Pyritosnub tesseract

Pyritosnub tesseract
	(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Pysnet
Coxeter diagram	x4s3s3s ()
Elements
Cells	96 skewed wedges, 32 triangular prisms, 24 rectangular trapezoprisms, 16 snub tetrahedra, 8 pyritosnub cubes
Faces	192 scalene triangles, 64+64 triangles, 96+96 isosceles trapezoids, 48+96 rectangles
Edges	96+96+96+192+192
Vertices	192
Vertex figure	Polyhedron with 1 pentagon, 1 tetragon, and 5 triangles
Measures (as derived from unit-edge great disprismatotesseractihexadecachoron)
Edge lengths	Remaining edges from class being alternated (96): 1
	Edges from diagonals of original squares (96):
	Edges of equilateral triangles (192+192):
	Long edges of rectangles (96):
Circumradius
Central density	1
Related polytopes
Army	Pysnet
Regiment	Pysnet
Dual	Heptahedral hecatonenneacontadichoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B4/2, order 192
Convex	Yes
Nature	Tame

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[edit | edit source]

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:

$\left(±\frac{1+3\sqrt2}{2}\,±\frac{1+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$

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:

$\left(±\frac12,\,±1,\,±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4}\right).$

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

$\left(±\frac12,\,±c_1,\,±c_2,\,±c_3\right),$

where

$c_1=\text{root}(592x^6-400x^5-324x^4+80x^3+80x^2+16x+1,\ 3) ≈ 0.7451616366591140373440626,$
$c_2=\text{root}(2368x^6-3392x^5+160x^4+384x^3-56x^2-16x+1,\ 4) ≈ 1.2970597497521540982365781,$
$c_3=\text{root}(2368x^6-3264x^5-2848x^4+288x^3+56x^2-112x+1,\ 4) ≈ 1.9603061382916052138473956.$