9-simplex

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Convex regular 9-polytope

title: "9-simplex" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["9-polytopes"] description: "Convex regular 9-polytope" topic_path: "general/9-polytopes" source: "https://en.wikipedia.org/wiki/9-simplex" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Convex regular 9-polytope ::

::data[format=table]

Regular decayotton(9-simplex)
[[Image:9-simplex_t0.svg
Type
Family
Schläfli symbol
Coxeter-Dynkindiagram
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Petrie polygon
Coxeter group
Dual
Properties
::

In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces. Its dihedral angle is cos−1(1/9), or approximately 83.62°.

It can also be called a decayotton, or deca-9-tope, as a 10-facetted polytope in 9-dimensions. The name decayotton is derived from deca for ten facets in Greek and yotta (a variation of "oct" for eight), having 8-dimensional facets, and -on.

Jonathan Bowers gives it acronym day.

Coordinates

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

:\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right) :\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right) :\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right) :\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right) :\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right) :\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right) :\left(\sqrt{1/45},\ 1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right) :\left(\sqrt{1/45},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right) :\left(-3\sqrt{1/5},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

More simply, the vertices of the 9-simplex can be positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,0,1). These are the vertices of one Facet of the 10-orthoplex.

Images

References

References

  1. {{harvnb. Klitzing

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