Catalan solid

Triakis tetrahedron, pentagonal icositetrahedron and disdyakis triacontahedron. The first and the last one can be described as the smallest and the biggest Catalan solid.

The solids above (dark) shown together with their duals (light). The visible parts of the Catalan solids are regular pyramids.
A rhombic dodecahedron with its face configuration.


In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865.

The Catalan solids are all convex. They are face-transitive but not vertex-transitive. This is because the dual Archimedean solids are vertex-transitive and not face-transitive. Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons. However, the vertex figures of Catalan solids are regular, and they have constant dihedral angles. Being face-transitive, Catalan solids are isohedra.

Additionally, two of the Catalan solids are edge-transitive: the rhombic dodecahedron and the rhombic triacontahedron. These are the duals of the two quasi-regular Archimedean solids.

Just as prisms and antiprisms are generally not considered Archimedean solids, so bipyramids and trapezohedra are generally not considered Catalan solids, despite being face-transitive.

Two of the Catalan solids are chiral: the pentagonal icositetrahedron and the pentagonal hexecontahedron, dual to the chiral snub cube and snub dodecahedron. These each come in two enantiomorphs. Not counting the enantiomorphs, bipyramids, and trapezohedra, there are a total of 13 Catalan solids.

nArchimedean solidCatalan solid
1 truncated tetrahedrontriakis tetrahedron
2 truncated cubetriakis octahedron
3 truncated cuboctahedrondisdyakis dodecahedron
4 truncated octahedrontetrakis hexahedron
5 truncated dodecahedrontriakis icosahedron
6 truncated icosidodecahedrondisdyakis triacontahedron
7 truncated icosahedronpentakis dodecahedron
8 cuboctahedronrhombic dodecahedron
9 icosidodecahedronrhombic triacontahedron
10 rhombicuboctahedrondeltoidal icositetrahedron
11 rhombicosidodecahedrondeltoidal hexecontahedron
12 snub cubepentagonal icositetrahedron
13 snub dodecahedronpentagonal hexecontahedron

Symmetry

The Catalan solids, along with their dual Archimedean solids, can be grouped by their symmetry: tetrahedral, octahedral, and icosahedral. There are 6 forms per symmetry, while the self-symmetric tetrahedral group only has three unique forms and two of those are duplicated with octahedral symmetry.


Tetrahedral symmetry
Archimedean
Catalan
Octahedral symmetry
Archimedean
Catalan
Icosahedral symmetry
Archimedean
Catalan

List

Name
(Dual name)
Conway name		 Pictures Orthogonal
wireframes		 Face
polygon		 Faces Edges Vert. Sym.
triakis tetrahedron
(truncated tetrahedron)
"kT"		 Isosceles

V3.6.6		 12 18 8 Td
rhombic dodecahedron
(cuboctahedron)
"jC"		 Rhombus

V3.4.3.4		 12 24 14 Oh
triakis octahedron
(truncated cube)
"kO"		 Isosceles

V3.8.8		 24 36 14 Oh
tetrakis hexahedron
(truncated octahedron)
"kC"		 Isosceles

V4.6.6		 24 36 14 Oh
deltoidal icositetrahedron
(rhombicuboctahedron)
"oC"		 Kite

V3.4.4.4		 24 48 26 Oh
disdyakis dodecahedron
(truncated cuboctahedron)
"mC"		 Scalene

V4.6.8		 48 72 26 Oh
pentagonal icositetrahedron
(snub cube)
"gC"		 Pentagon

V3.3.3.3.4		 24 60 38 O
rhombic triacontahedron
(icosidodecahedron)
"jD"		 Rhombus

V3.5.3.5		 30 60 32 Ih
triakis icosahedron
(truncated dodecahedron)
"kI"		 Isosceles

V3.10.10		 60 90 32 Ih
pentakis dodecahedron
(truncated icosahedron)
"kD"		 Isosceles

V5.6.6		 60 90 32 Ih
deltoidal hexecontahedron
(rhombicosidodecahedron)
"oD"		 Kite

V3.4.5.4		 60 120 62 Ih
disdyakis triacontahedron
(truncated icosidodecahedron)
"mD"		 Scalene

V4.6.10		 120 180 62 Ih
pentagonal hexecontahedron
(snub dodecahedron)
"gD"		 Pentagon

V3.3.3.3.5		 60 150 92 I

See also

References

  • Eugène Catalan Mémoire sur la Théorie des Polyèdres. J. l'École Polytechnique (Paris) 41, 1-71, 1865.
  • Alan Holden Shapes, Space, and Symmetry. New York: Dover, 1991.
  • Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 (The thirteen semiregular convex polyhedra and their duals)
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
  • Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms
  • Weisstein, Eric W. "Catalan Solids". MathWorld.
  • Weisstein, Eric W. "Isohedron". MathWorld.
  • Archimedean duals – at Virtual Reality Polyhedra
  • Interactive Catalan Solid in Java
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