10-simplex
Convex regular 10-polytope
| Regular hendecaxennon(10-simplex) | |
|---|---|
| Orthogonal projectioninside Petrie polygon | |
| Type | Regular 10-polytope |
| Family | simplex |
| Schläfli symbol | {3,3,3,3,3,3,3,3,3} |
| Coxeter-Dynkindiagram | |
| 9-faces | 11 9-simplex |
| 8-faces | 55 8-simplex |
| 7-faces | 165 7-simplex |
| 6-faces | 330 6-simplex |
| 5-faces | 462 5-simplex |
| 4-faces | 462 5-cell |
| Cells | 330 tetrahedron |
| Faces | 165 triangle |
| Edges | 55 |
| Vertices | 11 |
| Vertex figure | 9-simplex |
| Petrie polygon | hendecagon |
| Coxeter group | A10 [3,3,3,3,3,3,3,3,3] |
| Dual | Self-dual |
| Properties | convex |
In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces. Its dihedral angle is cos−1(1/10), or approximately 84.26°.
It can also be called a hendecaxennon, or hendeca-10-tope, as an 11-facetted polytope in 10-dimensions. Acronym: ux
The name hendecaxennon is derived from hendeca for 11 facets in Greek and -xenn (variation of ennea for nine), having 9-dimensional facets, and -on.
The Cartesian coordinates of the vertices of an origin-centered regular 10-simplex having edge length 2 are:
(
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{\displaystyle \left({\sqrt {1/55}},\ {\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)}
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{\displaystyle \left({\sqrt {1/55}},\ {\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)}
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{\displaystyle \left({\sqrt {1/55}},\ {\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ -{\sqrt {3/2}},\ 0,\ 0\right)}
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{\displaystyle \left({\sqrt {1/55}},\ {\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)}
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{\displaystyle \left({\sqrt {1/55}},\ {\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)}
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{\displaystyle \left({\sqrt {1/55}},\ {\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ -{\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)}
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{\displaystyle \left({\sqrt {1/55}},\ {\sqrt {1/45}},\ 1/6,\ -{\sqrt {7/4}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}
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{\displaystyle \left({\sqrt {1/55}},\ {\sqrt {1/45}},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}
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{\displaystyle \left({\sqrt {1/55}},\ -3{\sqrt {1/5}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}
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{\displaystyle \left(-{\sqrt {20/11}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}
More simply, the vertices of the 10-simplex can be positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,0,1). This construction is based on facets of the 11-orthoplex.
| Ak Coxeter plane | A10 | A9 | A8 |
|---|---|---|---|
| [11] | [10] | [9] | |
| [8] | [7] | [6] | |
| [5] | [4] | [3] |
The 2-skeleton of the 10-simplex is topologically related to the 11-cell abstract regular polychoron which has the same 11 vertices, 55 edges, but only 1/3 the faces (55).
.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}body.skin-vector-2022 .mw-parser-output .reflist-columns-2{column-width:27em}body.skin-vector-2022 .mw-parser-output .reflist-columns-3{column-width:22.5em}.mw-parser-output .references[data-mw-group=upper-alpha]{list-style-type:upper-alpha}.mw-parser-output .references[data-mw-group=upper-roman]{list-style-type:upper-roman}.mw-parser-output .references[data-mw-group=lower-alpha]{list-style-type:lower-alpha}.mw-parser-output .references[data-mw-group=lower-greek]{list-style-type:lower-greek}.mw-parser-output .references[data-mw-group=lower-roman]{list-style-type:lower-roman}.mw-parser-output div.reflist-liststyle-upper-alpha .references{list-style-type:upper-alpha}.mw-parser-output div.reflist-liststyle-upper-roman .references{list-style-type:upper-roman}.mw-parser-output div.reflist-liststyle-lower-alpha .references{list-style-type:lower-alpha}.mw-parser-output div.reflist-liststyle-lower-greek .references{list-style-type:lower-greek}.mw-parser-output div.reflist-liststyle-lower-roman .references{list-style-type:lower-roman}
-
Coxeter, H.S.M.:
- .mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#bf3c2c)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#bf3c2c)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}— (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)". Regular Polytopes (3rd ed.). Dover. pp. 296. ISBN 0-486-61480-8.
- Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Wiley. ISBN 978-0-471-01003-6.
- (Paper 22) — (1940). "Regular and Semi-Regular Polytopes I". Math. Zeit. 46: 380–407. doi:10.1007/BF01181449. S2CID 186237114.
- (Paper 23) — (1985). "Regular and Semi-Regular Polytopes II". Math. Zeit. 188 (4): 559–591. doi:10.1007/BF01161657. S2CID 120429557.
- (Paper 24) — (1988). "Regular and Semi-Regular Polytopes III". Math. Zeit. 200: 3–45. doi:10.1007/BF01161745. S2CID 186237142.
-
Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "26. Hemicubes: 1n1". The Symmetries of Things. p. 409. ISBN 978-1-56881-220-5.
-
Johnson, Norman (1991). Uniform Polytopes (Manuscript).
- Johnson, N.W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD). University of Toronto. OCLC 258527038.
-
Klitzing, Richard. "10D uniform polytopes (polyxenna)". x3o3o3o3o3o3o3o3o3o – ux
-
Glossary for hyperspace, George Olshevsky.
-
Polytopes of Various Dimensions
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Multi-dimensional Glossary
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|---|---|---|---|---|
| An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn |
| Triangle | Square | p-gon | Hexagon | Pentagon |
| Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | |
| Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell |
| 5-simplex | 5-orthoplex • 5-cube | 5-demicube | ||
| 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | |
| 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | |
| 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | |
| 9-simplex | 9-orthoplex • 9-cube | 9-demicube | ||
| 10-simplex | 10-orthoplex • 10-cube | 10-demicube | ||
| n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope |