Hexadecachoron

Hexadecachoron
(OFF file)
Rank	4
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Hex
Coxeter diagram	o4o3o3x ()
Schläfli symbol	{3,3,4}
Bracket notation	<IIII>
Elements
Cells	16 tetrahedra
Faces	32 triangles
Edges	24
Vertices	8
Vertex figure	Octahedron, edge length 1
Edge figure	tet 3 tet 3 tet 3 tet 3
Petrie polygons	Octagonal-octagrammic coil: $\left\{ \frac{8}{1,3} \right\}$
Measures (edge length 1)
Circumradius	$\frac{\sqrt2}{2} \approx 0.70711$
Edge radius	$\frac12 = 0.5$
Face radius	$\frac{\sqrt6}{6} \approx 0.40825$
Inradius	$\frac{\sqrt2}{4} \approx 0.35355$
Hypervolume	$\frac16 \approx 0.16667$
Dichoral angle	120°
Interior angles	At triangle: $\frac13$ At edge: $\frac16$ At vertex: $\frac{1}{24}$
Height	$\frac{\sqrt2}{2} \approx 0.70711$
Central density	1
Number of external pieces	16
Level of complexity	1
Related polytopes
Army	Hex
Regiment	Hex
Dual	Tesseract
Conjugate	None
Abstract & topological properties
Flag count	384
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B4, order 384
Convex	Yes
Net count	261
Nature	Tame

The hexadecachoron, or hex, also commonly called the 16-cell, is one of the 6 convex regular polychora. It has 16 regular tetrahedra as cells, joining 4 to an edge and 8 to a vertex in an octahedral arrangement. It is the 4-dimensional orthoplex.

It is also the square duotegum, the digonal duoantiprism, the digonal diswirlprism, the digonal duotransitionaltertegum, and the 8-3 step prism. It is the first in an infinite family of isogonal tetrahedral swirlchora, the first in an infinite family of isogonal square hosohedral swirlchora and also the first in an infinite family of isochoric square dihedral swirlchora. It can also be seen as a tetrahedral antiprism in two senses, being both a segmentochoron of a tetrahedron atop dual tetrahedron (being designated K-4.2 in Richard Klitzing's list of convex segmentochora) and the alternated cubic prism (that is a tesseract). It is also a regular-faced octahedral tegum.

It and the great duoantiprism are the only uniform duoantiprisms, and it is the only one that is convex and regular.

It is one of the three regular polychora that can tile 4D space, the others being the tesseract and the icositetrachoron. Tiling it results in the hexadecachoric tetracomb.

The hexadecachoron company also contains the tesseractihemioctachoron.

If the vertex coordinates are mapped from $\mathbb{R}^4$ to the complex coordinate space $\mathbb{C}^2$ in the obvious way, the resulting set of points is precisely the vertices of the Möbius-Kantor polygon, which is a complex polygon. The containing planes of the hexadecachoron's triangular faces become the edges of the Möbius-Kantor polygon (which are not edges in the traditional sense, but instead complex 1-spaces that contain three points each).

Gallery

• 

  Rotating hexadecachoron

• 

  Net

• 

  Cross-sections

• 

  Cross-section animation

• 

  Segmentochoron display, tetrahedron atop dual tetrahedron

• 

  Wireframe, cell, net

Vertex coordinates

The vertices of a regular hexadecachoron of edge length 1, centered at the origin, are given by all permutations of:

• $\left(\pm\frac{\sqrt2}{2},\,0,\,0,\,0\right)$.

They can also be given as the even changes of sign of:

• $\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right)$.

These are formed by alternating the vertices of a tesseract.

Representations

A hexadecachoron has the following Coxeter diagrams:

• o4o3o3x () (full symmetry)
• x3o3o *b3o () (D₄ symmetry, as demitesseract)
• s4o3o3o () (as alternated tesseract)
• s2s4o3o () (as tetrahedral antiprism/alternated cubic prism)
• s4o2s4o () (as digonal duoantiprism)
• s2s2s4o () (as disphenoidal antiprism)
• s2s2s2s () (as alternated 4D block)
• xo3oo3ox&#x (A₃ axial, tetrahedron atop dual tetrahedron)
• ooo4ooo3oxo&#xt (B₃ axial, as octahedral tegum)
• ooo3oxo3ooo&#xt (A₃ axial, as tetratetrahedral tegum)
• o(qo)o o(ox)o4o(oo)o&#xt (as square tegmatic tegum)
• o(qoo)o o(oqo)o o(ooq)o&#xt (as rhombic tegmatic tegum)
• xox oxo4ooo&#xt (B₂×A₁ axial, edge-first)
• xox oxo oxo&#xt (K₂ axial, edge-first)
• xoo3oox oqo&#xt (A₂×A₁ axial, face-first)
• oxoo3ooox&#xr (A₂ axial)
• xo4oo ox4oo&#zx (B₂×B₂ symmetry, as square duotegum)
• xo xo ox4oo&#zx (as square-rectangular duotegum)
• xo xo ox ox&#zx (as rectangular duotegum)
• xoxo oxox&#xr (K₂ axial)
• qo oo4oo3ox&#zx (B₂×A₁ symmetry)
• qo oo3ox3oo&#zx (A₃×A₁ symmetry)
• qo os2os3os&#zx (as triangular antiprismatic tegum)
• qooo oqoo ooqo oooq&#zx (K₄ symmetry)
• qoo oqo oox4ooo&#zx (B₂×A₁×A₁ symmetry)

