Great snub dodecicosidodecahedron

The great snub dodecicosidodecahedron, or gisdid, is a uniform polyhedron. It consists of 60 snub triangles, 20 more triangles, and 24 pentagrams that fall in coplanar pairs of one prograde, one retrograde. Four triangles and two pentagrams meet at each vertex.

Great snub dodecicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Gisdid
Coxeter diagram	s5/3s5/2s3*a ()
Elements
Faces	20+60 triangles, 24 pentagrams
Edges	60+60+60
Vertices	60
Vertex figure	Irregular hexagon, edge lengths 1, 1, 1, (√5–1)/2, 1, (√5–1)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt2}{2} ≈ 0.70711$
Volume	$\frac{5\sqrt2}{3} ≈ 2.35702$
Dihedral angles	5/2–3 #1: $\arccos\left(-\sqrt{\frac{5+2\sqrt5-4\sqrt{5\sqrt5-10}}{15}}\right) ≈ 125.77490^\circ$
	3–3: $\arccos\left(-\frac13\right) ≈ 109.47122^\circ$
	5/2–3 #2: $\arccos\left(\sqrt{\frac{5+2\sqrt5+4\sqrt{5\sqrt5-10}}{15}}\right) ≈ 16.30368^\circ$
Central density	10
Number of external pieces	660
Level of complexity	42
Related polytopes
Army	Semi-uniform srid, edge lengths $\sqrt{\frac{3-\sqrt5-\sqrt{10\sqrt5-22}}{2}}$ (pentagons), $\sqrt{\frac{\sqrt5-1-2\sqrt{\sqrt5-2}}{2}}$ (triangles)
Regiment	Gisdid
Dual	Great hexagonal hexecontahedron
Conjugate	Great snub dodecicosidodecahedron
Abstract & topological properties
Flag count	720
Euler characteristic	-16
Orientable	Yes
Genus	9
Properties
Symmetry	H3+, order 60
Convex	No
Nature	Tame

It is the only chiral uniform polyhedron with an achiral convex hull. As such, it cannot be made into a compound with its reflection. If the pentagrams are removed, however, the disnub icosahedron is formed.

This polyhedron's edges are a subset of those of the great dirhombicosidodecahedron, and it shares the same vertices.

Vertex coordinates

A great snub dodecicosidodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

• $\left(±\sqrt{\frac{\sqrt5-1-2\sqrt{\sqrt5-2}}{8}},\,±\sqrt{\frac{3-\sqrt5-\sqrt{10\sqrt5-22}}{8}},\,±\sqrt{\frac{2+\sqrt{2\sqrt5-2}}{8}}\right),$
• $\left(0,\,±\frac{\sqrt{3-\sqrt5}}{2},\,±\frac{\sqrt{\sqrt5-1}}{2}\right),$
• $\left(±\sqrt{\frac{3-\sqrt5+\sqrt{10\sqrt5-22}}{8}},\,±\sqrt{\frac{2-\sqrt{2\sqrt5-2}}{8}},\,±\sqrt{\frac{\sqrt5-1+2\sqrt{\sqrt5-2}}{8}}\right).$

Related polyhedra

o5/3o5/2o3*a truncations

Name	OBSA	CD diagram	Picture
Great complex icosidodecahedron (degenerate, sissid+gike)	gacid	x5/3o5/2o3*a ()
Great dodecicosidodecahedron	gaddid	x5/3x5/2o3*a ()
(degenerate, double cover of gissid)		o5/3x5/2o3*a ()
(degenerate, ditdid+gidtid)		o5/3x5/2x3*a ()
Great complex icosidodecahedron (degenerate, sissid+gike)	gacid	o5/3o5/2x3*a ()
(degenerate, double cover of sidhei)		x5/3o5/2x3*a ()
(degenerate, giddy+12(10/2))		x5/3x5/2x3*a ()
Great snub dodecicosidodecahedron	gisdid	s5/3s5/2s2*a ()

External links

• Bowers, Jonathan. "Polyhedron Category 6: Snubs" (#72).

• Klitzing, Richard. "gisdid".

• Wikipedia Contributors. "Great snub dodecicosidodecahedron".
• McCooey, David. "Great Snub Dodecicosidodecahedron"