Snub dodecateron

Snub dodecateron
File:Snub dodecateron.png (OFF file)
Rank	5
Type	Isogonal
Notation
Bowers style acronym	Snod
Coxeter diagram	s3s3s3s3s
Elements
Tera	360 phyllic disphenoidal pyramids, 20 triangular duoantiprisms, 30 snub tetrahedral antiprisms, 12 snub pentachora
Cells	720+720 irregular tetrahedra, 360 phyllic disphenoids, 90 tetragonal disphenoids, 120+120+120 triangular gyroprisms, 60 snub tetrahedra, 30 pyritohedral icosahedra
Faces	720+720+720+720 scalene triangles, 360+360 isosceles triangles, 240+240 triangles
Edges	180+180+360+360+720+720
Vertices	360
Measures (based on uniform great cellidodecateron of edge lengt 1)
Edge lengths	Diagonals of squares (180+180+360+360): ${\sqrt {2}}\approx 1.41421$
	Edges of triangles (720+720): ${\sqrt {3}}\approx 1.73205$
Circumradius	${\sqrt {5}}\approx 2.23607$
Central density	1
Related polytopes
Dual	Tetradecachoric triacosihexecontachoron
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	(A5×2)+, order 720
Convex	Yes
Nature	Tame

The snub dodecateron or snod, also commonly called the omninsub hexateron or omnisnub 5-simplex, is a convex isogonal polyteron that consists of 12 snub pentachora, 30 snub tetrahedral antiprisms, 20 triangular duoantiprisms, and 360 phyllic disphenoidal pyramids. 2 snub pentachora, 2 snub tetrahedral antiprisms, 1 triangular duoantiprism, and 5 phyllic disphenoidal pyramids join at each vertex. It can be obtained through the process of alternating the great cellidodecateron. However, it cannot be made uniform.

Unlike the related icosahedron and snub decachoron, which both possess central inversion symmetry, this polyteron has no such symmetry.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {\sqrt {6}}{2}}$ ≈ 1:1.22474.

Vertex coordinates

The vertices of a snub dodecateron can be given in six dimensions for an optimized snub dodecateron, as all even permutations of:

• $\left(0,\,{\frac {1}{2}},\,{\frac {14+{\sqrt {238-34{\sqrt {17}}}}}{28}},\,{\frac {14+{\sqrt {266+42{\sqrt {37}}}}}{28}},\,{\frac {7+{\sqrt {182+21{\sqrt {37}}}}}{14}},\,{\frac {14+{\sqrt {182+21{\sqrt {37}}}}}{14}}\right),$

using the absolute value method, or:

• $\left(0,\,{\frac {1}{2}},\,1,\,{\frac {3}{2}},\,2,\,{\frac {5}{2}}\right),$

using the ratio method (corresponding to the alternation of the uniform great cellidodecateron).

External links

• Wikipedia contributors. "Full snub 5-simplex".
• Klitzing, Richard. "snod".