Catalan solid

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A rhombic dodecahedron

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.

Name(s)
(Dual name)
Conway name
Pictures Orthogonal
wireframes
Faces Edges Vertices Face polygon Symmetry
triakis tetrahedron
(truncated tetrahedron)
"kT"
Triakis tetrahedronTriakis tetrahedron Dual tetrahedron t01.pngDual tetrahedron t01 A2.png 12 18 8 isosceles triangle
V3.6.6
Td
rhombic dodecahedron
(cuboctahedron)
"jC"
Rhombic dodecahedronRhombic dodecahedron 12 24 14 rhombus
V3.4.3.4
Oh
triakis octahedron
(truncated cube)
"kO"
Triakis octahedronTriakis octahedron Dual truncated cube t01.pngDual truncated cube t01 B2.png 24 36 14 isosceles triangle
V3.8.8
Oh
tetrakis hexahedron
(truncated octahedron)
"kC"
Tetrakis hexahedronTetrakis hexahedron Dual cube t12.pngDual cube t12 B2.png 24 36 14 isosceles triangle
V4.6.6
Oh
deltoidal icositetrahedron
(rhombicuboctahedron)
"oC"
Deltoidal icositetrahedronDeltoidal icositetrahedron 24 48 26 kite
V3.4.4.4
Oh
disdyakis dodecahedron
(truncated cuboctahedron)
"mC"
Disdyakis dodecahedronDisdyakis dodecahedron 48 72 26 scalene triangle
V4.6.8
Oh
pentagonal icositetrahedron
(snub cube)
"gC"
Pentagonal icositetrahedronPentagonal icositetrahedron (Ccw) 24 60 38 irr. pentagon
V3.3.3.3.4
O
rhombic triacontahedron
(icosidodecahedron)
"jD"
Rhombic triacontahedronRhombic triacontahedron 30 60 32 rhombus
V3.5.3.5
Ih
triakis icosahedron
(truncated dodecahedron)
"kI"
Triakis icosahedronTriakis icosahedron 60 90 32 isosceles triangle
V3.10.10
Ih
pentakis dodecahedron
(truncated icosahedron)
"kD"
Pentakis dodecahedronPentakis dodecahedron 60 90 32 isosceles triangle
V5.6.6
Ih
deltoidal hexecontahedron
(rhombicosidodecahedron)
"oD"
Deltoidal hexecontahedronDeltoidal hexecontahedron 60 120 62 kite
V3.4.5.4
Ih
disdyakis triacontahedron
(truncated icosidodecahedron)
"mD"
Disdyakis triacontahedronDisdyakis triacontahedron 120 180 62 scalene triangle
V4.6.10
Ih
pentagonal hexecontahedron
(snub dodecahedron)
"gD"
Pentagonal hexecontahedronPentagonal hexecontahedron (Ccw) 60 150 92 irr. pentagon
V3.3.3.3.5
I

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