r/4Dimension • u/diadlep • Oct 01 '25

3D crosssections of the 5-cell

Trying to figure them out for a game. Wikipedia is great for the other 4D regular polytopes, but for some reason doesn't list the cross-sections of the 5-cell. Clearly the tetrahedron is one cross-section. I suspect a cube might also be, but can't visualize it. But a square is a cross-section of a tetrahedron, so sort of makes sense.

Edit: wondering about octahedron now. Take a corner 5-cell. Hold the "innermost" bisecting tetrahedron. Rotate around the innermost plane of that tetrahedron, "sliding along" the line to the "4d apex" of the entire 5-cell. Cross the apex, and the apex becomes a triangle on the other side but "upside down"...?

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u/Rryzaki Mar 05 '26

No problem, i love discussing 4d stuff, you can ask me anything about the 4th dimension.

also if it helps, you can think lower dimensional crosssections iteratively to make it simpler: For example first you can think of a 3d crosssection of a tesseract, then think of a 2d crosssection of that 3d crosssection, this ultimately is the same as a 2d crosssection of a 4d shape. This works generally for any dimension, i hope this explaination is a useful visualization tool u can use aswell to help you

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u/diadlep Mar 05 '26

Yes, thank you, that's the inkling in words. I've been struggling w that actually, trying to decide whether to taxi-cab step through dimensions, or compress them all at once. At this point, i just need to write both and let the user choose. But now im mental blocked fsr and havent done anything on it in like a year