Hexagonal-snub dodecahedral duoprism

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Hexagonal-snub dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHasnid
Coxeter diagramx6o s5s3s
Elements
Tera20+60 triangular-hexagonal duoprisms, 12 pentagonal-hexagonal duoprisms, 6 snub dodecahedral prisms
Cells120+360 triangular prisms, 72 pentagonal prisms, 30+60+60 hexagonal prisms, 6 snub dodecahedra
Faces120+360 triangles, 180+360+360 squares, 72 pentagons, 60 hexagons
Edges180+360+360+360
Vertices360
Vertex figureMirror-symmetric pentagonal scalene, edge lengths 1, 1, 1, 1, (1+5)/2 (base pentagon), 3 (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius≈ 2.37648
Hypervolume≈ 97.73092
Diteral anglesThiddip–hip–thiddip: ≈ 164.17537°
 Thiddip–hip–phiddip: ≈ 152.92992°
 Sniddip–snid–sniddip: 120°
 Thiddip–trip–sniddip: 90°
 Phiddip–pip–sniddip: 90°
Central density1
Number of external pieces98
Level of complexity50
Related polytopes
ArmyHasnid
RegimentHasnid
DualHexagonal-pentagonal hexecontahedral duotegum
ConjugatesHexagonal-great snub icosidodecahedral duoprism, Hexagonal-great inverted snub icosidodecahedral duoprism, Hexagonal-great inverted retrosnub icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3+×G2, order 720
ConvexYes
NatureTame

The hexagonal-snub dodecahedral duoprism or hasnid is a convex uniform duoprism that consists of 6 snub dodecahedral prisms, 12 pentagonal-hexagonal duoprisms, and 80 triangular-hexagonal duoprisms of two kinds. Each vertex joins 2 snub dodecahedral prisms, 4 triangular-hexagonal duoprisms, and 1 pentagonal-hexagonal duoprism.

Vertex coordinates[edit | edit source]

The vertices of a hexagonal-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:

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

where