Variations

Besides the regular hexadecachoron, other types of polychora with 16 tetrahedral cells exist:

• Demitesseract - tetrahedra as two types, but no metrical variations - alternated tesseract
• Rectangular duotegum - 4 sets of 4 phyllic disphenoids, isogonal
• Tetrahedral antiprism - 2 tetrahedra, 8 triangular pyramids, and 6 tetragonal disphenoids, isogonal
• Tetrahedral antipodium - 2 different sized base tetrahedra, 2 sets of 4 triangular pyramid sides, and 6 digonal disphenoids
• Octahedral tegum - 16 identical triangular pyramids
• Square duotegum - 16 identical tetragonal disphenoids, noble
• Square-square duotegum: 16 identical digonal disphenoids
• Square-rhombic duotegum - 16 identical sphenoids
• Rhombic duotegum - 16 identical phyllic disphenoids
• Rhombic-rhombic duotegum - 16 identical irregular tetrahedra
• 8-3 step prism - isogonal with rhombic and phyllic dispenoids
• Digonal duoantiprism - isogonal, 2 sets of 8 tetragonal disphenoids
• Tetragonal disphenoidal antiprism, alternated square-rectangle duoprism, 4 rhombic and 4 digonal disphenoids, 8 sphenoids
• Rhombic disphenoidal antiprism - 4 pairs of opposite rhombic disphenoids connected by irregular tetrahedra
• Difold tetraswirlchoron - tetrahedra as digonal antiprisms, no metrical variations

Related polychora

The hexadecachoron is the colonel of a two-member regiment that also includes the tesseractihemioctachoron.

A hexadecachoron can be cut in half to produce 2 octahedral pyramids. Each of these can further be cut in half to produce 2 square scalenes, so a hexadecachoron can be consructed by joining four square scalenes together.

Uniform polychoron compounds composed of hexadecachora include:

• Demidistesseract (2)
• Stellated icositetrachoron (3)
• Chiricosahedral antiprism (5)
• Decachoric compound of 5 hexadecachora (5)
• Small stellated tetracontoctachoron (6)
• Snubahedral antiprism (6)
• Small snubahedral antiprism (6)
• Chirocrossifissal hexacontatetrachoron (8)
• Compound of 8 hexadecachora (8)
• Chirotrigonal antiprismatic enneacontahexachoron (12)
• Disnubahedral antiprism (12)
• Stellated chirohecatonicosachoron (15)
• Crossifissal dishexacontatetrachoron (16)
• Compound of 18 hexadecachora (18)
• Bitrigonal antiprismatic enneacontahexachoron (24)
• Stellated diacositetracontachoron (30)
• Stellated chirodishecatonicosachoron (30)
• Stellated triacosihexacontachoron (45)
• Stellated chirotrishecatonicosachoron (45)
• Compound of 50 hexadecachora (50)
• Stellated tetracosioctacontachoron (60)
• Stellated chirotetrishecatonicosachoron (60)
• Stellated dodecahedronary hexacosichoron (75)
• Stellated hexacosichoron (75)
• Compound of 300 hexadecachora (300)
• Stellated snub dodecahedronary pentishecatonicosachoron (375)
• Dodecahedronary disnubachoron (600)
• Compound of 675 hexadecachora (675)
• Several infinite families based on treating the hexadecachoron as a square duotegum, digonal duoantiprism, and 8-3 step prism

o4o3o3o truncations

Name	OBSA	CD diagram	Picture
Tesseract	tes	x4o3o3o ()
Truncated tesseract	tat	x4x3o3o ()
Rectified tesseract	rit	o4x3o3o ()
Tesseractihexadecachoron	tah	o4x3x3o ()
Rectified hexadecachoron = Icositetrachoron	ico	o4o3x3o ()
Truncated hexadecachoron	thex	o4o3x3x ()
Hexadecachoron	hex	o4o3o3x ()
Small rhombated tesseract	srit	x4o3x3o ()
Great rhombated tesseract	grit	x4x3x3o ()
Small rhombated hexadecachoron = Rectified icositetrachoron	rico	o4x3o3x ()
Great rhombated hexadecachoron = Truncated icositetrachoron	tico	o4x3x3x ()
Small disprismatotesseractihexadecachoron	sidpith	x4o3o3x ()
Prismatorhombated hexadecachoron	proh	x4x3o3x ()
Prismatorhombated tesseract	prit	x4o3x3x ()
Great disprismatotesseractihexadecachoron	gidpith	x4x3x3x ()

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Tetrahedron (16): Tesseract
• Triangle (32): Rectified tesseract
• Edge (24): Icositetrachoron

External links

• Bowers, Jonathan. "Category 1: Regular Polychora" (#3).

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "Hex".

• Quickfur. "The 16-Cell".

• Wikipedia Contributors. "16-cell".
• Hi.gher.Space Wiki Contributors. "Aerochoron".

• Hartley, Michael. "{3,3,4}*384".