Icositetrachoron

Icositetrachoron
Dimensionality	4
Schläfli Symbol	'"`UNIQ--postMath-00000001-QINU`"'
Symmetry	Full icositetrachoric symmetry
Coxeter–Dynkin diagram
Bowers acronym	ico
Subfacets
Vertices	24 points
Edges	96 line segments
Faces	96 triangles
Cells	24 octahedra
Tera	1 icositetrachoron
Hypervolumes
Vertex Count	'"`UNIQ--postMath-00000002-QINU`"'
Edge Length	'"`UNIQ--postMath-00000003-QINU`"'
Surface Area	'"`UNIQ--postMath-00000004-QINU`"'
Surcell Volume	'"`UNIQ--postMath-00000005-QINU`"'
Surteron Bulk	'"`UNIQ--postMath-00000006-QINU`"'

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

• 24 points (0D)
• 96 line segments (1D)
• 96 triangles (2D)
• 24 octahedra (3D)
• 1 icositetrachoron

Hypervolumes

• vertex count = $24$
• edge length = $96l$
• surface area = $24\sqrt{3}l^2$
• surcell volume = $8\sqrt{2}l^3$
• surteron bulk = $2l^4$

Vertex Coordinates

The vertices of an icositetrachoron, of edge length $\sqrt{2}$, can be denoted on the 4D Euclidean flune by the permutations of (±1, ±1, 0, 0). It can also be seen as the superposition of a tesseract and a hexadecachoron, giving the vertices (±1, ±1, ±1, ±1), along with the rotations of (±2, 0, 0, 0).

See also

Icositetrachoron regiment
Have Full Icositetrachoric symmetry ico · ihi Have Full Hexadecachoric symmetry: oh · shahoh · huhoh · hohoh · ohuhoh · ghahoh Have Full Demitesseractic symmetry: ratho · odho · hodho · hoh honho · dod honho · doh honho

$\{3,4,2\}$	$\{3,4,3\}$	$\{3,4,4\}$
Octahedral dichoron	Icositetrachoron	Order-4 octahedral honeycomb

Template:Ariant Nav X 4 3

	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