# Small disnub dodecahedron

Small disnub dodecahedron Rank3
TypeUniform
SpaceSpherical
Notation
Elements
Components12 pentagrammic antiprisms
Faces120 triangles, 24 pentagrams as 12 stellated decagrams
Edges120+120
Vertices60
Vertex figureIsosceles trapezoid, edge length 1, 1, 1, (5–1)/2
Measures (edge length 1)
Circumradius$\sqrt{\frac{15+\sqrt5}{40}} \approx 0.65643$ Volume$2\sqrt{5\sqrt5} \approx 6.68740$ Dihedral angles5/2–3: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}3}\right) \approx 114.80110^\circ$ 3–3: $\arccos\left(\frac{2-\sqrt5}3\right) \approx 94.51323^\circ$ Central density24
Number of external pieces1380
Level of complexity82
Related polytopes
ArmySemi-uniform Grid
DualCompound of twelve pentagrammic antitegums
ConjugateSmall disnub dodecahedron
Convex coreDisdyakis triacontahedron
Abstract & topological properties
Flag count960
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The small disnub dodecahedron, sadsid, or compound of twelve pentagrammic antiprisms is a uniform polyhedron compound. It consists of 120 triangles and 24 pentagrams (which fall in pairs into the same planes combining into 12 stellated decagrams), with one pentagram and three triangles joining at a vertex.

It can be formed by combining the two chiral forms of the small snub dodecahedron.

Its quotient prismatic equivalent is the pentagrammic antiprismatic dodecadakoorthowedge, which is fourteen-dimensional.

## Vertex coordinates

The vertices of a small disnub dodecahedron of edge length 1 are given by all even permutations of:

• $\left(\pm\sqrt{\frac{\sqrt5+\sqrt{5(\sqrt5-2)}}{20}},\,\pm\sqrt{\frac{5-2\sqrt5}{20}},\,\pm\sqrt{\frac{5+\sqrt5-2\sqrt{10(\sqrt5-1)}}{40}}\right),$ • $\left(\pm\sqrt{\frac{3\sqrt5-5}{40}},\,\pm\sqrt{\frac{5-\sqrt5}{10}},\,\pm\frac{\sqrt{5\sqrt5}}{10}\right),$ • $\left(\pm\sqrt{\frac{2\sqrt5+\sqrt{10(\sqrt5-1)}}{20}},\,\pm\sqrt{\frac{5-2\sqrt5}{20}},\,\pm\sqrt{\frac{5+\sqrt5+2\sqrt{10(\sqrt5-1)}}{40}}\right),$ • $\left(\pm\sqrt{\frac{\sqrt5-\sqrt{5(\sqrt5-2)}}{20}},\,\pm\sqrt{\frac{5+\sqrt5}{40}},\,\pm\sqrt{\frac{5-\sqrt5+2\sqrt{15(\sqrt5-2)}}{20}}\right),$ • $\left(\sqrt{\frac{\sqrt5+2\sqrt{5(\sqrt5-2)}}{20}},\,\pm\sqrt{\frac{5+\sqrt5}{40}},\,\pm\sqrt{\frac{5-\sqrt5-2\sqrt{5(\sqrt5-2)}}{20}}\right).$ 