Fun With Hyperbolic Space

I Love the Fractals

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00:00Here in hyperbolic space, we can fit more than four cubes around each edge,

00:04as long as the cubes are big enough.

00:06And here the cubes are quite large, and we, in fact,

00:10have six cubes around each edge.

00:13So if I just try and kind of move around the edge,

00:16I will go through six cubes to get back to where I started.

00:22And it can be kind of hard to see what's going on, or how many cubes.

00:25How do you count them?

00:26And here I'm just going to put my face in an edge

00:32so I can look down and see that, indeed, there are six cubes.

00:39And my aim is not very good here, but yep.

00:47So that's what's going on here with these hyperbolic cubes in space.

00:51And to see, here we have the same kind of thing going on,

00:58except the cubes have the corners cut off.

01:00They are truncated cubes.

01:03So now wherever the cubes meet at the corners, we have these triangles instead.

01:10But otherwise, it's the same.

01:12There are six around each edge.

01:15So here's an interesting question in this particular space,

01:19is what is this shape at the corner?

01:26It might look a little like an icosahedron to those familiar with that shape,

01:30but an icosahedron has five equilateral triangles around each vertex.

01:36If we were in Euclidean space, and we had our normal tiling of cubes,

01:41where we had four cubes around each edge,

01:44then we'd end up with four triangles meeting at each vertex,

01:47and eight cubes around each corner means that if we truncated that,

01:51we would get an octahedron in Euclidean space.

01:53But in this particular hyperbolic tiling,

01:56we have six equilateral triangles around each corner of this shape.

02:02And you can see, we can only see the cells nearest to us,

02:06and they'll kind of render in as we get closer.

02:11So we can't see through to the other side of this shape,

02:15just because it'll only render the closest ones.

02:17So the answer is that, well, we have six equilateral triangles around each vertex,

02:24and that shape is a Euclidean plane.

02:29And well, it's only going to render the closest ones,

02:33so it's going to kind of dance around.

02:35But if we were rendering all the cells,

02:36we would see stretched out around us an entire Euclidean plane of triangles,

02:42equilateral triangles.

02:45And that's just fascinating to think about,

02:49to imagine that each one of these corners here is a Euclidean plane.

02:54And I want to think of it as a shape that I can get into the middle of.

02:59But in fact, the center of this shape is at infinity,

03:04because it's a Euclidean plane.

03:08So here's a view of only those corner bits.

03:13And again, we're only rendering the closest ones.

03:15And as we get closer, we'll...

03:18But each one of these sets of triangles is an entire infinite Euclidean plane

03:25that, of course, don't touch each other.

03:27And when I'm moving through this space,

03:30I always want to get around the plane.

03:34I'm trying to get around this shape.

03:36I want to go all the way around it.

03:37But of course, I can't go around it.

03:39It's infinite.

03:40It's a plane.

03:44And because of hyperbolic space, they don't intersect.

03:48So it's awesome to be able to create these spaces

03:56and explore them, especially in virtual reality.

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