Catalan solid

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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.


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.

n Archimedean solid Catalan solid
1 truncated tetrahedron triakis tetrahedron
2 truncated cube triakis octahedron
3 truncated cuboctahedron disdyakis dodecahedron
4 truncated octahedron tetrakis hexahedron
5 truncated dodecahedron triakis icosahedron
6 truncated icosidodecahedron disdyakis triacontahedron
7 truncated icosahedron pentakis dodecahedron
8 cuboctahedron rhombic dodecahedron
9 icosidodecahedron rhombic triacontahedron
10 rhombicuboctahedron deltoidal icositetrahedron
11 rhombicosidodecahedron deltoidal hexecontahedron
12 snub cube pentagonal icositetrahedron
13 snub dodecahedron pentagonal hexecontahedron

Symmetry[edit]

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 Polyhedron truncated 4a max.png Polyhedron small rhombi 4-4 max.png Polyhedron great rhombi 4-4 max.png
Catalan Polyhedron truncated 4a dual max.png Polyhedron small rhombi 4-4 dual max.png Polyhedron great rhombi 4-4 dual max.png
Octahedral symmetry
Archimedean Polyhedron 6-8 max.png Polyhedron truncated 6 max.png Polyhedron truncated 8 max.png Polyhedron small rhombi 6-8 max.png Polyhedron great rhombi 6-8 max.png Polyhedron snub 6-8 left max.png
Catalan Polyhedron 6-8 dual blue.png Polyhedron truncated 6 dual.png Polyhedron truncated 8 dual max.png Polyhedron small rhombi 6-8 dual max.png Polyhedron great rhombi 6-8 dual max.png Polyhedron snub 6-8 left dual max.png
Icosahedral symmetry
Archimedean Polyhedron 12-20 max.png Polyhedron truncated 12 max.png Polyhedron truncated 20 max.png Polyhedron small rhombi 12-20 max.png Polyhedron great rhombi 12-20 max.png Polyhedron snub 12-20 left max.png
Catalan Polyhedron 12-20 dual max.png Polyhedron truncated 12 dual max.png Polyhedron truncated 20 dual max.png Polyhedron small rhombi 12-20 dual max.png Polyhedron great rhombi 12-20 dual max.png Polyhedron snub 12-20 left dual max.png

List[edit]

Name
(Dual name)
Conway name
Pictures Orthogonal
wireframes
Face
polygon
Faces Edges Vert. Sym.
triakis tetrahedron
(truncated tetrahedron)
"kT"
Triakis tetrahedronTriakis tetrahedron Dual tetrahedron t01 ae.pngDual tetrahedron t01 A2.pngDual tetrahedron t01.png Isosceles
DU02 facets.png
V3.6.6
12 18 8 Td
rhombic dodecahedron
(cuboctahedron)
"jC"
Rhombic dodecahedronRhombic dodecahedron Dual cube t1 v.png Dual cube t1.pngDual cube t1 B2.png Rhombus
DU07 facets.png
V3.4.3.4
12 24 14 Oh
triakis octahedron
(truncated cube)
"kO"
Triakis octahedronTriakis octahedron Dual truncated cube t01 e88.pngDual truncated cube t01.pngDual truncated cube t01 B2.png Isosceles
DU09 facets.png
V3.8.8
24 36 14 Oh
tetrakis hexahedron
(truncated octahedron)
"kC"
Tetrakis hexahedronTetrakis hexahedron Dual cube t12 e66.pngDual cube t12.pngDual cube t12 B2.png Isosceles
DU08 facets.png
V4.6.6
24 36 14 Oh
deltoidal icositetrahedron
(rhombicuboctahedron)
"oC"
Deltoidal icositetrahedronDeltoidal icositetrahedron Dual cube t02 f4b.pngDual cube t02.pngDual cube t02 B2.png Kite
DU10 facets.png
V3.4.4.4
24 48 26 Oh
disdyakis dodecahedron
(truncated cuboctahedron)
"mC"
Disdyakis dodecahedronDisdyakis dodecahedron Dual cube t012 f4.pngDual cube t012.pngDual cube t012 B2.png Scalene
DU11 facets.png
V4.6.8
48 72 26 Oh
pentagonal icositetrahedron
(snub cube)
"gC"
Pentagonal icositetrahedronPentagonal icositetrahedron (Ccw) Dual snub cube e1.pngDual snub cube A2.pngDual snub cube B2.png Pentagon
DU12 facets.png
V3.3.3.3.4
24 60 38 O
rhombic triacontahedron
(icosidodecahedron)
"jD"
Rhombic triacontahedronRhombic triacontahedron Dual dodecahedron t1 e.pngDual dodecahedron t1 A2.pngDual dodecahedron t1 H3.png Rhombus
DU24 facets.png
V3.5.3.5
30 60 32 Ih
triakis icosahedron
(truncated dodecahedron)
"kI"
Triakis icosahedronTriakis icosahedron Dual dodecahedron t12 exx.pngDual dodecahedron t12 A2.pngDual dodecahedron t12 H3.png Isosceles
DU26 facets.png
V3.10.10
60 90 32 Ih
pentakis dodecahedron
(truncated icosahedron)
"kD"
Pentakis dodecahedronPentakis dodecahedron Dual dodecahedron t01 e66.pngDual dodecahedron t01 A2.pngDual dodecahedron t01 H3.png Isosceles
DU25 facets.png
V5.6.6
60 90 32 Ih
deltoidal hexecontahedron
(rhombicosidodecahedron)
"oD"
Deltoidal hexecontahedronDeltoidal hexecontahedron Dual dodecahedron t02 f4.pngDual dodecahedron t02 A2.pngDual dodecahedron t02 H3.png Kite
DU27 facets.png
V3.4.5.4
60 120 62 Ih
disdyakis triacontahedron
(truncated icosidodecahedron)
"mD"
Disdyakis triacontahedronDisdyakis triacontahedron Dual dodecahedron t012 f4.pngDual dodecahedron t012 A2.pngDual dodecahedron t012 H3.png Scalene
DU28 facets.png
V4.6.10
120 180 62 Ih
pentagonal hexecontahedron
(snub dodecahedron)
"gD"
Pentagonal hexecontahedronPentagonal hexecontahedron (Ccw) Dual snub dodecahedron e1.pngDual snub dodecahedron A2.pngDual snub dodecahedron H2.png Pentagon
DU29 facets.png
V3.3.3.3.5
60 150 92 I

See also[edit]

References[edit]

  • 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

External links[edit]