[The intro]
One of my favourite things about shapes is all the different shapes they come in.
This is a regular icosahedron. It has 20 faces and all of them are equilateral triangles.
But if you saw my previous video on platonic solids you know that there are many other solids also made out of equilateral triangles.
Another one is the tetrahedron. It has just four faces but all of them are equilateral triangles.
You can make one by making a net of four triangles like this, lifting a couple up and then having them do a little bit of what I call "The Tetrahedral Hug".
It's really more of a kiss, isn't it?
One more platonic solid is made of nothing but these beautiful three sided shapes.
But it has a lot of them. We gotta have eight for it.
If I take this little llama- there... There's the llama!
Uhhh- what sound do llamas make? Umm...
BRLLBHUALlama
If I take this llama and I fold together the four here in my right hand to form a square pyramid,
and I make a square pyramid out of these four as well, I can join both of these pyramids base to base
into an octahedron.
Ohh so very platonic and so very much only made of equilateral triangles.
A solid made entirely out of equilateral triangles is called a deltahedron.
And there are an infinite number of them.
But there are only 8 that are strictly convex.
Meaning they have no concavities. Strictly means that we don't even allow the triangles to share the same plane.
They cannot be coplanar.
We've already got three of them right here so we have five more to build.
But in order to appreciate their names let's talk about some other stuff first.
The first thing I wanna talk about is pyramids.
The tetrahedron is already a pyramid, we're taking a polygon base and connecting that polygon to a point.
This is what a pyramid is but a pyramid doesn't have to a have a triangular base.
You could have for instance a square pyramid.
We take the square as a base and we connect that square base to a point.
There's a square pyramid the same could be done with a pentagon-
Ohoho.
But only if you're really good with- uh- your hands.
There we go! A pentagonal pyramid, how beautiful.
Now another thing that we can do is connect a polygon to, not a point, but another polygon just like it that has simply
been translated away but not rotated. If we do that what we are making is called a prism.
I'll connect these two squares with other squares and I will make actually a very Special Kind
of square prism: A Cube™
A Cube™.
You can make a prism out of anything.
This is what a prism-
Well you can't make a prism out of Anything like you couldn't make a prism out of- you know- Justice
I just mean that you can make a prism using any polygon base.
So here's a pentagon.
And if I put squares all the way around I can connect that pentagon with another pentagon.
Boom.
A pentagonal prism.
But there's another way to connect to parallel polygons
That doesn't involve squares. Instead it involves The Shape of the Day: The Equilateral Triangle™.
Not just any arrangement of triangles, but an alternating arrangement of triangles that go up...
And then, down...
And then up, and then down
And then up and down all the way around the shape.
The top of this chain is a plane.
And, as you can see, it has the exact shape required for me to fit another square
But this time the square is rotated a bit
So, here we have a square prism
(also known as a cube since this one is actually made of faces that are all identical)
And then here, we have a square antiprism.
The next preliminary thing we want to do is talk about combining pyramids.
Two pyramids.
If I do that, what I've created is a bipyramid.
Let's begin with the tetrahedron.
If I take two --
triangular prisms, there's one.
Let me get some room, so we've got a nice view.
Ahh, yeah, there he is.
And then, I'm gonna make another triangular pyramid
*pyramid abuse*
I can join them at their bases to make a triangular bipyramid.
Now, this shape is not a platonic solid because it's not regular enough.
We have a vertex here where three faces meet, and a vertex here where four faces meet.
But that's fine. Because -- we're not talking about platonic solids today,
we're talking about strictly convex deltahedra.
And this is another one.
So, gosh, we've already got four of the eight. We're halfway done.
The next modification you can make to solids that we're going to need to discuss to move further,
is elongation and gyroelongation.
This process is somewhat analogous to the construction of a prism and an antiprism.
Except, instead of connecting polygons, we're going to be connecting solids.
I'll show you what I mean.
Let's take the octahedron, not this one though, because that's my example of an octahedron.
I have previously prepared one right here.
