# Hexacontatetrapeton

(Redirected from Gee)
Hexacontatetrapeton
Rank6
TypeRegular
Notation
Bowers style acronymGee
Coxeter diagramx3o3o3o3o4o ()
Schläfli symbol{3,3,3,3,4}
Bracket notation<IIIIII>
Elements
Peta64 hexatera
Tera192 pentachora
Cells240 tetrahedra
Faces160 triangles
Edges60
Vertices12
Petrie polygons${\displaystyle \left\{{\frac {12}{1,3,5}}\right\}}$
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Edge radius${\displaystyle {\frac {1}{2}}=0.5}$
Face radius${\displaystyle {\frac {\sqrt {6}}{6}}\approx 0.40825}$
Cell radius${\displaystyle {\frac {\sqrt {2}}{4}}\approx 0.35355}$
Teron radius${\displaystyle {\frac {\sqrt {10}}{10}}\approx 0.31623}$
Inradius${\displaystyle {\frac {\sqrt {3}}{6}}\approx 0.28868}$
Hypervolume${\displaystyle {\frac {1}{90}}\approx 0.011111}$
Dipetal angle${\displaystyle \arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }}$
Height${\displaystyle {\frac {\sqrt {3}}{3}}\approx 0.57735}$
Central density1
Number of external pieces64
Level of complexity1
Related polytopes
ArmyGee
RegimentGee
DualHexeract
ConjugateNone
Abstract & topological properties
Flag count46080
Euler characteristic0
OrientableYes
Properties
SymmetryB6, order 46080
ConvexYes
Net count502110
NatureTame

The hexacontatetrapeton, or gee, also called the hexacross or 6-orthoplex, is a regular polypeton. It has 64 regular hexatera as facets, joining 3 to a tetrahedron peak and 32 to a vertex in a triacontaditeral arrangement. It is the 6-dimensional orthoplex. It is also an octahedral duotegum and square triotegum, triacontaditeric tegum, icosahedron-great icosahedron step prism, and 12-3-5 step prism.

It can also be seen as a segmentopeton as a hexateric antiprism.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0,\,0\right)}$.

## Representations

A hexacontatetrapeton has the following Coxeter diagrams:

• x3o3o3o3o4o () (full symmetry)
• x3o3o3o3o *d3o () (D6 symmetry)
• xo3oo3oo3oo3ox&#x (A5 axial, hexateric antiprism)
• ooo4ooo3ooo3ooo3oxo&#xt (B5 axial, triacontaditeric bipyramid)
• qo oo4oo3oo3oo3ox&#zx (B5×A1 symmetry)
• oo3ooo3ooo *b3ooo3oxo&#xt (D5 axial, still triacontaditeric bipyramid)
• qo oo3oo3oo *c3oo3ox&#zx (D5×A1 symmety)
• oxoo3oooo3oooo3ooox&#x (A4 axial)
• oqo xoo3ooo3ooo3oox&#xt (A4×A1 axial, pentachoron-first)
• xox ooo4ooo3ooo3oxo&#xt (B4×A1 symmetry, edge-first)
• xox oxo3ooo3ooo *c3ooo&#xt (D4×A1 axial, still edge-first)
• xo4oo oo4oo3oo3ox&#zx (B4×B2 symmetry, square-hexadecachoron duotegum)
• xo xo ox3oo3oo *d3oo&#zx (D4×A1×A1 symmetry, rectangle-demitesseract duotegum)
• oxo4ooo xoo3ooo3oox&#xt (A3×B2 symmetry, tetrahedron-first)
• oxo oxo xoo3ooo3oox&#xt (A3×A1×A1 symmetry, still tetrahedron-first)
• xoxo oxoo3oooo3ooox&#xr (A3×A1 axial)
• xoo3oox ooo4ooo3oox&#xt (B3×A2 axial, triangle-first)
• xoo3oox ooo3oxo3ooo&#xt (A3×A2 axial, triangle-first)
• oo4oo3xo oo4oo3ox&#zx (B3×B3 symmetry, octahedral duotegum)
• oo3xo3oo oo3ox3oo&#zx (A3×A3 symmetry, tetratetrahedral duotegum)
• xooo3ooxo oxoo3ooox&#xr (A2×A2 symmetry)
• xoo4ooo oxo4ooo oox4ooo&#zx (B2×B2×B2 symetry, square triotegum)
• xoo xoo oxo oxo oox oox&#zx (rectangular triotegum)

## Related polytopes

The regiment of the hexacontatetrapeton includes a total of 13 known uniform members, including itself, 1 with D6 symmetry (the triacontadihemihexeract), 4 with hexateric antiprism symmetry, 2 with doubled icosahedral step prism symmetry, 1 with icosahedral step prism symmetry, and 4 with triangular disphenoidal antiprismatic symmetry. The regiment also includes a number of scaliforms.

o4o3o3o3o3o truncations
Name OBSA CD diagram
Hexeract ax
Rectified hexeract rax
Birectified hexeract brox
Birectified hexacontatetrapeton brag
Rectified hexacontatetrapeton rag
Hexacontatetrapeton gee
Truncated hexeract tox
Bitruncated hexeract botox
Hexeractihexacontatetrapeton xog
Bitruncated hexacontatetrapeton botag
Truncated hexacontatetrapeton tag
Small rhombated hexeract srox
Small birhombated hexeract saborx
Small birhombated hexacontatetrapeton siborg
Small rhombated hexacontatetrapeton srog
Great rhombated hexeract grox
Great birhombated hexeract gaborx
Great birhombated hexacontatetrapeton gaborg
Great rhombated hexacontatetrapeton grog
Small prismated hexeract spox
Small biprismated hexeract sobpoxog
Small prismated hexacontatetrapeton spog
Prismatotruncated hexeract potax
Biprismatorhombated hexacontatetrapeton boprag
Prismatorhombated hexacontatetrapeton prog
Prismatorhombated hexeract prox
Biprismatorhombated hexeract boprax
Prismatotruncated hexacontatetrapeton potag
Great prismated hexeract gippox
Great biprismated hexeract gobpoxog
Great prismated hexacontatetrapeton gopog
Small cellated hexeract scox
Small cellated hexacontatetrapeton scag
Cellitruncated hexeract catax
Celliprismated hexacontatetrapeton copog
Cellirhombated hexeract crax
Cellirhombated hexacontatetrapeton crag
Celligreatorhombated hexeract cagorx
Celliprismatorhombated hexacontatetrapeton coprag
Celliprismated hexeract copox
Cellitruncated hexacontatetrapeton catog
Celliprismatotruncated hexeract captix
Celliprismatotruncated hexacontatetrapeton captog
Celliprismatorhombated hexeract coprix
Celligreatorhombated hexacontatetrapeton cagorg
Great cellated hexeract gocax
Great cellated hexacontatetrapeton gocog
Small terated hexeract stoxog
Tericellated hexacontatetrapeton tacog
Teriprismated hexacontatetrapeton topag
Terigreatorhombated hexeract togrix
Teriprismated hexeract tapox
Tericellirhombated hexacontatetrapeton tocrag
Teriprismatorhombated hexeract tiprixog
Terigreatoprismated hexeract tagpox
Tericellated hexeract tacox
Tericellitruncated hexeract tactaxog
Tericellirhombated hexeract tocrax
Tericelligreatorhombated hexeract tocagrax
Terigreatorhombated hexacontatetrapeton togrig
Tericelligreatorhombated hexacontatetrapeton tecagorg
Terigreatoprismated hexacontatetrapeton tagpog
Great terated hexeract gotaxog