle. For this size or A four size, I recommend that we start with
a two centimeter radius so measure out 2 centimeters because we're going
to build this pattern from the middle outwards. I'm going to start with
the central circle in the middle on the
axis that we drew. Now on the right
hand side of this, where we have made an
intersection of the circle with the line with
the same radius, we're going to draw a new
circle which should go through the center of the original
and then two more. 12. Then the same on the other side, we need three more circles, starting with the first
intersection to the left, going through the original
center here, and two more. Be careful when you put
your point on the point, that's quite important, that it doesn't slip and that
it's in the right place. Now we have seven circles. Below that, there are now six intersections between
each two circles that we drew and that's where
we're going to do another six circles
with the same radius. I'll start on this side. Again, nice and careful because the accuracy here will
depend on the points and this should now go through the two centers above creating this pretty
little Rule triangle, a curve triangle, we're going to start seeing
these petals now. There's the other
intersection next to it. Again, with the same
radius and repeat this six times along this
horizontal row. Now we're going to draw
five more circles from the intersections
of the six we just made in the same exact way. Same radius, going through
the two centers above. All the petals should
be the same size. The final row on the
bottom consists of four of those same circles going
through the centers above. I'm now going to rotate
the page and repeat exactly the same rows that are here on the bottom to create
a symmetrical pattern. Now that the grid is
fully constructed, we have to draw some
lines to see where the intersections and
the straight lines of those cubes are
starting to appear. We're going to start
with vertical lines because there's
going to be lines in three different directions. I'm going to start building up the lines going
outwards because we know from the design
that there's going to be a little star that
forms in the middle. The very central circle
where we started, there are these two petal shapes and the first one
on the right is where I'm going to put my line and my ruler and
align these two. Now, to be most accurate, it's always best to separate the points that
we're aligning further apart. I'm going to take these
two centers into account. Not the first one in the very
center that we started off, but the one that goes
through the first petal to the right through
the vertices of that petal and going through each of those points there and I'm going to just
draw a vertical line. Now I'm going to draw
a similar one parallel to this every other petal. We just use this petal here. Now we're going to move on to
the next one which is here. That means here and here. Again, I'm going to
separate this and align to the furthest possible
points because that makes the lines
more accurate and again, a vertical line that's parallel. That should be the distance, one of the small little
petals apart. That's correct. Now we're going to go to
the next one to the right. Again, align these two. Look at those two points that
are a bit further apart. And go ahead. Make sure these extend a little bit beyond what
we actually need. Finally, this one that
connects those two. We have four lines on the
right hand side of the center. We're going to repeat the
same on the left hand side. The left of the center, we
see this almond shape line separate as far as possible using the centers
of the circles of the grid and make four
more lines on that side. Now we're going to
repeat the same thing, but going in the
other two directions. What I mean by this
is at the moment, we use this vertical direction using those two
big almond shapes. If you look at going
left to right, there are two more going in that direction and two more
going in that direction. Now, I like to
rotate so that I can imagine this so rotate this so that this is in the
center and you could see two vertical almond shapes on either side of the center. We're going to repeat
exactly the same, starting on the
right of the center, the first almond shape using the two vertices
above and below, and then separate apart. Then repeat that line
parallel four times. Make sure you extend long
enough so that it crosses with these lines here
because there will be some angles on the
edge here that we need. Moving to the next almond
to the right, just here. Use the centers of the circles of the grid
that you've drawn and make sure you extend out to complete all the
corners that you can see. Two more on that side. And the same for on the left
hand side of the center. Now repeat exactly the same, but going in the
other direction, bottom right to top left by
rotating this and finding the vertical almonds that go parallel to the
center on either side. If the centers of the almonds are going
vertically either side, start with drawing four on one side and four on the other, going through all the centers
and making sure that you extend so that it crosses
on the outer corners. Here we have it. This is all the
lines we need now to trace out the cubes in
the correct positions.
