Heptapeton

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Heptapeton
6-simplex t0.svg
Rank6
TypeRegular
SpaceSpherical
Bowers style acronymHop
Info
Coxeter diagramx3o3o3o3o3o
Schläfli symbol{3,3,3,3,3}
SymmetryA6, order 5040
ArmyHop
RegimentHop
Elements
Vertex figureHexateron, edge length 1
Peta7 hexatera
Tera21 pentachora
Cells35 tetrahedra
Faces35 triangles
Edges21
Vertices7
Measures (edge length 1)
Circumradius21/7 ≈ 0.65465
Inradius21/42 ≈ 0.10911
Hypervolume7/5760 ≈ 0.00045933
Dipetal angleacos(1/6) ≈ 80.40593°
HeightsPoint atop hix: 21/6 ≈ 0.76376
 Dyad atop perp pen: 35/10 ≈ 0.59161
 Trig atop perp tet: 42/12 ≈ 0.54006
Central density1
Euler characteristic0
Related polytopes
DualHeptapeton
ConjugateHeptapeton
Properties
ConvexYes
OrientableYes
NatureTame

The heptapeton, or hop, also commonly called the 6-simplex, is the simplest possible non-degenerate polypeton. The full symmetry version has 7 regular hexatera as facets, joining 3 to a tetrahedron peak and 6 to a vertex, and is one of the 3 regular polypeta. It is the 6-dimensional simplex. It is one of two uniform self-dual polypeta, the other being the great icosiheptapeton. It is also the 7-2-3 step prism and gyropeton.

It can be obtained as a segmentopeton in three ways: as a hexateric pyramid, dyad atop perpendicular pentachoron, or triangle atop perpendicular tetrahedron.

Vertex coordinates[edit | edit source]

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

  • (±1/2, –3/6, –6/12, –10/20, –15/30, –21/42),
  • (0, 3/3, –6/12, –10/20, –15/30, –21/42),
  • (0, 0, 6/4, –10/20, –15/30, –21/42),
  • (0, 0, 0, 10/5, –15/30, –21/42),
  • (0, 0, 0, 0, 15/6, –21/42),
  • (0, 0, 0, 0, 0, 21/7).

Much simpler coordinates can be given in seven dimensions, as all permutations of:

  • (2/2, 0, 0, 0, 0, 0, 0).

Related polytopes[edit | edit source]

o3o3o3o3o3o truncations
Name OBSA CD diagram Picture
Heptapeton hop CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t0.svg
Rectified heptapeton ril CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t1.svg
Birectified heptapeton bril CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t2.svg
Truncated heptapeton til CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t01.svg
Bitruncated heptapeton batal CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t12.svg
Tetradecapeton fe CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t23.svg
Small rhombated heptapeton sril CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t02.svg
Small birhombated heptapeton sabril CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t13.svg
Great rhombated heptapeton gril CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t012.svg
Great birhombated heptapeton gabril CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t123.svg
Small prismated heptapeton spil CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t03.svg
Small biprismatotetradecapeton sibpof CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t14.svg
Prismatotruncated heptapeton patal CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t013.svg
Biprismatorhombated heptapeton bapril CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t124.svg
Prismatorhombated heptapeton pril CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t023.svg
Great prismated heptapeton gapil CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t0123.svg
Great biprismatotetradecapeton gibpof CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t1234.svg
Small cellated heptapeton scal CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t04.svg
Cellitruncated heptapeton catal CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t014.svg
Cellirhombated heptapeton cral CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t024.svg
Celligreatorhombated heptapeton cagral CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t0124.svg
Celliprismated heptapeton copal CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t034.svg
Celliprismatotruncated heptapeton captal CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t0134.svg
Celliprismatorhombated heptapeton copril CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t0234.svg
Great cellated heptapeton gacal CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t01234.svg
Small teritetradecapeton staf CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t05.svg
Tericellated heptapeton tocal CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t015.svg
Teriprismated heptapeton topal CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t025.svg
Terigreatorhombated heptapeton togral CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t0125.svg
Tericellirhombated heptapeton tocral CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t0135.svg
Teriprismatorhombitetradecapeton taporf CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t0235.svg
Terigreatoprismated heptapeton tagopal CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t01235.svg
Tericellitruncatotetradecapeton tactaf CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
6-simplex t0145.svg
Tericelligreatorhombated heptapeton tacogral CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
6-simplex t01245.svg
Great teritetradecapeton gotaf CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
6-simplex t012345.svg

Representations[edit | edit source]

A regular heptapeton has the following Coxeter diagrams:

  • x3o3o3o3o3o (full symmetry)
  • ox3oo3oo3oo3oo&#x (A5 axial, hexateric pyarmid)
  • xo ox3oo3oo3oo&#x (A4×A1 axial, pentachric scalene)
  • xo3oo ox3oo3oo&#x (A3×A2 axial, tetrahedral tettene)
  • oxo3ooo3ooo3ooo&#x (A4 only, pentachoric pyramidal pyramid)
  • oxo oox3ooo3ooo&#xt (A3×A1 axial, tetrahedral scalenic pyramid)
  • oxo3ooo oox3ooo&#x (A2×A2 axial, triangular disphenoidal pyramid)
  • xoo oxo oox3ooo&#x (A2×A1×A1 axial, triangular scalenic scalene)


External links[edit | edit source]

  • Klitzing, Richard. "Hop".