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Catalan solid

13 polyhedra; duals of the Archimedean solids

Summary

13 polyhedra; duals of the Archimedean solids

The Catalan solids are the dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to the vertices of Archimedean solids, and vice versa.

Description

The Catalan solids are face-transitive or isohedral, meaning that their faces are symmetric to one another, but they are not vertex-transitive because their vertices are not symmetric. Their duals, the Archimedean solids, are vertex-transitive but not face-transitive. Each Catalan solid has constant dihedral angles, meaning the angle between any two adjacent faces is the same. Additionally, two Catalan solids, the rhombic dodecahedron and rhombic triacontahedron, are edge-transitive, meaning their edges are symmetric to each other. Some Catalan solids were discovered by Johannes Kepler during his study of zonohedra, and Eugene Catalan completed the list of the thirteen solids in 1865.{{multiref | |

The [[rhombic dodecahedron]]'s construction, the dual polyhedron of a [[cuboctahedron]], by [[Dorman Luke construction

In general, each face of a dual uniform polyhedron (including the Catalan solid) can be constructed by using the Dorman Luke construction.{{multiref | | | |

Two Catalan solids, the pentagonal icositetrahedron and the pentagonal hexecontahedron, are chiral, meaning that these two solids are not their own mirror images. They are dual to the snub cube and snub dodecahedron, respectively, which are also chiral.

Eleven of the thirteen Catalan solids are known to have the Rupert property, which means that a copy of the same solid can be passed through a hole in the solid.

Name	Image	Faces	Edges	Vertices	Dihedral angle	Point group
triakis tetrahedron	[[Image:triakistetrahedron.svg	70px	Triakis tetrahedron]]	12 isosceles triangles	18	8	129.521°	Td
rhombic dodecahedron	[[Image:rhombicdodecahedron.jpg	70px	Rhombic dodecahedron]]	12 rhombi	24	14	120°	Oh
triakis octahedron	[[Image:triakisoctahedron.jpg	70px	Triakis octahedron]]	24 isosceles triangles	36	14	147.350°	Oh
tetrakis hexahedron	[[Image:tetrakishexahedron.jpg	70px	Tetrakis hexahedron]]	24 isosceles triangles	36	14	143.130°	Oh
deltoidal icositetrahedron	[[Image:deltoidalicositetrahedron.jpg	70px	Deltoidal icositetrahedron]]	24 kites	48	26	138.118°	Oh
disdyakis dodecahedron	[[Image:disdyakisdodecahedron.jpg	70px	Disdyakis dodecahedron]]	48 scalene triangles	72	26	155.082°	Oh
pentagonal icositetrahedron	[[Image:pentagonalicositetrahedronccw.jpg	70px	Pentagonal icositetrahedron (Ccw)]]	24 pentagons	60	38	136.309°	O
rhombic triacontahedron	[[Image:rhombictriacontahedron.svg	70px	Rhombic triacontahedron]]	30 rhombi	60	32	144°	Ih
triakis icosahedron	[[Image:triakisicosahedron.jpg	70px	Triakis icosahedron]]	60 isosceles triangles	90	32	160.613°	Ih
pentakis dodecahedron	[[Image:pentakisdodecahedron.jpg	70px	Pentakis dodecahedron]]	60 isosceles triangles	90	32	156.719°	Ih
deltoidal hexecontahedron	[[Image:deltoidalhexecontahedron.jpg	70px	Deltoidal hexecontahedron]]	60 kites	120	62	154.121°	Ih
disdyakis triacontahedron	[[Image:disdyakistriacontahedron.jpg	70px	Disdyakis triacontahedron]]	120 scalene triangles	180	62	164.888°	Ih
pentagonal hexecontahedron	[[Image:pentagonalhexecontahedronccw.jpg	70px	Pentagonal hexecontahedron (Ccw)]]	60 pentagons	150	92	153.179°	I

References

Footnotes

Works cited

{{citation | editor1-last = Emmer | editor1-first = Michele | editor2-last = Abate | editor2-first = Marco | hdl-access = free
{{citation | doi-broken-date = 1 July 2025
{{citation | title-link = Mathematical Models (Cundy and Rollett)
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{{citation | editor-last1 = Gruber | editor-first = P. M. | editor-last2 = Wills | editor-first2 = J. M.
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(Section 3-9)

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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