Small disprismatotesseractihexadecachoron

The small disprismatotesseractihexadecachoron, square duoexpandoprism, or sidpith, also commonly called the runcinated tesseract, is a convex uniform polychoron that consists of 16 regular tetrahedra, 32 triangular prisms, and 8+24 cubes. 1 tetrahedron, 3 triangular prisms, and 1+3 cubes join at each vertex. It is the result of expanding the cells of either a tesseract or a hexadecachoron outwards, and thus could also be called the runcinated 16-cell.

It is the second in the series of duoexpandoprisms formed as the convex hull of a compound of two perpendicular square-octagonal duoprisms and the only uniform one. Blending it with these duoprisms produces the inverted quasiprismatodishexadecachoron.

Vertex coordinates
Coordinates for the vertices of a small disprismatotesseractihexadecachoron with edge length 1 are given by all permutations of:
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12,\,±\frac12\right).$$

Representations
A small disprismatotesseractihexadecachoron has the following Coxeter diagrams:


 * x4o3o3x (full symmetry)
 * xxxx4oooo3oxxo&#xt (BC3 axial, cube-first)
 * xxxowoqo3ooqooqoo3oqowoxxx&#xt (A3 axial, tetrahedron-first)
 * xowqwox xxxxxxx4oxoxoxo&#xt (BC2×A1 symmetry, cube-first)
 * qo3oo3oq *b3xx&#zx (D4 symmetry)
 * wx xx4oo3ox&#zx (BC3×A1 symmetry)
 * xo4xx ox4xx&#zx (BC3×BC2 symmetry, square duoexpandoprism)
 * wxxx xwxx xxwx xxxw&#zx (A1×A1×A1×A1 symmetry)

Semi-uniform variant
The small disprismatotesseractihexadecachoron has a semi-uniform variant of the form x4o3o3y that maintains its full symmetry. This variant uses 8 cubes of size x, 16 tetrahedra of size y, 32 semi-uniform triangular prisms of form x y3o, and 24 semi-uniform square prisms of form y x4o as cells, with 2 edge lengths.

With edges of length a (of cubes) and b (of tetrahedra), its circumradius is given by $$\sqrt{\frac{2a^2+b^2+ab\sqrt2}{2}}$$ and its hypervolume is given by $$\frac{6a^4+36a^2b^2+b^4+(24a^3b+8ab^3)\sqrt2}{6}$$.

It has coordinates given by all permutations of:


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

Variations
The uniform small disprismatotesseractihexadecachoron, and indeed the semi-uniform variants, are a special class of the general isogonal square duoexpandoprism. This generall variant has as cells 2 sets of 8 square prisms, 16 rectangular trapezoprisms, 32 wedges, and 16 tetragonal disphenoids as cells. They can generally be obtained as the convex hull of 2 orthogonal square-ditetragonal duoprisms.

This is one of a total of five polychora that can be obtained as the convex hull of two orthogonal square-ditetragonal duoprisms. To produce variants of this polychoron, if the polychoron is written as ao4bc oa4cb&#zy, c must be in the range $$c < b+\frac{a}{\sqrt2}$$. It generally has circumradius $$\sqrt{\frac{2a^2+2b^2+2ab\sqrt2+c^2}{2}}$$.

Related polychora
The small dispirsmatotesseractihexadecachoron is the colonel of a regiment with a total of 8 uniform, 1 uniform compound, and 6 scaliform members. Of these members, 6 have full tesseractic symmetry, namely the small tesseractifaceted cubitesseractihexadecachoron, the small hexadecafaceted prismatotesseractioctachoron, the small tesseractitesseractihexadecachoron, the disprismatotesseract, the small spinoprismatotesseractihexadecachoron, and the [small spinoprismatotesseractioctachoron. The eighth member, the inverted quasiprismatodishexadecachoron, has square duoprism symmetry, as does the compound octagonal diorthoprismand the 6 scaliform members.

A small disprismatotesseractihexadecachoron can be constructed from a small rhombicuboctahedral prism by attaching cube atop small rhombicuboctahedron segmentochora to its bases. The cap from the set of 24 cubes is the square cupofastegium.