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The icositetrachoron is a four-dimensional regular polytope composed of 24 octahedral cells, three of which join at each edge. It is also called a 24-cell because of this. Its Bowers acronym is "ico", and it can also be called a xylochoron. It is one of the three regular polychora that tile 4-dimensional space, leading to the icositetrachoric tetracomb.

Construction

One way to construct the icositetrachoron is to augment cubic pyramids onto the cells of the tesseract. If the sides of the pyramids are exactly the same edge length as the tesseract, the pyramidal cells merge into octahedra.

Another way is via the rectification of the hexadecachoron. The tetrahedral cells will rectify into octahedra, while the vertices become further octahedral cells. This can be represented as $ r\{3,3,4\} $.

Structure and Sections

Structure

The icositetrachoron is composed of octahedra joining six to a vertex. Seen from a cell, we first have one, then the eight joining to it. The next layer is six octahedra sharing only a vertex with the top, then another eight, and finally the opposite cell.

Subfacets

Hypervolumes

See also

Regular polychora (+ tho)
Convex regular polychora: pen · tes · hex · ico · hi · ex

Self-intersecting regular polychora: fix · gohi · gahi · sishi · gaghi · gishi · gashi · gofix · gax · gogishi

Tesseractihemioctachoron: tho

$ \{3,4,2\} $ $ \{3,4,3\} $ $ \{3,4,4\} $
Octahedral dichoron Icositetrachoron Order-4 octahedral honeycomb
$ \{2,4,3\} $ $ \{3,4,3\} $ $ \{4,4,3\} $
Cubic hosochoron Icositetrachoron Order-3 square tiling honeycomb
Regular
$ t_0 \{3,4,3\} $
Rectified
$ t_1 \{3,4,3\} $
Birectified
$ t_2 \{3,4,3\} $
Trirectified
$ t_3 \{3,4,3\} $
Truncated
$ t_{0,1} \{3,4,3\} $
Bitruncated
$ t_{1,2} \{3,4,3\} $
Tritruncated
$ t_{2,3} \{3,4,3\} $
Icositetrachoron Rectified icositetrachoron Rectified icositetrachoron Icositetrachoron Truncated icositetrachoron Bitruncated icositetrachoron Truncated icositetrachoron
Cantellated
$ t_{0,2} \{3,4,3\} $
Bicantellated
$ t_{1,3} \{3,4,3\} $
Cantitruncated
$ t_{0,1,2} \{3,4,3\} $
Bicantitruncated
$ t_{1,2,3} \{3,4,3\} $
Runcinated
$ t_{0,3} \{3,4,3\} $
Runcicantellated
$ t_{0,2,3} \{3,4,3\} $
Runcitruncated
$ t_{0,1,3} \{3,4,3\} $
Runcicantitruncated
$ t_{0,1,2,3} \{3,4,3\} $
Cantellated pentachoron Cantellated icositetrachoron Cantitruncated icositetrachoron Cantitruncated icositetrachoron Runcinated icositetrachoron Runcitruncated icositetrachoron Runcitruncated icositetrachoron Omnitruncated icositetrachoron
Regular
$ t_0 \{4,3,3\} $
Rectified
$ t_1 \{4,3,3\} $
Birectified
$ t_2 \{4,3,3\} $
Trirectified
$ t_3 \{4,3,3\} $
Truncated
$ t_{0,1} \{4,3,3\} $
Bitruncated
$ t_{1,2} \{4,3,3\} $
Tritruncated
$ t_{2,3} \{4,3,3\} $
Tesseract Rectified tesseract Icositetrachoron Hexadecachoron Truncated tesseract Bitruncated tesseract Truncated hexadecachoron
Cantellated
$ t_{0,2} \{4,3,3\} $
Bicantellated
$ t_{1,3} \{4,3,3\} $
Cantitruncated
$ t_{0,1,2} \{4,3,3\} $
Bicantitruncated
$ t_{1,2,3} \{4,3,3\} $
Runcinated
$ t_{0,3} \{4,3,3\} $
Runcicantellated
$ t_{0,2,3} \{4,3,3\} $
Runcitruncated
$ t_{0,1,3} \{4,3,3\} $
Runcicantitruncated
$ t_{0,1,2,3} \{4,3,3\} $
Cantellated tesseract Rectified icositetrachoron Cantitruncated tesseract Truncated icositetrachoron Runcinated tesseract Runcitruncated hexadecachoron Runcitruncated tesseract Omnitruncated tesseract
Regular
$ t_0 \{3,3,b3\} $
Rectified
$ t_1 \{3,3,b3\} $
Birectified
$ t_2 \{3,3,b3\} $
Trirectified
$ t_3 \{3,3,b3\} $
Truncated
$ t_{0,1} \{3,3,b3\} $
Bitruncated
$ t_{1,2} \{3,3,b3\} $
Tritruncated
$ t_{2,3} \{3,3,b3\} $
Hexadecachoron Icositetrachoron Hexadecachoron Hexadecachoron Truncated Hexadecachoron Truncated Hexadecachoron Rectified tesseract
Cantellated
$ t_{0,2} \{3,3,b3\} $
Bicantellated
$ t_{1,3} \{3,3,b3\} $
Cantitruncated
$ t_{0,1,2} \{3,3,b3\} $
Bicantitruncated
$ t_{1,2,3} \{3,3,b3\} $
Runcinated
$ t_{0,3} \{3,3,b3\} $
Runcicantellated
$ t_{0,2,3} \{3,3,b3\} $
Runcitruncated
$ t_{0,1,3} \{3,3,b3\} $
Runcicantitruncated
$ t_{0,1,2,3} \{3,3,b3\} $
Rectified tesseract Truncated hexadecachoron Bitruncated tesseract Bitruncated tesseract Rectified tesseract Rectified icositetrachoron Bitruncated tesseract Truncated icositetrachoron
$ t_1 \{2,3,4\} $ $ t_1 \{3,3,4\} $ $ t_1 \{4,3,4\} $ $ t_1 \{5,3,4\} $ $ t_1 \{6,3,4\} $
Rectified Octahedral hosochoron Icositetrachoron Rectified cubic honeycomb Rectified order-4 dodecahedral honeycomb Rectified order-4 Hexagonal tiling honeycomb
$ t_1 \{3,3,2\} $ $ t_1 \{3,3,3\} $ $ t_1 \{3,3,4\} $ $ t_1 \{3,3,5\} $ $ t_1 \{3,3,6\} $
Rectified tetrahedral dichoron Rectified pentachoron Icositetrachoron Rectified hexacosichoron Rectified order-6 tetrahedral honeycomb