Truncated tesseract

The truncated tesseract, or tat, is a convex uniform polychoron that consists of 16 regular tetrahedra and 8 truncated cubes. 1 tetrahedron and three truncated cubes join at each vertex. As the name suggests, it can be obtained by truncating the tesseract.

As the truncated tesseract, it is the square member of an infinite family of isogonal truncated duoprisms, and could be called the truncated square duoprism.

Vertex coordinates
The vertices of a truncated tesseract of edge length 1 are given by all permutations of:
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$$

Representations
A truncated tesseract has the following Coxeter diagramms:


 * x4x3o3o (full symmetry)
 * xwwx4xoox3oooo&#xt (B3 symmetry, truncated cube-first)
 * xwwxoooo3ooxwwxoo3ooooxwwx&#xt (A3 axial, tetrahedron-first)
 * wx3oo3xw *b3oo&#zx (D4 symmetry)
 * wx xw4xo3oo&#zx (B3×A1 symmetry)
 * ox4wx xo4xw&#zx (B2×B2 symmetry, truncated square duoprism)
 * xwww wxww wwxw wwwx&#zx (A1×A1×A1×A1 symmetry)

Semi-uniform variant
The truncated tesseract has a semi-uniform variant of the form x4y3o3o that maintains its full symmetry. This variant uses 16 tetrahedra of size y and 8 semi-uniform truncated cubes of form x4y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated cubes only) and b (of tetrahedra), its circumradius is given by $$\sqrt{\frac{2a^2+3b^2+3ab\sqrt2}{2}}$$ and its hypervolume is given by $$\frac{6a^4+72a^2b^2+23b^4+(24a^3b+48ab^3)\sqrt2}{6}$$.

It has coordinates given by all permutations of:


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

Related polychora
Uniform polychoron compounds composed of truncated tesseracts include:


 * Truncated great icositetrachoron (3)
 * Truncated great stellated tetracontoctachoron (6)
 * Truncated dodecahedronary cubichoron (75)
 * Truncated cubichoron (75)