Hexacontatetrapeton

The hexacontatetrapeton, or gee, also called the hexacross or 6-orthoplex, is one of the 3 regular polypeta. 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:
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,0\right).$$

Representations
A hexacontatetrapeton has the following Coxeter diagrams:


 * o4o3o3o3o3x (full symmetry)
 * o3o3o *b3o3o3x (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,3 have icosahedral step prism symmetry, and the last 4 have lower subsymmetries. The regiment also includes a number of scaliforms.