Snub decachoron
Snub decachoron | |
---|---|
Rank | 4 |
Type | Isogonal |
Notation | |
Bowers style acronym | Snad |
Coxeter diagram | s3s3s3s () |
Elements | |
Cells | 60 phyllic disphenoids, 20 triangular gyroprisms, 10 snub tetrahedra |
Faces | 120+120 scalene triangles, 20+40 triangles |
Edges | 30+60+60+120 |
Vertices | 60 |
Vertex figure | Polyhedron with 2 pentagons, 2 tetragons, and 4 triangles |
Measures (as derived from unit-edge great prismatodecachoron) | |
Edge lengths | Edges from diagonals of original squares (30+60): |
Edges of equilateral triangles (60+120): | |
Circumradius | |
Central density | 1 |
Related polytopes | |
Army | Snad |
Regiment | Snad |
Dual | Enneahedral hexecontachoron |
Abstract & topological properties | |
Euler characteristic | 0 |
Orientable | Yes |
Properties | |
Symmetry | (A4×2)+, order 120 |
Convex | Yes |
Nature | Tame |
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]
- Bowers, Jonathan. "Pennic and Decaic Isogonals".
- Klitzing, Richard. "snad".
- Wikipedia contributors. "Full snub 5-cell".