Small dodecicosidodecahedron

Template:Short description Template:Uniform polyhedra db File:Small dodecicosidodecahedron.stl 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).

Rhombicosidodecahedron	Small dodecicosidodecahedron	Small rhombidodecahedron
Small stellated truncated dodecahedron	Compound of six pentagrammic prisms	Compound of twelve pentagrammic prisms

Dual

Template:Uniform polyhedra db File:Small dodecacronic hexecontahedron.stl 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 $\frac{7+\sqrt{5}}{6}\approx 1.53934466292$.

References

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External links

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