If I take an octahedron, and I separate it -- like I'm cracking open an egg --
Ohh yea- oh -- okay.
Well, I wasn't able to do it, but, I have two square pyramids.
And what I can do, is instead of combining them base to base, what if I --
Separate them a bit, and put, like, squares in between?
Well if I do that, I will have built myself what is called an elongated square bipyramid.
Which, by the way, an octohedron is just a square bipyramid.
An octohedron is just two square pyramids put together.
But we usually don't call it a square bipyramid because octohedron is a much cooler name. Apparently.
Alright, now, if I take a square bipyramid and put squares in between them, there we go.
So, I will now put the second square pyramid on top, there it is!
An elongated square bipyramid.
It's not, obviously, a deltahedron because, well, we have square faces.
But, they don't need to be squares.
Because: there's another way to elongate two solids, and that can be done by what is called gyroelongation.
Let me remove these square pyramids, and instead of using squares -- to separate them,
I'm going to do what I did with the antiprism, and I'm going to use equilateral triangles.
So, I'll grab these three, and, I will start building an alternating band of up and down equilateral triangles.
Alright, yeah o-okay, good.
Woah!
Alright, you know how that saying goes:
You can't make a gyroelongated square bipyramid without droppin' a few tiles.
Got me through a lot of rough parts in my life.
Okay.
Here we go, OH!
*visible defeat*
This is startin' to feel like it might make a better timelapse than a real thing.
[Music plays over timelapse footage]
We've got it. We've got the alternating band of triangles,
and their hole, as you can see, is a square, which is the base of our square pyramid.
So, we just need to gingerly pick that up -- and, ta-da!
Oh no. Yes. There it is!
Ahh, yeah. This is a gyroelongated square bipyramid, and it is a strictly convex deltahedra.
Wonderful, so what've we got here? We finished five of them, we have three more to build.
Let's try to gyroelongate the triangular bipyramid.
So I'll need to make two triangular pyramids.
There's one, and here's two.
Now, I'm gonna gyroelongate them with that alternating band of triangles.
du-du-da, du-da-daa
Ah, this is a lot easier than gyroelongating a square pyramid.
Alright, here we go. Nice! Nice!
NOT NICE.
This is not strictly convex. Notice that we have two triangles that are coplanar.
This, my friends, is a rhombohedron. It's a cube, except that the faces of this "cube" are not squares, they're rhombuses.
So:
*Michael enacts his wrath*
It's gotta die.
Alright, so we can't gyroelongate --
the triangular bipyramid and get a deltahedron.
The other kind of pyramid that we can make besides the square pyramid or the triangular pyramid
is: a pentagonal pyramid.
There we go. That's the pentagonal pyramid. I built it earlier and I actually put a pentagon base on there.
But what we wanna do now is gyroelongate th- oh.
[background music cuts out]
Actually, why gyroelongate it when we can just make it a bipyramid?
Yeah, let's do that one first. Okay, umm, let's see here. So if I take two pentagonal pyramids,
and put them together *click*
I have a pentagonal bipyramid. This ten sided shape is just a beaut, and it is a strictly convex deltahedra.
So I'll put it, like, I wanna make sure it's still in the frame.
Beautiful.
*a strictly convex set of coughs*
Now,
gyroelongating the pentagonal pyramid is not a task for the faint of heart.
But if you do that, you wind up with an icosahedron.
Look at that!
If you look closely, you can see that I have a pentagonal pyramid right here, and I have one here on the other side,
and separating them, I have an alternating band of equilateral triangles.
So folks, if you wanna be really cool you can call the icosahedron a Gyroelongated Pentagonal Bipyramid™.
But you probably don't because it's a real mouthful to say.
So we've got now: one, two, three, four, five, SIX strictly convex deltahedra.
We've got two more to make.
And, I think the next thing we should do --
is that we should start talking about yet another geometric thing you can do to solids to make new solids,
that I hadn't discussed previously, and it's called augmenting
Now, if you augment a shape, you replace one of its faces with a solid.
