# Pyritosnub tesseract

Pyritosnub tesseract
Rank4
TypeIsogonal
Notation
Bowers style acronymPysnet
Coxeter diagramx4s3s3s ()
Elements
Cells96 skewed wedges, 32 triangular prisms, 24 rectangular trapezoprisms, 16 snub tetrahedra, 8 pyritosnub cubes
Faces192 scalene triangles, 64+64 triangles, 96+96 isosceles trapezoids, 48+96 rectangles
Edges96+96+96+192+192
Vertices192
Vertex figurePolyhedron with 1 pentagon, 1 tetragon, and 5 triangles
Measures (as derived from unit-edge great disprismatotesseractihexadecachoron)
Edge lengthsRemaining edges from class being alternated (96): 1
Edges from diagonals of original squares (96): ${\displaystyle {\sqrt {2}}\approx 1.41421}$
Edges of equilateral triangles (192+192): ${\displaystyle {\sqrt {3}}\approx 1.73205}$
Long edges of rectangles (96): ${\displaystyle 1+{\sqrt {2}}\approx 2.41421}$
Circumradius${\displaystyle {\sqrt {8+3{\sqrt {2}}}}\approx 3.49895}$
Central density1
Related polytopes
ArmyPysnet
RegimentPysnet
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB4/2, order 192
ConvexYes
NatureTame

The pyritosnub tesseract or pysnet, also known as the edge-snub hexadecachoron, is a convex isogonal polychoron that consists of 8 pyritosnub cubes, 16 snub tetrahedra, 24 rectangular trapezoprisms, 32 triangular prisms, and 96 skewed wedges. 3 wedges and one of each of the other 4 types of cells join at each vertex. It can be obtained through the process of alternating one set of 192 edges of the great disprismatotesseractihexadecachoron in such a way that the octagonal faces turn into rectangles. However, it cannot be made uniform, as it generally has 5 different edge lengths, which can be minimized to no more than 2 different sizes.

A variant with 8 regular icosahedra and 32 uniform triangular prisms can be vertex-inscribed into a prismatorhombisnub icositetrachoron.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:a ≈ 1:1.49032, where a is the second largest real root of 37x6-50x5-81x4+40x3+80x2+32x+4.

## Vertex coordinates

Vertex coordinates for a pyritosnub tesseract, created from the vertices of a great disprismatotesseractihexadecachoron of edge length 1, are given by all even permutations of:

• ${\displaystyle \left(\pm {\frac {1+3{\sqrt {2}}}{2}},\pm {\frac {1+2{\sqrt {2}}}{2}},\pm {\frac {1+{\sqrt {2}}}{2}},\pm {\frac {1}{2}}\right).}$

A variant using regular icosahedra and uniform triangular prisms of edge length 1, centered at the origin, has vertices given by all even permutations of:

• ${\displaystyle \left(\pm {\frac {1}{2}},\pm 1,\pm {\frac {3+{\sqrt {5}}}{4}},\pm {\frac {5+{\sqrt {5}}}{4}}\right).}$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by all even permutations of:

• ${\displaystyle \left(\pm {\frac {1}{2}},\pm c_{1},\pm c_{2},\pm c_{3}\right),}$

where

• ${\displaystyle c_{1}={\text{root}}(592x^{6}-400x^{5}-324x^{4}+80x^{3}+80x^{2}+16x+1,\ 3)\approx 0.7451616366591140373440626,}$
• ${\displaystyle c_{2}={\text{root}}(2368x^{6}-3392x^{5}+160x^{4}+384x^{3}-56x^{2}-16x+1,\ 4)\approx 1.2970597497521540982365781,}$
• ${\displaystyle c_{3}={\text{root}}(2368x^{6}-3264x^{5}-2848x^{4}+288x^{3}+56x^{2}-112x+1,\ 4)\approx 1.9603061382916052138473956.}$