Snub decachoron

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Snub decachoron
Rank4
TypeIsogonal
Notation
Bowers style acronymSnad
Coxeter diagrams3s3s3s ()
Elements
Cells60 phyllic disphenoids, 20 triangular gyroprisms, 10 snub tetrahedra
Faces120+120 scalene triangles, 20+40 triangles
Edges30+60+60+120
Vertices60
Vertex figurePolyhedron with 2 pentagons, 2 tetragons, and 4 triangles
Measures (as derived from unit-edge great prismatodecachoron)
Edge lengthsEdges from diagonals of original squares (30+60):
 Edges of equilateral triangles (60+120):
Circumradius
Central density1
Related polytopes
ArmySnad
RegimentSnad
DualEnneahedral hexecontachoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(A4×2)+, order 120
ConvexYes
NatureTame

The snub decachoron, or snad, also commonly called the omnisnub pentachoron or omnisnub 5-cell, is a convex isogonal polychoron that consists of 10 snub tetrahedra, 20 triangular gyroprisms, and 60 phyllic disphenoids. 2 snub tetrahedra, 2 triangular antiprisms, and 4 disphenoids join at each vertex. It can be obtained through the process of alternating the great prismatodecachoron. However, it cannot be made uniform, as it generally has 4 edge lengths, which can be minimized to no fewer than 2 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.21301.

This polychoron generally has a doubled symmetry. A variant without this extended symmetry also exists, known as the snub pentachoron.

Vertex coordinates[edit | edit source]

A snub decachoron formed directly from alternating a great prismatodecachoron of edge length 1 has coordinates in 5 dimensions given by all even permutations of:

An optimized snub decachoron using the absolute value method, where the phyllic disphenoids become rhombic disphenoids, is given by all even permutations of:

Finally, a variant optimized by the ratio method is given by all even permutations of:

External links[edit | edit source]