Catalan solid

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A rhombic dodecahedron with its face configuration
The disdyakis triacontahedron, with face configuration V4.6.10, is the largest catalan solid with 120 faces.

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids 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. 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
Uniform polyhedron-33-t01.png Uniform polyhedron-33-t02.png Uniform polyhedron-33-t012.png
Catalans
Triakistetrahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg
Octahedral symmetry
Archimedean
Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg Uniform polyhedron-43-t12.svg Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png
Catalans
Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg
Icosahedral symmetry
Archimedean
Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.svg Uniform polyhedron-53-t12.svg Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
Catalans
Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg

List[edit]

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

See also[edit]

References[edit]

External links[edit]