Modeling a Hypercube with Tetrahedra

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Modeling a Hypercube with Tetrahedra

Postby PatrickPowers » Sun Sep 01, 2019 6:35 am

I don't much care for the usual models of hypercubes. Can I do better?

Most models use topological cubes. The eight cubes which are the faces each have eight verticies and six sides, but that's it. That's not so bad. What really bothers me is that the topo cubes change shape as the cube rotates. The rotations are elastic. Is it possible to do it with a rigid rotation?

Yes. The trick is to use two solid tetrahedra. There is a rule that they always rotate in exactly the same way. The only difference is their initial position. We'll get to that soon, but for now note that two tetrahedra have a total of eight faces. Each face corresponds to one of the hypercube. We will arrange the two tetrahedra so that each face has six neighbors, just like a hypercube, and rotates in more or less the same way.

Start by holding a tetrahedron in such a way that only one face can be seen. Our modeling rule is that we can "see" a face only if it is perpendicular to us. We can't see it at an angle. So hold it with one face perpendicular to you. You can "see" that face.

Take the next tetra and hold it so that you are looking at the pointy end. One face will be distal. Our second rule is that this distal face on the second terta is considered to be the opposite face from the first. You can't see the opposite face at all.

Hold that second tetra in a way that is as opposite as possible as the first. If the first tetra had a horizontal edge at the bottom, the second tetra should be held with a horizontal edge at the top.

Now let's find the first plane of rotation of 4D. The first, rather humdrum one is to "barrel roll" each tetrahedron so that on the first the same face always stays in view, on the second you are always staring at a pointy corner. Not very exciting.

Next are the other three of the the four basic planes of rotation of a 4D cube. Consider a triangle. Three planes may be found by drawing a line from a vertex to the center of the opposite side. Choose one of those three planes. Rotate the two tetras in that plane. (Well, not really in the same plane, but you get the idea. Just rotate them the same way.) The sides will appear in the correct order: neighboring side, opposite side, anti-neighboring side, original. Two of the sides will appear on one of the tetras, the other two sides on the other.

I made some tetrahedra and found it it was difficult to do this. Hands and arms just aren't built for it, and you can't rest the tetras on a flat surface very well. They are required to have opposite orientation, so at least one will fall over once you let go of it.

Hypercube topology with a rigid 3D rotation! It might be possible to extend to arbitrary planes of rotation via some tinkering with the rules. But can't be bothered to investigate this right now.

As a historical note, the oldest known board game from 3000 BC used four tetrahedral dice. On each die, two of the verticies were colored. Each die counts one if a colored vertex is up, else zero. You may roll a number from zero to four, with two the most likely.
Last edited by PatrickPowers on Wed Sep 04, 2019 1:02 pm, edited 1 time in total.
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Re: Modeling a Hypercube with Tetrahedra

Postby PatrickPowers » Sun Sep 01, 2019 11:50 am

Deleted.
Last edited by PatrickPowers on Tue Sep 03, 2019 8:44 am, edited 1 time in total.
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Re: Modeling a Hypercube with Tetrahedra

Postby PatrickPowers » Sun Sep 01, 2019 11:52 am

Doing the same thing with two cubes instead of two tetrahedra models a 5D cube. The five rotational planes are the "barrel roll" plane, two going from corner to opposite corner on a cube face, and two going from the center of an edge to center of the opposite edge.

Does using two 4D cubes model a 7D cube? Maybe two quintahedra model 6D. And so forth.
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Re: Modeling a Hypercube with Tetrahedra

Postby mr_e_man » Tue Sep 03, 2019 9:19 pm

I don't understand what you're doing. :\

PatrickPowers wrote:Next are the other three of the the four basic planes of rotation of a 4D cube.


PatrickPowers wrote:[...] a 5D cube. The five rotational planes [...]


Shouldn't there be 6 planes and 10 planes, respectively?
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Re: Modeling a Hypercube with Tetrahedra

Postby PatrickPowers » Wed Sep 04, 2019 1:23 pm

mr_e_man wrote:I don't understand what you're doing. :\

PatrickPowers wrote:Next are the other three of the the four basic planes of rotation of a 4D cube.


PatrickPowers wrote:[...] a 5D cube. The five rotational planes [...]


Shouldn't there be 6 planes and 10 planes, respectively?


Oh right you are! I was thinking that all the rotations that don't involve forward-back look the same. All you can see is the same face. But they would look somewhat different. The 3D cube that is the face of the 4D die would rotate differently. So for the 4D cube there are three "barrel roll" rotations.

The model might still work though. The three barrel roll rotations could be

clockwise on both tetra
clockwise on one tetra, anticlockwise on the other
clockwise on one tetra, no rotation on the other.

Looking at it this way, a double rotation involving all four dimensions just might be possible. I'll look at the models when I get home.

It is hard to get the idea across in English. A video would be a lot better. But rotating them by hand is too awkward. I'll try sticking toothpicks into the tetra and twirling them.

Right now I'm getting ready to go to Tirta Ganga, so it might be a week or so to make the vid. Assuming it all works.
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Re: Modeling a Hypercube with Tetrahedra

Postby PatrickPowers » Wed Sep 04, 2019 10:49 pm

The idea is this: relax the requirement that the dimensions be 90 degrees apart. Then it's possible to fit in more. The four vertices of the tetrahedron correspond to four dimensions.

A tetrahedron has six edges. Each may be used as a axis of rotation, analogous to the six planes of 4D.

I looked further at my model and managed to salvage it, but feel it is too idiosyncratic and confusing to explain to be of general interest.
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