Small disprismatotesseractihexadecachoron
Small disprismatotesseractihexadecachoron  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Sidpith 
Coxeter diagram  x4o3o3x () 
Elements  
Cells  16 tetrahedra, 32 triangular prisms, 8+24 cubes 
Faces  64 triangles, 48+96 squares 
Edges  96+96 
Vertices  64 
Vertex figure  Triangular antipodium, edge lengths 1 (small base), and √2 (large base and sides) 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Tet–3–trip: 150° 
Cube–4–trip:  
Cube–4–cube: 135°  
Central density  1 
Number of external pieces  80 
Level of complexity  8 
Related polytopes  
Army  Sidpith 
Regiment  Sidpith 
Dual  Triangularantitegmatic hexacontatetrachoron 
Conjugate  Quasidisprismatotesseractihexadecachoron 
Abstract & topological properties  
Flag count  3072 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  B_{4}, order 384 
Flag orbits  8 
Convex  Yes 
Nature  Tame 
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 16cell.
It is the third in the series of duoexpandoprisms formed as the convex hull of a compound of two perpendicular squareoctagonal duoprisms and the only uniform one. Blending it with these duoprisms produces the inverted quasiprismatodishexadecachoron. It additionally contains the vertices and edges of the small rhombicuboctahedral prism.
Gallery[edit  edit source]

B4 Coxeter plane projection

Wireframe

Net

Crosssection animation
Vertex coordinates[edit  edit source]
Coordinates for the vertices of a small disprismatotesseractihexadecachoron with edge length 1 are given by all permutations of:
 .
Representations[edit  edit source]
A small disprismatotesseractihexadecachoron has the following Coxeter diagrams:
 x4o3o3x () (full symmetry)
 xxxx4oooo3oxxo&#xt (B_{3} axial, cubefirst)
 xxxowoqo3ooqooqoo3oqowoxxx&#xt (A_{3} axial, tetrahedronfirst)
 xowqwox xxxxxxx4oxoxoxo&#xt (B_{2}×A_{1} symmetry, cubefirst)
 qo3oo3oq *b3xx&#zx (D_{4} symmetry)
 wx xx4oo3ox&#zx (B_{3}×A_{1} symmetry)
 xo4xx ox4xx&#zx (B_{3}×B_{2} symmetry, square duoexpandoprism)
 wxxx xwxx xxwx xxxw&#zx (K_{4} symmetry)
Semiuniform variant[edit  edit source]
The small disprismatotesseractihexadecachoron has a semiuniform variant of the form x4o3o3y that maintains its full symmetry. This variant uses 8 cubes of size x, 16 tetrahedra of size y, 32 semiuniform triangular prisms of form x y3o, and 24 semiuniform 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 and its hypervolume is given by .
It has coordinates given by all permutations of:
 .
Variations[edit  edit source]
The uniform small disprismatotesseractihexadecachoron, and indeed the semiuniform variants, are a special class of the general isogonal square duoexpandoprism. This general variant has 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 squareditetragonal duoprisms.
This is one of a total of five polychora that can be obtained as the convex hull of two orthogonal squareditetragonal duoprisms. To produce variants of this polychoron, if the polychoron is written as ao4bc oa4cb&#zy, c must be in the range . It generally has circumradius .
Related polychora[edit  edit source]
The small disprismatotesseractihexadecachoron is the colonel of a regiment with a total of 8 uniform and 6 scaliform members. Of these members, 7 have full tesseractic symmetry, namely sidpith, stefacoth, shafipto, steth, dippit, snappoth, and snipto. The eighth uniform member iquipadah and the 6 scaliform members have square duoprism symmetry. The compound octagonal diorthoprism can be edgeinscribed in this polychoron.
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.
Uniform polychoron compounds composed of small disprismatotesseractihexadecachora include:
 Small prismated great icositetrachoron (3)
 Small prismated great stellated tetracontoctachoron (6)
 Small prismated dodecahedronary cubichoron (75)
External links[edit  edit source]
 Bowers, Jonathan. "Category 11: Antipodiumverts" (#447).
 Bowers, Jonathan. "Tessic Isogonals".
 Klitzing, Richard. "sidpith".
 Quickfur. "The Runcinated Tesseract".
 Wikipedia contributors. "Runcinated tesseract".