3. Outlining the Verticals: The outlining is going to
be the trickiest part. Here's one ready made variation. We can use that to help us. I also have this grid that I'm going to use
to help me as well. The first thing I would say is that right here in the
middle of the design, there's this star
six pointed star, which is exactly here. Inside it, we can see this little hexagon just
around the central point. I think working from there would be a really
good reference point. I'm going to start with
the vertical lines, looking to go to the
left and then of course, everything on the
right hand side of the center will be the same. You can see here that
I've also kept some of the pretty curves from the background, but we
don't need to do that. We're going to focus on
finding these lines. We're going to start with
these two here going down. Here's the center and here's the little hexagon
in the central part. I'm going to just align
a vertical line through all these points that go there, but I don't need the entire
line, just a few segments. What we need first
is immediately above the central hexagon, we have a cube sitting
into it, of course, the lower part of the cube is where it has a line segment. Starting from the top of
this hexagon in the center, we're going to draw a line up. Here's the central hexagon. From here, a line below this, there's also one of the cube sitting just below the hexagon. However, the top
part of the cube is that roof side where
we don't need a segment, so we need to skip that
and then draw a segment. From the center, we're going to skip one, two distances away. Here we are in the
center, we're skipping one, two distances away. That distance is the roof of the lower cube, and
here we have it. Now, I'm using quite a thick
marker so you guys can see, it might be slightly off, but these are the two
lines that we need so far. Go to gently move only that distance
from the center to the edge of that
central hexagon here. Now I'm going to align
the ruler with that side. Using the line from
the grid underneath. This will be a little bit
trickier now, more lines. But let's locate
this line that we've already drawn, which
is this line here. This line, now we need
one that's parallel, just sliding down, parallel to it to the nearest left of it. Sliding slightly down along the lines and
parallel to this one. This creates this gap
of this first cube. Now from here, I'm going to
skip an entire distance. I'm going to skip
this entire distance here just to do this
little part here, which is half a distance. We've done this
one and this one. Now from here, I can see that I just need to go
around that central star. At the moment we've done
this part, which is here, we're going to skip half of
that and then draw only half. Skipping half of
that, in other words, we're skipping the side
of the inner hexagon, we're skipping that and then
just half a distance below. Skip that and then
half distance below. Then skip an entire distance. And then just draw one
below that, which is again, parallel, it relates to these two in the same
way as these two. Parallel to this one just slightly further,
slightly lower down. This is now done. That was quite tricky
because of course, some of those parts needed to go behind the central cubes. Now I'm actually going to leave these two
because they're trickier. Let's do this main part. This will be a good one to do. I'm going to find the
first line we did was at the top and then it's
parallel was next to it. Now I'm going to move one
full distance away from that. Now we are looking at
this vertical line. In fact, I'm going to start from the very bottom because look, it's easy to align looking
at that lowest line here. We have these two lines
already on our grid. Then we're going to skip
a whole distance here, and then I'm going to go
upwards by doing full length, skip half, full length,
skip half, and so on. Let's do it. Following
this full length. Skip half, which is that
little bit full length. Skip half full length. Skip half full length, and that's four of
them, isn't there? I only need four
here and then that leaves some space here
for the top of this cube. Again, I'm going to skip
that little part here. These little ones, I think we should worry
about them later. Then now we're going
to do this one, which again is that
distance away, a whole little hexagon away. This one really roughly should mirror this one,
the one we did last. This one mirrors this one. We just have to be careful here because there
are differences. This one mirrors this
one on the other side, we go down and up from
this corner here, that is its corresponding corner and that segment
mirrors this one. That's a full segment. Now we located our first full segment. Full, skip half full
full, skip half, four. Skip a whole one,
then draw a half, skip a half, draw a full one. Again, from here, skip a
full one, that's a full one, the only draw a half to here, skip a half and at the top, draw the full one and that
one now mirrors this one. So we've done that side. Now, let's start here, that end, we can't get it wrong. It's exactly at the very edge of the outermost circle going
through the middle axis. There's the axis, there's
the outermost circle. We just need to join
these two vertices that just touch that
circle on the outside. Only one segment here. Now to find this one, which I just skipped
because it's a single one. We need a whole distance apart. We could just trace
this diagonal here. That's this diagonal here, it should be parallel to this
line which we already have. We're drawing this
one parallel to this one slightly up from it. Just this Okay. Let's recap going from the left. We have these two. We have all of those, but we
didn't draw this one. Now let's locate this one will
be a little bit trickier. This one is close to the
second one down from the top. First one, second one
down, parallel to it. Go down but only do half. It's this second one going down, just do half. That's that. Then those four in these four, which alternate, short, long, and we're back into the middle. Now we need to make
the right hand side a perfect mirror image of that. To do that, I'm just
going to slightly slip. I'm going to start from
that side going up because we can see now we see
exactly what that should look like on the other side of that outermost
edge, it should be here. That's why I'm starting from
the furthest away corner that we can see what's happening on the left and that is what we're drawing here,
this side one. Just from here to here. Going to now move a whole distance in. There's nothing actually
here. There's nothing here. I'm moving a whole distance in. It's just a line. It should be the mirror image of the mirror image of this. This is the only one going
here from that side. It's just the other side of this edge just
slightly further down. We've done these two and the next one is half
a distance away. This will be slightly
more tricky now. We've done these two and now we need to recreate that one, which is also the mirror