So I'll take two triangles, this is the first one as the base, and I'm gonna connect it up to another triangle.
There it is, a triangular prism.
But, I can augment it and get rid of these square faces and replace them with equilateral triangles
As you can see, I've built --
umm --
three square pyramids, and I'm gonna replace the square faces on this prism
with these square pyramids.
That's the way to do it. Now, let's augment this face, just remove the square, we don't want you. We're trying to make deltahedra here.
And, here we go, lift this one up, and throw it right there.
And then finally, I'm gonna take away that face, and grab this square pyramid, and boom!
A triaugmented triangular prism! The seventh strictly convex deltahedra that we have built today.
The final shape, the final solid we wanna make today, is
Well, it has the best name. It's called the snub disphenoid.
Yeah, umm, its name is a little bit nutty, umm, and I'm gonna describe why it is called a snub disphenoid
after this video, because, if you construct it according to its name, it's incredibly difficult.
But! There's an easier way to make it, and it involves using a square antiprism. Remember this friend from earlier that we made?
Well, we're gonna dissect him.
What we need to do is replace the square on the top and at the base, with two equilateral triangles.
So, I will remove this square --
*click*
and I'm gonna replace it with these two triangles. Boink! Just like that.
Now, you'll notice that when I replaced the square with two equilateral triangles, I actually have a choice.
I can put the triangles in this way,
or this way.
For that reason, the snub disphenoid actually has two different versions that are mirror images of each other.
I'm just gonna randomly choose that direction.
Look at that! That is a snub disphenoid, and it is the eighth and final possible strictly convex deltahedra that can exist.
These are all of them.
Equilateral triangles can make three kinds of pyramids: triangular, square, and pentagonal.
The triangular is a tetrahedron and already meets the criteria for strictly convex deltahedra, so, perfect.
Make two share a face? Still strictly convex, still only made of equilateral triangles.
Now, the other two consist of only equilateral triangles if we make them bipyramids.
More equilateral triangles can be added by gyroelongating, and doing that to the triangular bipyramid gives us, well, a rhombohedron. Not good.
But! Gyroelongating the square bipyramid fit our criteria.
Gyroelongating the pentagonal bipyramid gave us an icosahedron, which was great.
Now, gyroelongating more than once would just leave us with coplanar triangles, so we stopped there.
The only augmented prism that consists of only equilateral triangles and does not become concave --
is a triaugmented triangular prism.
And, the only snubification procedure that results in only equilateral triangles is snubifying the right kind of disphenoid.
And there we have it: all eight strictly convex deltahedra.
Hah. Geometry? More like ge --
and as always, thanks for watching.
Okay, so now that we know how to construct all eight strictly convex deltahedra,
and we know why almost all of them have the names that they have, let's revisit the snub disphenoid;
whose name we didn't really break into. First of all, disphenoid.
Disphenoid means "wedge shape"
Funny enough, the tetrahedron is a disphenoid, but it's usually not called that.
The regular tetrahedron, that is. It's just called, y'know, a regular tetrahedron.
But, let me, let me do this. Let me break apart a tetrahedron into a net
Like this.
Oops.
There we go, okay, and now let's build this same net, but not using equilateral triangles, let's use --
isoscoles triangles. So, the one in the middle points down, and we put two on the side --
I'm gonna not use the one that disappears into the table.
There and there, and there we go. Now, if I fold the triangles up, I can make a tetrahedron.
This will also be a tetrahedron but it will not be a regular one. But both of them are called disphenoids because, well,
in this case you can really see why it's called disphenoid, it definitely looks like a wedge.
Wedge, right? But --
When you snubify a shape, you're doing a very very interesting maneuver. But before we can talk about snubifying,
we have to talk about expansion. Okay, we'll begin with what I think is an easier way to visualize the procedure,
we're gonna begin with a cube. So, let me find enough squares to make -- oh, there's already a cube over here.
Okay, so, here's a cube.
