Cool fact about 4D

Higher-dimensional geometry (previously "Polyshapes").

Cool fact about 4D

Postby quickfur » Mon Jun 06, 2005 6:23 am

I just discovered an awesomely cool fact about 4D...

The cell-first projection of the tetracube has the same envelope as the cell-first projection of the 16-cell: a 3D cube!

I stumbled upon this curious fact when doodling around with various projections. I observed that the cell-first projection of the octahedron consists of two interpenetrating triangles inside a hexagonal envelope. So I wondered what 4D object would project to two interpenetrating tetrahedra.

To make a long story short, the answer is: the 16-cell, projected cell-first.

This particular projection is very fascinating, since it is closely related to the geometry of the 3D cube. There are precisely two ways to inscribe a tetrahedron inside a cube. The edges of the tetrahedron would lie on the diagonals of the cube's faces: each face would have one diagonal that coincides with one of the tetrahedron's edges. If you take the set of diagonals not coincident with the tetrahedron's edges, you get a dual tetrahedron: precisely the other possible way to inscribe the tetrahedron.

These two possible ways to inscribe a tetrahedron in a cube corresponds with the closest and farthest cell of the 16-cell, in its cell-first projection to 3D. Let's call them F and B (for "front" and "back"). Now here's the interesting bit: there are 4 cells that share a face with F, and these cells project precisely into the volume between the inscribed tetrahedron and the cubical envelope in the projection. This is also true with cell B. This gives us 10 of the 16-cell's cells.

Now notice that the tetrahedra around F don't share a face with the tetrahedra around B. They correspond with two different diagonals on each face of the cubical envelope. But notice that a square with two diagonals is actually a projection of a tetrahedron! This means that the 6 faces of the cubical envelope correspond with 6 tetrahedral cells. So we have precisely 16 cells!

Note furthermore an unusual property of this projection: all of the edges of the 16-cell project onto the surface of the cubical envelope! There are no edges that lie inside the cubical envelope. The edges are projected precisely to the 12 edges of the cubical envelope plus the diagonals on each face.

So, the cell-first projections of the 4D cross polytope and the 4D measure polytope have identical envelopes. This is also true of the 2D cross and measure polytopes (although they happen to be the same thing in 2D), but it's not true in 3D: the cell-first projection of an octahedron is a hexagon, but the cell-first projection of a cube is a square. It seems that there is something special about 4D that makes the cross and the measure project cell-first to the same envelope. Does anyone know if this is true in all even dimensions, or is it really something unique to 4D?
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Postby wendy » Mon Jun 06, 2005 6:33 am

The vertex first projection of the cube and 24-cell are also identical: a rhombic dodecahedron!

It's probably true of only 4 dimensions, unless you loose a lot of dimensions in the process: consider the number of vertices.

cross-polytope has 2n vertices

cube has 2^n/2 vertices.

In 4d, both of these are 8. In five dimensions, the cross polytope has 10, while the 4d cube has 16. You don't get 16 vertices on a cross until 8 dimensions.
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Postby quickfur » Mon Jun 06, 2005 4:56 pm

wendy wrote:The vertex first projection of the cube and 24-cell are also identical: a rhombic dodecahedron!

Yep, that's another cool thing about 4D.

Another cool thing about this is that the 24-cell itself is actually a 4D rhombic dodecahedron, in the sense that just as the rhombdode is formed by joining the 6 vertices lying on the 3D coordinate axes to the corners of a cube, the 24-cell is formed by joining the 8 vertices lying on the 4D coordinate axes to the corners of the tetracube.

It's probably true of only 4 dimensions, unless you loose a lot of dimensions in the process: consider the number of vertices.

cross-polytope has 2n vertices

cube has 2^n/2 vertices.

In 4d, both of these are 8.

Oh? I thought the tetracube has 16 vertices... Just that 8 of them happen to project onto the other 8 in the cell-first projection.

In five dimensions, the cross polytope has 10, while the 4d cube has 16. You don't get 16 vertices on a cross until 8 dimensions.

Well, something's definitely special about 4D, in that the number of vertices of the cross is exactly half the number of vertices of the measure. The cell-first projection of a hypercube always maps to an n-1 hypercube where half the vertices are coincident with the other half, so you're right, having both the cross and the hypercube project into the same (n-1) envelope is unique to 4D.

I've also been noticing that one of the factors that makes the 24-cell regular is the fact that the cubical pyramid of height 1 has edges of precisely length 2 (both the apex-to-base edges and the intra-base edges). This curious coincidence---that the height is precisely half of the length of the edges---causes regular octahedra with edge length of 2 to form when you attach cubical pyramids to the facets of a tetracube. And the reason this comes about is because the left-hand side of the 4D Pythagorean equation x<sup>2</sup>+y<sup>2</sup>+z<sup>2</sup>+w<sup>2</sup>=h<sup>2</sup> happens to collapse nicely when edges are unit length into 1<sup>2</sup>+1<sup>2</sup>+1<sup>2</sup>+1<sup>2</sup>=4, which is a perfect square that happens to be equal to 2<sup>2</sup>. So one of the reasons the 24-cell exists is because 4*(1<sup>2</sup>)=2<sup>2</sup>, which is a curious identity that only happens with the number 2. And 4 happens to be 2<sup>2</sup>.
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Postby wendy » Mon Jun 06, 2005 11:41 pm

Actually, the rhombic dodecahedron has two different forms in 4d.

