5-dimensional hypercube

This picture illustrates the vertices and edges of a 5-dimensional hypercube, also known as a penteract. It has 32 vertices, 80 edges, 80 two-dimensional faces, 40 three-dimensional faces and 10 four-dimensional faces.

Hypercubes are examples of regular polytopes. These are multidimensional analogues of the familiar Platonic solids in three dimensions, namely the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. In dimensions five and higher, there exist analogues of the tetrahedron, of the cube (pictured), and of the octahedron, but there are no other regular polytopes. In dimension four, things are more interesting: there exist analogues of the dodecahedron and icosahedron, as well as the 24-cell, which has no analogue in any other dimension. I wrote a detailed post about this in March 2013, which you can find here: https://plus.google.com/101584889282878921052/posts/HJxUtVJ16zn

The illustration is an excerpt from a very nice video by Oliver Knill (polytopes.mov). The video illustrates all of the regular polytopes in four and five dimensions by using successive stereographic projections. If you like polytopes, this video is well worth seven minutes of your time.

#mathematics #scienceeveryday
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