image of this one. We can see how that
distance should be here. There's the intersection here, and we need a full
distance down. Then we skip half here and
then draw half going down, skip a whole distance here. So it's lower than that one and goes a full distance down. It's next to that one the same way as this is
next to that one. So far so good. Then
one more at the bottom, we skipping half a distance here and then just adjust a little. Skip that and then this
full distance here, which is the mirror
image of that. Excellent. That was this one. Now I'm going to move
half a distance only. And now we can make
that tiny little bit. Now we can see where it is
because of the mirror image, it's much less scary
ones than the first one. Let me just align this well and then I'll move my
fingers so we can see. This is the segment we
want, but on this side, it's clearly going to
start here and just half. That was that now moving
half a distance here. These will be the four distances that we're
going to do here. They will be the main edges that are pointing forward
in that main column. I see these as two main columns. We could start from here a line and see that it
starts there from there, full distance, just like here, skip a half, then full distance. These two are there. Then again, skip a half. Then
a full distance. We know we are
correct because look, all of these are aligned within the same two parallel lines SLOs the final one
at the bottom, which is corresponding
to this one is from this center to this
little vertex here. We've done these. Now, this will get a little tricky because
I need to overlap it. We need to recreate that
last one over here. If I did it this way, you
could see what we need. We need half full. I could actually just rotate it. I will rotate it, and
this is the very center. In fact, we could have done the entire side going this way. This corresponds to this. Then this little
segment across here, this long segment here, and this little one at the edge. I'm going to turn it back around and these are all the
lines going down.
4. Outlining the Diagonals: Going to do some diagonals
and normally I would rotate. But actually, in this case, I feel like seeing the orientations of
the cubes will help. I want to now focus on mainly the roofs and I'm going to
be going in this direction, starting at the top right. We can see that
we've already got this line here and we
just need to join that. Where do we join it with
that intersection there? That's the first segment
here, it's the roof. Then of course, it's parallel should be here
joining those two. This is much easier to see now, hopefully you will agree because it's starting
to shape up. That's why I didn't
want to rotate. We've built almost
or at least we can visualize almost all of the roof the top of
that cube there. Now half a distance away, again, using that parallel line. Let's see what we need
to do. I'm going to start from down here
because it's closer. It's from the very edge of that outer line
we've done, again, a full distance going it's that to where we
meet the next line. That's how I know where to stop. We're skipping that little
bit here and then we only need half of a distance
after we skip. From here, half a
distance because that will be the line of the
bottom of this cube. Okay. Now we're skipping
this and now we want to find the line parallel to these
two and that we've got, and that's the next one we want. Again, we're just
making sure we're using the two parallel lines within
which our cubes are lying. Then the top one from the
corner we've already drawn, which is here all the way to the top, all the way to here. Of course, that
should correspond to that, which we managed to do. I'm going to leave
this again and just skip an entire distance because just visualizing those bigger
spaces makes it easier. Starting from the
bottom again at the top of this line
here, which is there, we're going to draw this line parallel to its opposite here. That's the full
length, then skip a half full full, half full. We don't have to worry
about the pattern here is fully the four segments. There's one from here
to here. Is the next. Skip from here to here, is the next, and finally
from here to here. Again, we could
see how these are all pairs of parallel lines, except for this one which
tucks behind the other one. The only one we skipped
was this one here. Which is at the
bottom of that first. I'm going to go back and do it now as I can visualize it more. It's at the bottom of
that top right cube and it's only half a distance up to this just two there. Can you see this is starting
to shape up like this? Which actually is telling me
we haven't done that line. That segment here
completes this shape. We did this one here, that's the second full
row and then moving. Let's do the one that's parallel and part of the inner star. I'm going to do this one a
full distance away like this. You can use those rhombuses that are shapen up to help us. This is the line
we're now doing. Starting from the bottom
corner of the rightmost cube, we could see now only
half a distance and stop because that
tucks behind here, then an entire distance away and then an entire distance in. In other words, this
rhombus here is completely uninterrupted
and it's part of the star. Here we have it. Then the next one is only half, but we could already see why
that is and the top one, again, going up like this, the top one is uninterrupted. Tricky, isn't it? It's
starting to shape up. Now, I feel like I want to
go in the middle now and just finish those quirky ones. Now we can see that we're
aiming for the shapes, these inner concave hexagons. I'm just going to go in
the middle and add those. This one completes
this shape, this one, that one and as before, you just need that little
segment going down, which is the mirror image of this one which we just added. We've done the top half of this. Now we're going to go on in the middle of
the middle star. A line as far apart as possible, making sure you're going
through all the points. Let's see what we have here. This cube here, which is there, we need to join
that all the way. That's that corner, which
takes us into the star, jump the entire star, then jump the entire rhombus, and then jump this rhombus, which is that side piece here, side face, and we just need that because that
is parallel to that, it's part of this here. Now they should start being the mirror images
of the ones before. But this is the mirror line. This side here should be
the same as this side here. What I mean is that
these segments here are identical to
these segments here. This is the middle through the
center, that's the middle. We want that to correspond. We want this to correspond
across to this. Then we skip that and then
we have that little segment which goes across
here back to here, skip above the hexagon, then a full length across
from here, full length. And skip a full length, and then a small segment across. There's this segment
across, so that's done. Okay, so now I'm
going to move half a distance away.