One of these is the "double-cube".

The other is the 20-sided figure m3o3o3m. This is the section of the 5d cube, projected vertex-first onto four dimensions.
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Postby quickfur » Tue Jun 07, 2005 12:32 am

wendy wrote:Actually, the rhombic dodecahedron has two different forms in 4d.

One of these is the "double-cube".

Never heard of it. Care to elaborate?

The other is the 20-sided figure m3o3o3m. This is the section of the 5d cube, projected vertex-first onto four dimensions.

Cool. I guess it's something common to all dimensions? -- that the n-hypercube projected vertex first into (n-1) dimensions will yield an (n-1)-space-tiling polytope analogous to the hexagon/rhombic dodecahedron.
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Postby brasileiro » Tue Jun 07, 2005 6:27 pm

Well, I suppose that this is something I have soooo much to learn from. I'll have to keep comin back to study you guys lol.
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Postby wendy » Wed Jun 08, 2005 4:45 am

A double-cube is formed by adding pyramids to a cube. If the height of the pyramid is set to half the edge length, the faces fall together by pairs, and you have one face per cube-margin.

The double-cube tiles in every dimension. It is also called an edge-tegmated cross-polytope, or o3m3o..4o.

The simplex-antitegmate cluster is formed in a different way, and it has 2^(n+1)-2 vertices. One can easily construct it by projecting the measure polytope down one of its long axies.

If one shrinks a measure polytope (eg a cube) along one of its axies, one can just put 4 around the obtuse corner. The outline of the four exposed faces forms an antitegmate cluster (since a cube is an antitegum), and the remaining faces form a tiling figure.

One can make this tiling from a higher dimension, by considering the points in the plane x+y+z+... = 0. These form a tiling with simplex symmetry. The Voronii cells (which is what you get when you expand the spheres to fill all space without intersection), is one of these simplex antitegum-cluster things.

In four and higher dimensions, the nature of these are different.

A sphere of sqrt(2) diameter, occupies a volume of 4 in the cubic tiling, of 2 in the tiling of double-cubes, and of sqrt(5) in the tegmate cluster thing. The last two are the most efficient tilings known in 4d.

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Postby quickfur » Wed Jun 08, 2005 4:14 pm

wendy wrote:A double-cube is formed by adding pyramids to a cube. If the height of the pyramid is set to half the edge length, the faces fall together by pairs, and you have one face per cube-margin.

The double-cube tiles in every dimension. It is also called an edge-tegmated cross-polytope, or o3m3o..4o.

Ah, I see. So a rhombic dodecahedron is a 3D double-cube? It's not surprising that it tiles n-space, since the half-height pyramids can be thought of as cutting an adjacent n-cube into pyramids, so the tiling is really just a modified tiling of the measure polytope.

The simplex-antitegmate cluster is formed in a different way, and it has 2^(n+1)-2 vertices. One can easily construct it by projecting the measure polytope down one of its long axies.

Vertex-first projection?

If one shrinks a measure polytope (eg a cube) along one of its axies, one can just put 4 around the obtuse corner. The outline of the four exposed faces forms an antitegmate cluster (since a cube is an antitegum), and the remaining faces form a tiling figure.

One can make this tiling from a higher dimension, by considering the points in the plane x+y+z+... = 0. These form a tiling with simplex symmetry. The Voronii cells (which is what you get when you expand the spheres to fill all space without intersection), is one of these simplex antitegum-cluster things.

In four and higher dimensions, the nature of these are different.
[...]

Fascinating. It seems again that 4D is special: the projection of the tetracube into 3D produces a rhombic dodecahedron, which is identical to the double-cube construction in 3D. But the 3D cube projects to a hexagon, which is different from the analogous 2D double-square construction (which is just another square). I would hazard to guess the vertex-first projection of the 5D cube into 4D would be different from the 24-cell, right? If so, 4D is truly unique in this respect (being the only dimension whose measure polytope projects into a double-cube).
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Postby wendy » Thu Jun 09, 2005 12:19 am

Four dimensions is not all that special. There are a lot of crossing threads in the low dimensions, because different dimensions preform different roles. Some of these crossing threads do not become distinct until you hit six dimensions.

The Gosset polytope, passes through 3d as a triangle-prism, and 4d as a rectified pentachoron. In 5d, it's a half-cube, and there-after it is distinct, rising to the spectular tiling in 8 dimensions.

In five dimensions, we see distinct, two new products.

The projection down one of the diagonals of the cube is a vertex-first projection.

I normally work with six dimensions, although i am still trying to build the space of great arrows and the space of rotations for five dimensions.


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