7-simplex

Regular octaexon (7-simplex)
Orthogonal projection inside Petrie polygon
Type	Regular 7-polytope
Family	simplex
Schläfli symbol	{3,3,3,3,3,3}
Coxeter-Dynkin diagram
6-faces	8 6-simplex
5-faces	28 5-simplex
4-faces	56 5-cell
Cells	70 tetrahedron
Faces	56 triangle
Edges	28
Vertices	8
Vertex figure	6-simplex
Petrie polygon	octagon
Coxeter group	A7 [3,3,3,3,3,3]
Dual	Self-dual
Properties	convex

In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos⁻¹(1/7), or approximately 81.79°.

Alternate names

It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions. The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives a octaexon the acronym oca.^[1]

Coordinates

The Cartesian coordinates of the vertices of an origin-centered regular octaexon having edge length 2 are:

$\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)$

$\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)$

$\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)$

$\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)$

$\left(\sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)$

$\left(\sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)$

$\left(-\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$

More simply, the vertices of the 7-simplex can be positioned in 8-space as permutations of (0,0,0,0,0,0,0,1). This construction is based on facets of the 8-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Related polytopes

This polytope is one of 71 uniform 7-polytopes with A₇ symmetry.

t0	t1	t2	t3	t0,1	t0,2	t1,2	t0,3
t1,3	t2,3	t0,4	t1,4	t2,4	t0,5	t1,5	t0,6
t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4	t0,2,4	t1,2,4	t0,3,4
t1,3,4	t2,3,4	t0,1,5	t0,2,5	t1,2,5	t0,3,5	t1,3,5	t0,4,5
t0,1,6	t0,2,6	t0,3,6	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,2,3,4	t1,2,3,4
t0,1,2,5	t0,1,3,5	t0,2,3,5	t1,2,3,5	t0,1,4,5	t0,2,4,5	t1,2,4,5	t0,3,4,5
t0,1,2,6	t0,1,3,6	t0,2,3,6	t0,1,4,6	t0,2,4,6	t0,1,5,6	t0,1,2,3,4	t0,1,2,3,5
t0,1,2,4,5	t0,1,3,4,5	t0,2,3,4,5	t1,2,3,4,5	t0,1,2,3,6	t0,1,2,4,6	t0,1,3,4,6	t0,2,3,4,6
t0,1,2,5,6	t0,1,3,5,6	t0,1,2,3,4,5	t0,1,2,3,4,6	t0,1,2,3,5,6	t0,1,2,4,5,6	t0,1,2,3,4,5,6

Notes

1. ^ Richard Klitzing, 7D uniform polytopes (polyexa), x3o3o3o3o3o - oca

External links

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

This geometry-related article is a stub. You can help Wikipedia by expanding it.v · Categories: 7-polytopes Geometry stubs Wikimedia Foundation. 2010. Игры ⚽ Нужен реферат? Octadecatrienoic acid Octagon, Alabama Look at other dictionaries: Simplex — Saltar a navegación, búsqueda Para el algoritmo del mismo nombre, véase Algoritmo simplex. Un 3 simplejo o tetraedro que puede pensarse como una región del espacio que consiste en la parte acotada por (y que también incluye) los cuatro puntos,… …   Wikipedia Español Simplex (disambiguation) — Simplex may refer to:*Simplex, a term in geometry *Simplex, a one way communications channel * Simplex was a trade name used by British railway locomotive manufacturers, The Motor Rail Tramcar Co Ltd *SimplexGrinnell, a brand of fire protection… …   Wikipedia Simplex noise — is a method for constructing an n dimensional noise function comparable to Perlin noise ( classic noise) but with a lower computational overhead, especially in larger dimensions. Ken Perlin designed the algorithm in 2001 [Ken Perlin, Noise… …   Wikipedia Simplex — steht für: Simplex (Mathematik), n dimensionales Polytop mit n+1 Ecken Simplex Verfahren, Algorithmus zur Lösung mathematischer Optimierungsprobleme Simplex (Grammatik), sprachwissenschaftlich bzw. grammatikalisch ein einfaches, d. h. nicht… …   Deutsch Wikipedia simplex — SÍMPLEX s.n. 1. Sistem de telecomunicaţii care asigură legătura bilaterală simultană între două posturi. 2. Ţesătură din tricot cu ambele părţi lucrate pe faţă. – Din fr. simplex. Trimis de IoanSoleriu, 24.07.2004. Sursa: DEX 98  símplex s. n.,… …   Dicționar Român simplex — see epidermolysis bullosa simplex, genital herpes simplex, HERPES SIMPLEX, ICHTHYOSIS SIMPLEX …   Medical dictionary Simplex communication — is a name for a type of communication circuit. There are two (contradictory) definitions that have been used for the term. When one definition is used for simplex , then the other definition is actually referred to as half duplex .One way only –… …   Wikipedia simplex — ● simplex nom masculin En télécommunications et en transmission de données, mode de transmission permettant le transfert d informations dans un seul sens (par opposition à duplex). simplex [sɛ̃plɛks] adj. invar. et n. m. ÉTYM. XXe; de simple. ❖ ♦ …   Encyclopédie Universelle Simplex Nthala — Pas d image ? Cliquez ici. Situation actuelle Club actuel …   Wikipédia en Français Simplex signaling — (SX): Signaling in which two conductors are used for a single channel, and a center tapped coil, or its equivalent, is used to split the signaling current equally between the two conductors. Note: SX signaling may be one way, for intra central… …   Wikipedia Simplex (Automobilhersteller) — Simplex war ein französischer Hersteller von Automobilen. Unternehmensgeschichte Das Unternehmen begann 1920 mit der Produktion von Automobilen. 1921 endete die Produktion. Fahrzeuge Das einzige Modell war mit einem Einzylindermotor mit 735 cm³… …   Deutsch Wikipedia 18+ © Academic, 2000-2025 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”