Through the centers. Now we've just done this one, and we want to mirror this one, which essentially is
just these two segments. This one should go on
the other side, one, two to the middle, one, two on the other
side, just half. That distance is there. Then down here, one, two to the middle, one, two
on the other side, just here. That here corresponds to that. Move again only half a
distance. Align carefully. Here is the situation
where we have just four straight lines
without interruptions, the same way as these
four full lengths, but on the opposite side. That one corresponds to
that skip half distance, skip half full
distance and the last. That corresponds to
that, and so on. Let's do the next distance. These are the four. We haven't
actually done those yet. There was only that one. Yes, we have. It's
this one here. One, two, three, four distances
away, one, two, three, four distances away,
and it's from here up, making sure it aligns. That's this one. That's the only
one we've drawn on that side, then distance away. So now we want this here. So except for the third one, the other ones are
whole distances. So that here completes the
rhombus, then another rhombus. Then the third one
is that quirky shape where it's only half, and then the fourth
rhombus here. Then that will be just the
side of this single cube here. Again, parallel to that within the same two lines and complete the
Rhombus on the bottom. Again use these
points otherwise. Okay, little scary, isn't it? This is the original
orientation. Now let's do some
going this way. Here on the bottom, it's
obvious, we're going to do this. Complete bottom. Each one should become
easier now because we can actually see where
the shapes are going. Complete the top
of that same face. This edge here is
there. Now move up. What do we have here? Full edge, half, and then two
other full ones. 4.5 that tucks behind this one in front,
and then two more. Then we have this
little quirky one here, which will go on there. I just here. Completes this the upside
down shape back here. This is now a nice one. This is those four uninterrupted edges. Full edge four times in a row. I'm going to move halfway
in because there are two short segments joining
those little shapes. Let's just see where they are. At the bottom of this
cube here, it's half. And then going further up there is just here at
the bottom of that keep. Then we're going to move
upwards into the star now and see what we have there around the
star and into the star. We're going to start
from the bottom edge and we have only
half a short edge. You can see it's
sticking out here. Then skip a big length and
complete this rhombus here, which is there, that
goes into the star, then another half
and another four. Half completes the star and
full completes the rhombus. We've done those four. I'm going to move on that
side so we can see now. Now let's do the central part, going through the star and
going through these rhombuses. It's these two cubes
to complete top one, bottom one up to
here, it's all done. The rest of it now, I'm
going to rotate this way and reflect all of
those lines on that side. So whatever goes down here needs to be the mirror
image of these line segments. This one is across here. Then this short one
is across here, then this long one
is across here. Finally this short
one is across here, much easier to see them now. Let's do half a distance
and see if we have any of those little quirky
half distances to be. It's just on this one. The first one on the side of the center. The second one, we have
this one and this one. This one here is there. This one here is here. We're clearly missing one there. We'll go back to complete
that in a moment. This is the four main
ones going this way. There will be four
uninterrupted segments. Then just half a distance away because I can see
it's just this one off that is here
corresponds to that. Then we're going to
do this long one. That will complete quite
a few of those cubes now. Long distances are short
and long too long. Then this short one here
and a long one the bottom. Here you can see it's clearly the edge of that cube that
will correspond to that. Then the final one
is down there. I'm going to turn it back to the original orientation and
just notice where we've got. I think it's just these
two segments that should correspond to these
that we haven't the. That is the whole
thing outlined. Now once that's nice and dry, we can rub off the
marks unless you choose to keep some
of them and then decorate it in a way that you're emphasizing the three
dimensions of each cube.
Diana Reeves, Geometric Artist & Educator
