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Catalan solid

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Title: Catalan solid  
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Subject: Polyhedron, Conway polyhedron notation, Deltoidal hexecontahedron, Deltoidal icositetrahedron, Disdyakis dodecahedron
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Catalan solid

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. 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

Contents

  • Symmetry 1
  • List 2
  • See also 3
  • References 4
  • External links 5

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
Catalans
Octahedral symmetry
Archimedean
Catalans
Icosahedral symmetry
Archimedean
Catalans

List

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

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

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

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

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

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

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

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

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

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

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

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

V4.6.10
120 180 62 Ih
pentagonal hexecontahedron
(snub dodecahedron)
"gD"
Pentagonal hexecontahedronPentagonal hexecontahedron (Ccw) 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.
  • (The thirteen semiregular convex polyhedra and their duals)
  • (Section 3-9)
  • Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms

External links

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