The Icositetrachoron and the Truncated Icositetrachoron, Rotating in Hyperspace

There are six regular, convex four-dimensional polytopes. Five of them correspond on a 1:1 basis with the Platonic solids (as the tesseract corresponds to the cube), leaving one four-dimensional polytope without a three-dimensional analogue among the Platonics. That polychoron is the icositetrachoron, also named the 24-cell and made of 24 octahedral cells. It also happens to be self-dual.

If, in hyperspace, the corners are cut off just right, new cells are created with 24 cubic cells created at the corners, with 24 truncated octahedral cells remaining from the original polychoron. This is the truncated icositetrachoron:

4-dimensional polytopes have 3-dimensional nets. These nets are shown below — first for the icositetrachoron, and then for the truncated icositetrachoron.

I used Stella 4d to create these images. You can try Stella for free at http://www.software3d.com/Stella.php.

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