🕸Fergus Duniho wrote on Wed, Dec 6, 2023 11:02 PM UTC:
I think that just numbering the faces is not good enough. The diagrams at the top are woefully inadequate, because they don't even make it clear what number each face has. So, I propose something more systematic. The tesseract image shows a smaller cube inside of a larger cube with lines between their corresponding corners. The faces of these two cubes do not count as faces of the tesseract, but the quadrilateral shapes that extend between the larger and smaller cubes are the faces of the tesseract. For each pair of parallel cube faces, the tesseract has four faces, and since each cube has six faces, we get six sets of four faces, which is a total of 24 faces. So, I propose naming each face with a pair of numbers. Use 1-6 to designate which cube faces the tesseract face is between. Let's assume it's like a die with opposite numbers adding up to 7.
Then use 1-4 to distinguish the faces on one side of the tesseract. Ideally, we could use the same number for faces that share the same cube corners, though I haven't worked it out to check whether this is possible. I think it is these two that you are saying a piece has a choice between moving to one or the other.
Let me try to work this out. Suppose 1-1 pairs with 5-1, 1-2 with 2-2, 1-3 with 3-3, and 1-4 with 4-4. With 2-2 paired with 1-2,
As I was trying to work this out, I got an idea that should prove easier. Each face of the tesseract abuts two different sides of each cube. One of these is primary, and the other one just indicates which of the four faces extending from the primary cube face is indicated. Numbering the cube like a die, the faces of the tesseract are 1-2, 1-3, 1-4, 1-5, 2-1, 2-3, 2-4, 2-6, 3-1, 3-2, 3-5, 3-6, 4-1, 4-2, 4-5, 4-6, 5-1, 5-3, 5-4, 5-6, 6-2, 6-3, 6-4, and 6-5. Pairs with the same number (1-1, 2-2, etc.) and pairs with numbers adding up to 7 (1-6, 2-5, etc.) are excluded. You could remove the hyphens and just use them as coded number pairs rather than true numerals. Using these makes visualization easier, because faces with the same numbers in the opposite order pair up, and neighboring faces will always share a number.
I think that just numbering the faces is not good enough. The diagrams at the top are woefully inadequate, because they don't even make it clear what number each face has. So, I propose something more systematic. The tesseract image shows a smaller cube inside of a larger cube with lines between their corresponding corners. The faces of these two cubes do not count as faces of the tesseract, but the quadrilateral shapes that extend between the larger and smaller cubes are the faces of the tesseract. For each pair of parallel cube faces, the tesseract has four faces, and since each cube has six faces, we get six sets of four faces, which is a total of 24 faces. So, I propose naming each face with a pair of numbers. Use 1-6 to designate which cube faces the tesseract face is between. Let's assume it's like a die with opposite numbers adding up to 7.
Then use 1-4 to distinguish the faces on one side of the tesseract. Ideally, we could use the same number for faces that share the same cube corners, though I haven't worked it out to check whether this is possible. I think it is these two that you are saying a piece has a choice between moving to one or the other.Let me try to work this out. Suppose 1-1 pairs with 5-1, 1-2 with 2-2, 1-3 with 3-3, and 1-4 with 4-4. With 2-2 paired with 1-2,As I was trying to work this out, I got an idea that should prove easier. Each face of the tesseract abuts two different sides of each cube. One of these is primary, and the other one just indicates which of the four faces extending from the primary cube face is indicated. Numbering the cube like a die, the faces of the tesseract are 1-2, 1-3, 1-4, 1-5, 2-1, 2-3, 2-4, 2-6, 3-1, 3-2, 3-5, 3-6, 4-1, 4-2, 4-5, 4-6, 5-1, 5-3, 5-4, 5-6, 6-2, 6-3, 6-4, and 6-5. Pairs with the same number (1-1, 2-2, etc.) and pairs with numbers adding up to 7 (1-6, 2-5, etc.) are excluded. You could remove the hyphens and just use them as coded number pairs rather than true numerals. Using these makes visualization easier, because faces with the same numbers in the opposite order pair up, and neighboring faces will always share a number.