Small dodecicosidodecahedron explained

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.^[1] 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).

Dual

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(	5	+
	8

	1
	8

\sqrt{5}) ≈ 25.24283296152^\circ

, one of

\arccos(-	1	+
	8

	9
	40

\sqrt{5}) ≈ 67.78301154744^\circ

and one of

360^\circ-\arccos(-

	1	-
	4

	1
	10

\sqrt{5}) ≈ 241.73132252952^\circ

. Its dihedral angles equal

\arccos({	-19-8\sqrt{5

}})\approx 154.121\,363\,125\,78^. The ratio between the lengths of the long and short edges is

	7+\sqrt{5

}\approx 1.539\,344\,662\,92.

References

Coxeter. H. S. M.. Harold Scott MacDonald Coxeter. Uniform Polyhedra. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 246. 916. May 13, 1954. 401–450. 10.1098/rsta.1954.0003.
Book: Wenninger, Magnus . Magnus Wenninger

. Magnus Wenninger . Polyhedron Models . Cambridge University Press . 1974 . 0-521-09859-9 . 1738087 .

Notes and References

Web site: 33: small dodecicosidodecahedron. Maeder. Roman. MathConsult.