Tesseractihexadecachoron

The tesseractihexadecachoron, or tah, also commonly called the bitruncated tesseract, is a convex uniform polychoron that consists of 16 truncated tetrahedra and 8 truncated octahedra. 2 truncated tetrahedra and 2 truncated octahedra join at each vertex. It is the medial stage of the truncation series between a tesseract and its dual hexadecachoron. As such is could also be called a bitruncated 16-cell.

Vertex coordinates
The vertices of a tesseractihexadecachoron of edge length 1 are all permutations of:
 * $$\left(±\sqrt2,\,±\sqrt2,\,±\frac{\sqrt2}{2},\,0\right).$$

Alternatively it can be given under D4 symmetry as all even sign changes of:
 * $$\left(\frac{5\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right).$$

Representations
The tesseractihexadecachoron has the following Coxeter diagrams:


 * o4x3x3o (full symmetry)
 * x3x3x *b3o (D4 symmetry, great rhombated demitesseract)
 * s4o3x3x (as runcicantic tesseract)
 * s4x3x3o (also as snub)
 * ooqoo4xuxux3xooox&#xt (BC3 axial, truncated octahedron-first)
 * xxuxoo3xuxxux3ooxuxx&#xt (A3 axial, truncated tetrahedron-first)
 * Qqo ooq4xux3xoo&#zx (BC3×A1 symmetry)

Semi-uniform variant
The tesseractihexadecachoron has a semi-uniform variant of the form o4x3y3o that maintains its full symmetry. This variant uses 16 semi-uniform truncated tetrahedra of form x3y3o and 8 semi-uniform truncated octahedra of form o4x3y as cells, with 2 edge lengths.

With edges of length a (of square faces) and b (of triangular faces), its circumradius is given by $$\sqrt{\frac{3a^2+2b^2+4ab}{2}}$$ and its hypervolume is given by $$\frac{23a^4+88a^3b+120a^2b^2+64ab^3+12b^4}{6}$$.

It has coordinates given by all permutations of:


 * $$\left(±(a+b)\frac{\sqrt2}{2},\,±(a+b)\frac{\sqrt2}{2},\,±a\frac{\sqrt2}{2},\,0\right).$$

There is also a semi-uniform variant with D4 symmetry known as the great rhombated demitesseract.

Related polychora
Uniform polychoron compounds composed of tesseractihexadecachora include:


 * Bitruncated demidistesseract (2)
 * Bitruncated great icositetrachoron (3)
 * Bitruncated great stellated tetracontoctachoron (6)
 * Bitruncated dodecahedronary hexacosichoron (75)