Tesseract Face Connectivity

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Tesseract Face Connectivity

Postby g3taso » Thu Jul 31, 2008 7:46 pm

Ladies & Gentlemen;

I was wondering if you could point me in the right direction. I have recently developed a keen interest in 4-cube (tesseract) research on a purely amateur level. Using sets of [8] dice I have experimented with “unrolling” the eight cubes and have a pretty solid handle on the concept. However, I am interested in knowing if it is possible to construct a diagram showing how each of the 24 faces of the cubes connect to the cubes on other faces. I have looked over what mathematical resources are available to us “civilians” out here and I have not been able to find such a chart or even guidance on how such a chart might be constructed. Ideally, I would like to be able to place this on a table with the various faces represented by small tiles connected by lines.

I was considering assigning a color to each face on each “die” and coding the connections by the protocol of “blue face, die 1 to blue face, die 4” and so forth and attempting to construct a small program to run it, but I am thoroughly stumped on how to create such an algorithm.

Can you provide any guidance on where I might be able to acquire the answer to my inquiry?
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Re: Tesseract Face Connectivity

Postby quickfur » Fri Aug 15, 2008 9:25 pm

It's very simple, there are 8 cubes in the tesseract, in 4 opposing pairs. Each cube is directly connected via its 6 square faces to the other 6 cubes that aren't its opposite. To draw a diagram of this, just start with 8 points representing 8 cubes, preferably place them in a circle, then draw a line between every pair of points except the one directly opposite it. This is the graph of the tesseract's faces.

Another way to see the connectivity of the cells in the tesseract is to imagine a cube within a cube. (This is a common depiction of the tesseract, and is the image of its cell-first perspective projection into 3-space.) The outer cube and the inner cube are on opposite sides of the tesseract (in 4-space, they are not touching each other). In between them are 6 frustum-shaped volumes, each corresponding to one of the remaining 6 cells of the tesseract. The outer cube is connected to each frustum, of which there are 6, and each frustum is connected to the outer cube and the inner cube plus the 4 other frustums surrounding it (6 in total). The inner cube, of course, is connected to the 6 frustums surrounding it.

So you see, the cells are all equivalent, they are all connected to 6 other cells. In the 3-space projection, of course, this connected appears to be different for the inner/outer cubes and the frustums, but this is just an artifact of projection. In 4-space, they are completely identical.
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Re: Tesseract Face Connectivity

Postby quickfur » Fri Aug 15, 2008 9:42 pm

Oh, nevermind, I misread your question. You want the connectivity between the 2D faces on the tetracube? It's not that hard... there is precisely 1 face between every two connected cubes. The best approach, I think, is to use the cube-within-a-cube projection:

There are 6 outer faces and 6 inner faces (for the outer/inner cubes, respectively), and 12 faces that each connect an outer edge to an inner edge.

Take the top outer face, for example. It is connected to 8 other faces: the 4 outer faces on the side (but not the bottom face), and the trapezoidal faces that connect from the top edges of the outer cube to the top edges of the inner cube. (Note, of course, that the trapezoidal faces are really squares in 4-space.)

As for drawing a diagram of this... have you ever taken a look at GraphViz? It's a very handy tool for doing graph layouts. Hmm, maybe I'll do a quick run of it and post the result...
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Re: Tesseract Face Connectivity

Postby g3taso » Tue Aug 26, 2008 1:23 am

Could you? That would be awesome!
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Re: Tesseract Face Connectivity

Postby g3taso » Tue Aug 26, 2008 1:28 am

I just checked out GraphViz. It requires knowledge of the DOT language. I don't know this language! Aaarrrrrgggggggghhhhhhhhhhhh!
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Re: Tesseract Face Connectivity

Postby quickfur » Tue Aug 26, 2008 6:02 pm

You could just google for graphviz documentation, the DOT language is fully explained there. It's really very simple, the most basic syntax looks like this:

Code: Select all
graph {
  face1 -- face2
  face1 -- face3
  face3 -- face4
  ...
}


The "face.." identifiers are just names for your nodes, and the "--" indicate edges between them. You can use any identifier for your nodes, call them "node1", "node2", "node3", or call them "a", "b", "c", whatever you like.

I started making the graph for all tetracube faces, but it became too tangly and not really that interesting. What exactly do you want the graph for? You probably want a representative subgraph of some sort rather than the full graph. Full graphs for these kinds of combinatorial objects tend to be very dense and hard to make much sense of.
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