Hecatonicosoctaexon

The hecatonicosoctaexon, or zee, also called the heptacross or 7-orthoplex, is one of the 3 regular polyexa. It has 128 regular heptapeta as facets, joining 4 to a pentachoron peak and 64 to a vertex in a hexacontatetrapetal arrangement. It is the 7-dimensional orthoplex. It is also a convex segmentoexon, as a heptapetal antiprism.

Vertex coordinates
The vertices of a regular hecatonicosoctaexon of edge length 1, centered at the origin, are given by all permutations of:
 * $$\left(\pm\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,0,\,0\right).$$

Representations
A hecatonicosoctaexon has the following Coxeter diagrams:


 * o4o3o3o3o3o3x (full symmetry)
 * o3o3o *b3o3o3o3x (D7 symmetry)
 * xo3oo3oo3oo3oo3ox&#x (A6 axial, heptapetal antiprism)
 * ooo4ooo3ooo3ooo3ooo3oxo&#xt (B6 axial, hexacontatetrapetal bipyramid)
 * qo oo4oo3oo3oo3oo3ox&#zx (B6×A1 symmetry)
 * xox ooo4ooo3ooo3ooo3oxo&#xt (B5×A1 axial, dege-first)
 * xox ooo3ooo3ooo *b3ooo3oxo&#xt (D5×A1 axial, edge-first with half symmetry)
 * xoox oxoo3oooo3oooo3ooxo&#xr (A4×A1 symmetry)
 * xoo3oox ooo4ooo3ooo3oox&#xt (B4×A2 axial, triangle-first)
 * xoo3oox oxo3ooo3ooo *d3ooo&#xt (D4×2 symmetry, triangle-first with half symmetry)
 * oxo4oooo xoo3ooo3ooo3oox&#xt (A4×B2 axial, pentachoron-first)
 * oxo oxo xoo3ooo3ooo3oox&#xt (A4×A1×A1 symmetry, pentachoron-first with half symmetry)
 * xo4oo oo4oo3oo3oo3ox&#zx (B5×B2 symmetry, square-triacontaditeral duotegum)
 * xoo3ooo3oox ooo4ooo3oxo&#xt (B3×A3 axial, tetrahedron-first)
 * xoo3ooo3oox ooo3oxo3ooo&#xt (A3×A3 axial, tetrahedron-first with half symmetry)
 * oo4oo3xo oo4oo3oo3ox&#zx (B4×B3 symmetry, octahedral-hexadecachoric duotegum)
 * oo3xo3oo ox3oo3oo *e3oo&#zx (D4×A3 symmetry, tetratetrahedral-demitesseractic duotegum)
 * oxoo3oooo3oooo3oooo3ooxo&#xr (A5 symmetry)
 * oqo xoo3ooo3ooo3ooo3oox&#xt (A5×A1 axial, hexateron-first)
 * xooo3ooxo oxoo3oooo3ooox&#xr (A3×A2 symmetry)
 * ooxoo4ooooo oxooo3ooooo3oooxo&#xcr (A3×B2 symmetry)
 * oxooo3oooxo ooxoo3ooooo4ooooo&#xcr (B3×A2 symmetry)
 * o(xoo)o o(xoo)o o(oxo)o o(oxo)o o(oox)o o(oox)o&#xt (square triotegum bipyramid)