Triacontaditeron

The triacontaditeron, or tac, also called the pentacross, 5-orthoplex, or square-octahedral duotegum, is one of the 3 regular polytera. It has 32 regular pentachora as facets, joining 16 to a vertex in an hexadecachoral arrangement. It is the 5-dimensional orthoplex.

It can also be seen as a convex segmentoteron, as a pentachoric antiprism. It is also the hexadecachoric tegum.

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

Representations
A triacontaditeron has the following Coxeter diagrams:


 * o4o3o3o3x (full symmetry)
 * o3o3o *b3o3x (D5 symmetry, h has demitesseract verf)
 * xo3oo3oo3ox&#x (A4 axial, pentachoric antiprism)
 * ooo4ooo3ooo3oxo&#xt (BC4 axial, as hexadecachoric bipyramid)
 * qo oo4oo3oo3ox&#zx (BC4×A1 symmetry, hexadecachoric bipyramid)
 * oxo3ooo3oo *b3oo&#zx (D4 symmetry, demitesseractic bipyramid)
 * qo ox3oo3oo *c3oo&#zx (D4×1 symmetry, demitesseractic bipyramid)
 * xox ooo4ooo3oxo&#xt (BC3×A1 symmetry, edge-first)
 * xox ooo3oxo3ooo&#xt (A3×A1 axial, edge-first)
 * xoo3oox oxo4ooo&#xt (BC2×A2 symmetry, face-first)
 * oxo oxo xoo3oox&#xt (BC2×A1×A1 axial, triangle-first)
 * xoo3ooo3oox oqo&#xt (A3×A1 symmetry, cell-first)
 * oxoo3oooo3ooox&#xr (A3 symmetry)
 * xoxo oxoo3ooox&#xr (A2×A1 symmetry)
 * xo4oo oo4oo3ox&#zx (BC3×BC2 symmetry, square-octahedral duotegum)
 * xo xo oo3ox3oo&#zx (A3×A1×A1 symmetry, square-octahedral duotegum)
 * o(xo)o o(xo)o o(ox)o o(ox)o&#xt (BC2×BC2 symmetry, square duotegmatic bipyramid)

Related polytopes
The regiment of the triacontaditeron contains 4 uniform members, including itself, one with D5 symmetry (the hexadecahemidecateron), and 2 with pentachoric antiprism symmetry (the pentachoric hemiantiprism and spinopentachoric hemiantiprism). THere are also 2 scaliform members known with 5-2 step prism alterprismatic symmetry.