6-simplex
6-simplex | |
---|---|
Type | uniform polypeton |
Schläfli symbol | {35} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Elements |
f5= 7,f4= 21,C= 35,F= 35,E= 21,V= 7 |
Coxeter group | A6,[35], order 5040 |
Bowers name and (acronym) |
Heptapeton (hop) |
Vertex figure | 5-simplex |
Circumradius | 0.654654[1] |
Properties | convex,isogonalself-dual |
Ingeometry,a 6-simplexis aself-dualregular6-polytope.It has 7vertices,21edges,35 trianglefaces,35tetrahedralcells,215-cell4-faces, and 75-simplex5-faces. Itsdihedral angleis cos−1(1/6), or approximately 80.41°.
Alternate names
[edit]It can also be called aheptapeton,orhepta-6-tope,as a 7-facettedpolytope in 6-dimensions. Thenameheptapetonis derived fromheptafor sevenfacetsinGreekand-petafor having five-dimensional facets, and-on.Jonathan Bowers gives a heptapeton the acronymhop.[2]
As a configuration
[edit]Thisconfiguration matrixrepresents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.[3][4]
Coordinates
[edit]TheCartesian coordinatesfor an origin-centered regular heptapeton having edge length 2 are:
The vertices of the6-simplexcan be more simply positioned in 7-space as permutations of:
- (0,0,0,0,0,0,1)
This construction is based onfacetsof the7-orthoplex.
Images
[edit]AkCoxeter plane | A6 | A5 | A4 |
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Graph | ![]() |
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Dihedral symmetry | [7] | [6] | [5] |
AkCoxeter plane | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [4] | [3] |
Related uniform 6-polytopes
[edit]The regular 6-simplex is one of 35uniform 6-polytopesbased on the [3,3,3,3,3]Coxeter group,all shown here in A6Coxeter planeorthographic projections.
Notes
[edit]- ^Klitzing, Richard."heptapeton".bendwavy.org.
- ^Klitzing, Richard."6D uniform polytopes (polypeta) x3o3o3o3o3o — hop".
- ^Coxeter 1973,§1.8 Configurations
- ^Coxeter, H.S.M. (1991).Regular Complex Polytopes(2nd ed.). Cambridge University Press. p. 117.ISBN9780521394901.
References
[edit]- Coxeter, H.S.M.:
- — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)".Regular Polytopes(3rd ed.). Dover. p. 296.ISBN0-486-61480-8.
- Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995).Kaleidoscopes: Selected Writings of H.S.M. Coxeter.Wiley.ISBN978-0-471-01003-6.
- (Paper 22)— (1940)."Regular and Semi Regular Polytopes I".Math. Zeit.46:380–407.doi:10.1007/BF01181449.S2CID186237114.
- (Paper 23)— (1985)."Regular and Semi-Regular Polytopes II".Math. Zeit.188(4): 559–591.doi:10.1007/BF01161657.S2CID120429557.
- (Paper 24)— (1988)."Regular and Semi-Regular Polytopes III".Math. Zeit.200:3–45.doi:10.1007/BF01161745.S2CID186237142.
- Conway, John H.;Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "26. Hemicubes: 1n1".The Symmetries of Things.p. 409.ISBN978-1-56881-220-5.
- Johnson, Norman(1991). "Uniform Polytopes" (Manuscript). Norman Johnson (mathematician).
- Johnson, N.W. (1966).The Theory of Uniform Polytopes and Honeycombs(PhD). University of Toronto.OCLC258527038.
External links
[edit]- Olshevsky, George."Simplex".Glossary for Hyperspace.Archived fromthe originalon 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary