Hexadecachoron

Hexadecachoron
Dimensionality	4
Schläfli Symbol	{3,3,4}
Symmetry	Full hexadecachoric symmetry
Coxeter–Dynkin diagram
Bowers acronym	hex
Subfacets
Vertices	8 points
Edges	24 line segments
Faces	32 triangles
Cells	16 tetrahedra
Tera	1 hexadecachoron
Hypervolumes
Vertex Count	'"`UNIQ--postMath-00000001-QINU`"'
Edge Length	'"`UNIQ--postMath-00000002-QINU`"'
Surface Area	'"`UNIQ--postMath-00000003-QINU`"'
Surcell Volume	'"`UNIQ--postMath-00000004-QINU`"'
Surteron Bulk	'"`UNIQ--postMath-00000005-QINU`"'

A hexadecachoron (also called tetracross) is the four-dimensional cross polytope and the dual of a tesseract. It is also called a 16-cell, due to its 16 tetrahedral cells. It has the schläfli symbol $\{3, 3, 4\}$, meaning that 4 tetrahedra meet at each edge. Its Bowers acronym is "hex". It is also called an aerochoron in the elemental naming scheme. It is one of three regular polychora that can tile 4-dimensional space, leading to the hexadecachoric tetracomb.

Properties

The hexadecachoron is the 4-D demihypercube, meaning it can be constructed by taking half the vertices of a tesseract. Because of this it can be called a "demitesseract". It is also a bipyramid of an octahedron, or a square duopyramid. Eight tetrahedra join at each vertex.

Symbols

Dynkin symbols for this shape include:

• o4o3o3x (regular)
• x3o3o *b3o (demitesseract)
• s4o3o3o, s2s4o3o s4o2s4o, s2s2s4o, s2s2s2s (alternated tesseract)
• xo3oo3ox&#xt (tetrahedral antiprism)
• oxo3ooo4ooo&#xt (octahedral bipyramid)
• xox oxo4ooo&#xt (edge first)
• xoo3oox oqo&#xt (triangle first)
• xo4oo ox4oo&#zx (square duopyramid)
• qo ox3oo4oo&#xt (octahedarl bipyramid)
• xoo4ooo oqo ooq&#zx (square bipyramidal bipyramid)
• qooo oqoo ooqo oooq&#zx (rhombic bipyramidal bipyramid)

Structure and Sections

Structure

When seen from one of its cells, the hexadecachoron first has one cell, then four attached to its faces. The next six share an edge with the top cell and are perpendicular to it. The remaining four side cells share only a vertex with the top cell, and are themselves joined to the final cell, in dual orientation to the top cell.

In vertex-first orientation, it is just two octahedral pyramids joined together.

 Hypervolumes

• vertex count =$8$
• edge length =$24 l$
• surface area =$8\sqrt{3}l^2$
• surcell volume =$4\frac{\sqrt{2}}{3}l^3$
• surteron bulk =$\frac{l^4}{6}$

Subfacets

• 8 points (0D)
• 24 line segments (1D)
• 32 triangles (2D)
• 16 tetrahedra (3D)

Radii

• Vertex radius:$\frac{\sqrt{2}}{2}l$
• Edge radius:$\frac{1}{2}l$
• Face radius:$\frac{\sqrt{6}}{6}l$
• Cell radius:$\frac{\sqrt{2}}{4}l$

Angles

• Dichoral angle: 120º

Equations

The surface of a hexadecachoron can be given by

$|x|+|y|+|z|+|w| = 1$

Vertex coordinates

The vertices of a hexadecachoron of side 2 are all permutations of (±√2,0,0,0).

Related shapes

• Dual: Tesseract
• Vertex figure: Octahedron, side length 1
• Diminishings/caps: Octahedral pyramid (single), Square pyramidal pyramid (two adjacent)
• Regiment members: 2 (other uniform member: Tesseractihemioctachoron)

See also

Dimensionality	Negative One	Zero	One	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	Eleven	Twelve	Thirteen	Fourteen	Fifteen	Sixteen	...	Aleph null
Simplex $\{3^{n-1}\}$	Null polytope $)($ $\emptyset$	Point $()$ $\mathbb{B}^0$	Line segment $\{\}$ $\mathbb{B}^1$	Triangle $\{3\}$	Tetrahedron $\{3^2\}$	Pentachoron $\{3^3\}$	Hexateron $\{3^4\}$	Heptapeton $\{3^5\}$	Octaexon $\{3^6\}$	Enneazetton $\{3^7\}$	Decayotton $\{3^8\}$	Hendecaxennon $\{3^9\}$	Dodecadakon $\{3^{10}\}$	Tridecahendon $\{3^{11}\}$	Tetradecadokon $\{3^{12}\}$	Pentadecatradakon $\{3^{13}\}$	Hexadecatedakon $\{3^{14}\}$	Heptadecapedakon $\{3^{15}\}$	...	Omegasimplex
Cross $\{3^{n-2},4\}$				Square $\{4\}$	Octahedron $\{3, 4\}$	Hexadecachoron $\{3^2, 4\}$	Pentacross $\{3^3, 4\}$	Hexacross $\{3^4, 4\}$	Heptacross $\{3^5, 4\}$	Octacross $\{3^6, 4\}$	Enneacross $\{3^7, 4\}$	Dekacross $\{3^8, 4\}$	Hendekacross $\{3^9, 4\}$	Dodekacross $\{3^{10}, 4\}$	Tridekacross $\{3^{11}, 4\}$	Tetradekacross $\{3^{12}, 4\}$	Pentadekacross $\{3^{13}, 4\}$	Hexadekacross $\{3^{14}, 4\}$	...	Omegacross
Hydrotopes $\{3^{n-2}, 5\}$				Pentagon $\{5\}$	Icosahedron $\{3, 5\}$	Hexacosichoron $\{3^2, 5\}$	Order-5 pentachoric tetracomb $\{3^3, 5\}$	Order-5 hexateric pentacomb $\{3^4, 5\}$	...
Hypercube $\{4, 3^{n-2}\}$				Square $\{4\}$	Cube $\{4,3\}$	Tesseract $\{4, 3^2\}$	Penteract $\{4, 3^3\}$	Hexeract $\{4, 3^4\}$	Hepteract $\{4, 3^5\}$	Octeract $\{4, 3^6\}$	Enneract $\{4, 3^7\}$	Dekeract $\{4, 3^8\}$	Hendekeract $\{4, 3^9\}$	Dodekeract $\{4, 3^{10}\}$	Tridekeract $\{4, 3^{11}\}$	Tetradekeract $\{4, 3^{12}\}$	Pentadekeract $\{4, 3^{13}\}$	Hexadekeract $\{4, 3^{14}\}$	...	Omegeract
Cosmotopes $\{5, 3^{n-2}\}$				Pentagon $\{5\}$	Dodecahedron $\{5, 3\}$	Hecatonicosachoron $\{5, 3^2\}$	Order-3 hecatonicosachoric tetracomb $\{5, 3^3\}$	Order-3-3 hecatonicosachoric pentacomb $\{5, 3^4\}$	...
Hyperball $\mathbb B^n$				Disk $\mathbb B^2$	Ball $\mathbb B^3$	Gongol $\mathbb B^4$	Pentorb $\mathbb B^5$	Hexorb $\mathbb B^6$	Heptorb $\mathbb B^7$	Octorb $\mathbb B^8$	Enneorb $\mathbb B^9$	Dekorb $\mathbb B^{10}$	Hendekorb $\mathbb B^{11}$	Dodekorb $\mathbb B^{12}$	Tridekorb $\mathbb B^{13}$	Tetradekorb $\mathbb B^{14}$	Pentadekorb $\mathbb B^{15}$	Hexadekorb $\mathbb B^{16}$	...	Omegaball $\mathbb B^{\aleph_0}$

$\{2,3,4\}$	$\{3, 3, 4\}$	$\{4,3,4\}$	$\{5,3,4\}$	$\{6,3,4\}$
Octahedral hosochoron	Hexadecachoron	Cubic honeycomb	Order-4 dodecahedral honeycomb	Order-4 Hexagonal tiling honeycomb

$\{3,3,2\}$	$\{3,3,3\}$	$\{3, 3, 4\}$	$\{3,3,5\}$	$\{3,3,6\}$
Tetrahedral dichoron	Pentachoron	Hexadecachoron	Hexacosichoron	Order-6 tetrahedral honeycomb

	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