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🕸Fergus Duniho wrote on Thu, Dec 7, 2023 01:58 AM UTC in reply to Ben Reiniger from Wed Dec 6 11:26 PM:

I think this is incorrect. The faces of the two cubes are 2d faces of the tesseract.

Okay, I was trying to make sense of this in a way that added up to 24 faces. Assuming that what I called 1-2 and 2-1 are the same face and so on, the faces I counted as 24 reduce to 12, there being one for each edge of a cube. Then the remaining faces would be the six faces of each cube. That makes more sense.

However, simply numbering them has not been a helpful way of keeping track of them, and the diagrams have not made it clear which numbers match which faces. So, I recommend going with something more systematic that can better convey relations between faces.

Ideally, the method should be neither redundant nor arbitrary. But I'll start by describing a redundant method. This would start by numbering the inner cube like a die. Using one side of the inner cube as a reference point, we could identify up to six different faces. For example, starting with 1, 1-0 would be the inner face, 1-2, 1-3, 1-4, and 1-5 would be neighboring faces, and 1-1 would be the outer face parallel with 1-0. This would be redundant, because 1-2 would be the same as 2-1, etc.

To make this non-redundant, I propose writing each designation with the lower digit first. So, then we would have, dispensing with the hyphen, 01, 02, 03, 04, 05, and 06 for the inner cube faces, 11, 22, 33, 44, 55, and 66 for the outer cube faces, and 12, 13, 14, 15, 23, 24, 26, 35, 36, 45, 46 and 56 for the other faces.


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