Now, when you expand something geometrically you take --
each of its sides or faces and you move them perpendicular to themselves away from the center.
So, if we think of these two sides, these two faces of the cube as being represented by these squares
Ooh, those magnets are quite -- doing what magnets do.
Okay, we could - they expand- no!
Okay, the faces can expand out like this.
If you imagine all six faces expanding out,  then you've got yourself an expanded cube. But we're not quite done because we wanna fill in the gaps.
and what we could do, is fill those gaps in with a rectangle.
But, if we put them just the right distance apart, the squares can be --
*tragedy strikes*
Shoot!
Would you bring me a -- ah, I think I have enough squares. I think I've got enough.
If we expand these faces just far enough away, they can be connected with another square. Okay? Perfect.
Now, let's take a look at this uhh, woahhh.
Let's take a look at this front face on the cube. If we expand that away,
just the right amount, we can connect it to the top face with a square.
Now, in between, we have a gap that can be perfectly filled with an equilateral triangle.
And, down here, with another square.
*audible distress*
Yep, yep there we go, come together, oh yeah!
Alright, wonderful. So, if I turn this around, you might be able to see more clearly what I've done.
I've got three -- woah, I've got three squares that represent the original faces of the cube,
this one,
this one,
and this one.
They've expanded away from the center and I've filled in their gaps with squares and triangles.
If I do this with one more face, it can actually become quite stable, and then I don't have to keep holding it.
So, I'll build this, I'll put an equilateral triangle right there, and I'll put a square right there.
Wooh! Now I don't have to hold it. So now we have this beautiful shape, if I put the cube inside, you can see what that
expansion has done. I've taken the top of the cube and expanded it up to here,
The sides are these two, and the back is back here, and I've filled in the gaps.
So that's expansion. But there's another way to fill in the gaps that uses not squares, but equilateral triangles.
And it's gonna look something like --
this.
I'll take twooo squares.
I'm runnin' low on squares, could you bring me a big stack of squares
You guys, you guys think I'm obsessed with shapes, well, just you wait. Alright, so:
We got lots more isosceles triangles,
we got lots more of the big squares I don't want.
Here we go! Here's some squares!
Cool.
So, let's snubify this cube.
What we, what we make by snubifying it is called a snub cube.
At these faces, I want to expand 'em out.
But instead of expanding them, and connecting them with another square,
like that, I'm instead -- oh goodness, I'm gonna connect them with two equilateral triangles.
So, the face that I've expanded out like this, I'll put a triangle on top of, 'kay?
And then I'm gonna put another triangle on its side, and the top face is going to connect...
There. Which means the top face is actually going to rotate a little bit.
Now, if you continue doing this, you create a snub cube.
And I'm not gonna have the dexterity to do this in real life but here's a fantastic animation of it happening.
Oooh, look at that snubification!
So if we snubify a tetrahedron, that is, we take the faces, expand them out, and connect them with equilateral triangles...
we will be snubbing a disphenoid, which is what a tetrahedron is.
And, we wind up with a snub disphenoid!
Wooh!
So there you go, that's why it's called what it is called. Snubification and expansion are fantastic, I've got some links down below to learn more about them.
And as always,
thanks for watching.
I'll take two...
squares. I'm runnin' low on squares.
Could you bring me a big stack of squares?
There's... oh my gosh.
There's a Whole Bunch O' Squares™ out on the- there's a Whole Bunch O' Squares™ on the
Trophy
shelf.
That's right we got a Trophy Shelf™.
It's where we keep our squares.
Because in reality, being square's the real prize.
I'm talkin about nerds.
I'm talkin bout bein a- being a nerd
I wouldn't know anything about being a nerd.
I'm not like... spending my days building shapes.
Eugh! Ghoy! Disphenoids phe-
Okay.
Look, we get it... four eyes.
That's what my wife calls me. She calls me Four Eyes and I'm like "What is this, 1920?"
Okay I- Hannah! - Alright bring them in.
Here you go!
You guys, you guys think I'm obsessed with shapes? Well, just you wait.
