{"id":424,"date":"2013-12-15T01:00:24","date_gmt":"2013-12-15T01:00:24","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=424"},"modified":"2020-01-06T01:55:30","modified_gmt":"2020-01-06T01:55:30","slug":"truncated-hypercube","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2013\/12\/15\/truncated-hypercube\/","title":{"rendered":"Truncated Hypercube"},"content":{"rendered":"
\"Truncated<\/a>

Truncated Hypercube – Jos Leys, www.josleys.com<\/p><\/div>\n

\n

This is a truncated 4-dimensional cube. You can take an ordinary 3-dimensional cube, cut off its corners, and get a uniform polyhedron<\/a> with $2 \\times 3 = 6$ octagonal faces and $2^3 = 8$ triangular faces. It’s called the truncated cube<\/a>. Similarly, you can take a 4-dimensional cube, cut off its corners, and get a 4d uniform polytope<\/a> with $2 \\times 4 = 8$ truncated cubes as facets and $2^4 = 16$ tetrahedral facets! It’s called the truncated 4-cube<\/b>.<\/p>\n

This particular truncated 4-cube was drawn in a curved style by Jos Leys<\/a>. You can see more of his 4d polytopes here:<\/p>\n

• Jos Leys, 4d Polychora<\/a>. <\/p>\n

A polychoron<\/a><\/b> is just another name for a 4-dimensional polytope. The truncated 4-cube is also called the truncated tesseract<\/b>, and you can learn more about it here:<\/p>\n

• Truncated tesseract<\/a>, Wikipedia. <\/p>\n

The Coxeter diagram<\/a> of the truncated 4d cube is<\/p>\n

\n●—4—●—3—o—3—o<\/b>\n<\/div>\n

 <\/p>\n

where the black dots are often drawn as dots with rings around them, and the white ones are often drawn as dots without rings. The unmarked diagram<\/p>\n

\no—4—o—3—o—3—o<\/b>\n<\/div>\n

 <\/p>\n

describes the symmetry group of the 4-cube, including both rotations and reflections. This group, called a Coxeter group<\/a>, has four generators $s_1, \\dots, s_4$ obeying relations that are encoded in the diagram:<\/p>\n

$$ (s_1 s_2)^4 = 1 $$
\n$$ (s_2 s_3)^3 = 1 $$
\n$$ (s_3 s_4)^3 = 1 $$<\/p>\n

together with relations <\/p>\n

$$s_i^2 = 1$$<\/p>\n

and <\/p>\n

$$ s_i s_j = s_j s_i \\; \\textrm{ if } \\; |i – j| > 1 $$<\/p>\n

Marking the Coxeter diagram lets us describe many uniform polytopes with the same symmetry group as the 4-cube. You can think of the 4 dots as corresponding to the vertices, edges, 2d faces and 3d facets of the cube. Blackening the vertex and edge dots:<\/p>\n

\n●—4—●—3—o—3—o<\/b>\n<\/div>\n

 <\/p>\n

is a way to indicate that the truncated 4-cube has a vertex for each vertex-edge flag<\/b>: that is, each pair consisting of a vertex and an edge of the 4-cube, where the vertex lies on the edge.<\/p>\n

All this generalizes from 4 dimensions to higher (or lower) dimensions. The truncated $n$-cube<\/b> has $2n$ truncated $(n-1)$-cubes and $2^n$ $(n-1)$-simplices as faces, and it is described by a Coxeter diagram just like the one above, but with $n$ dots. For example, the truncated 5-cube has this diagram:<\/p>\n

\n●—4—●—3—o—3—o—3—o<\/b>\n<\/div>\n

 <\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This is a truncated 4-dimensional cube, drawn in a curved style by Jos Leys<\/a>. You can take an ordinary 3-dimensional cube, cut off its corners and get a truncated cube. Similarly, you can take a 4-dimensional cube, cut off its corners, and get a 4-dimensional uniform polytope<\/a> with $2 \\times 4 = 8$ truncated cubes as faces and $2^4 = 16$ tetrahedral faces! It’s called the truncated 4-cube<\/b>.<\/p>\n

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