Small dodecicosidodecahedron

Small dodecicosidodecahedron
File:Small dodecicosidodecahedron.png
Type	Uniform star polyhedron
Elements	F = 44, E = 120 V = 60 (χ = −16)
Faces by sides	20{3}+12{5}+12{10}
Coxeter diagram	Template:CDD
Wythoff symbol	5 3 5/4 | 5
Symmetry group	Ih, [5,3], *532
Index references	U33, C42, W72
Dual polyhedron	Small dodecacronic hexecontahedron
Vertex figure	File:Small dodecicosidodecahedron vertfig.png 5.10.3/2.10
Bowers acronym	Saddid

File:Small dodecicosidodecahedron.stl

3D model of a small dodecicosidodecahedron

In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U₃₃. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.

Related polyhedra

It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the triangular and pentagonal faces in common), and with the small rhombidodecahedron (having the decagonal faces in common).

File:Small rhombicosidodecahedron.png Rhombicosidodecahedron	File:Small dodecicosidodecahedron.png Small dodecicosidodecahedron	File:Small rhombidodecahedron.png Small rhombidodecahedron
File:Small stellated truncated dodecahedron.png Small stellated truncated dodecahedron	File:UC36-6 pentagrammic prisms.png Compound of six pentagrammic prisms	File:UC37-12 pentagrammic prisms.png Compound of twelve pentagrammic prisms

Dual

File:Small dodecacronic hexecontahedron.stl

3D model of a small dodecacronic hexecontahedron

The dual polyhedron to the small dodecicosidodecahedron is the small dodecacronic hexecontahedron (or small sagittal ditriacontahedron). It is visually identical to the small rhombidodecacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models.

Proportions

Faces have two angles of $\arccos (\frac{5}{8}+\frac{1}{8}\sqrt{5})\approx 25.24283296152^{\circ }$, one of $\arccos (-\frac{1}{8}+\frac{9}{40}\sqrt{5})\approx 67.78301154744^{\circ }$ and one of $360^{\circ }-\arccos (-\frac{1}{4}-\frac{1}{10}\sqrt{5})\approx 241.73132252952^{\circ }$. Its dihedral angles equal $\arccos (\frac{-19-8\sqrt{5}}{41})\approx 154.12136312578^{\circ }$. The ratio between the lengths of the long and short edges is $\frac{7+\sqrt{5}}{6}\approx 1.53934466292$.

External links

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