Square-snub dodecahedral duoprism

Square-snub dodecahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Squasnid
Coxeter diagram	x4o s5s3s
Elements
Tera	20+60 triangular-square duoprisms, 12 square-pentagonal duoprisms, 4 snub dodecahedral prisms
Cells	80+240 triangular prisms, 30+60+60 cubes, 48 pentagonal prisms, 4 snub dodecahedra
Faces	80+240 triangles, 60+120+240+240 squares, 48 pentagons
Edges	120+240+240+240
Vertices	240
Vertex figure	Mirror-symmetric pentagonal scalene, edge lengths 1, 1, 1, 1, (1+√5)/2 (base pentagon), √2 (top and side edges)
Measures (edge length 1)
Circumradius	≈ 2.26884
Hypervolume	≈ 37.61665
Diteral angles	Tisdip–cube–tisdip: ≈ 164.17537°
	Tisdip–cube–squipdip: ≈ 152.92992°
	Sniddip–snid–sniddip: 90°
	Tisdip–trip–sniddip: 90°
	Squipdip–pip–sniddip: 90°
Height	1
Central density	1
Number of external pieces	96
Level of complexity	50
Related polytopes
Army	Squasnid
Regiment	Squasnid
Dual	Square-pentagonal hexecontahedral duotegum
Conjugates	Square-great snub icosidodecahedral duoprism, Square-great inverted snub icosidodecahedral duoprism, Square-great inverted retrosnub icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H3+×B2, order 480
Convex	Yes
Nature	Tame

The square-snub dodecahedral duoprism or squasnid is a convex uniform duoprism that consists of 4 snub dodecahedral prisms, 12 square-pentagonal duoprisms, and 80 triangular-square duoprisms of two kinds. Each vertex joins 2 snub dodecahedral prisms, 4 triangular-square duoprisms, and 1 square-pentagonal duoprism. It is a duoprism based on a square and a snub dodecahedron, which makes it a convex segmentoteron.

Vertex coordinates

The vertices of a square-snub dodecahedral duoprism of edge length 1 are given by all even permutations with an odd number of sign changes of the last three coordinates of:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\xi \phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +\phi )+1}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\phi {\sqrt {\xi (\xi +1)}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\xi ^{2}\phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi {\sqrt {\xi +1-\phi }}}{2}},\,{\frac {\sqrt {\xi ^{2}(1+2\phi )-\phi }}{2}}\right),$

as well as all even permutations with an even number of sign changes of the last three coordinates of:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\xi ^{2}\phi {\sqrt {3-\xi ^{2}}}}{2}},\,{\frac {\xi \phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}}{2}},\,{\frac {\phi ^{2}{\sqrt {\xi (\xi +\phi )+1}}}{2\xi }}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt {\phi (\xi +2)+2}}{2}},\,{\frac {\phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}}{2}},\,{\frac {\xi {\sqrt {\xi (1+\phi )-\phi }}}{2}}\right),$

where

• $\phi ={\frac {1+{\sqrt {5}}}{2}},$
• $\xi ={\sqrt[{3}]{\frac {\phi +{\sqrt {\phi -{\frac {5}{27}}}}}{2}}}+{\sqrt[{3}]{\frac {\phi -{\sqrt {\phi -{\frac {5}{27}}}}}{2}}}.$

External links

• Klitzing, Richard. "squasnid".