Category
📚
LearningTranscript
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.