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User:Tomruen/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.

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 there are a total of 13 Catalan solids.

Name(s)
Coxeter-Dynkin diagram(s)
Picture
Solid
Picture
Transparent
Net Dual (Archimedean solids) Faces Edges Vertices Face polygon Symmetry
triakis tetrahedron
Triakis tetrahedron Triakis tetrahedron
(Animation)
truncated tetrahedron
12 18 8 isosceles triangle
V3.6.6
Td
rhombic dodecahedron

Rhombic dodecahedron Rhombic dodecahedron
(Animation)
cuboctahedron

12 24 14 rhombus
V3.4.3.4
Oh
triakis octahedron
Triakis octahedron Triakis octahedron
(Animation)
truncated cube
24 36 14 isosceles triangle
V3.8.8
Oh
tetrakis hexahedron
(or disdyakis hexahedron or hexakis tetrahedron)

Tetrakis hexahedron Tetrakis hexahedron
(Animation)
truncated octahedron

24 36 14 isosceles triangle
V4.6.6
Oh
deltoidal icositetrahedron
(or trapezoidal icositetrahedron)
Deltoidal icositetrahedron Deltoidal icositetrahedron
(Animation)
rhombicuboctahedron
24 48 26 kite
V3.4.4.4
Oh
disdyakis dodecahedron
(or hexakis octahedron)
Disdyakis dodecahedron Disdyakis dodecahedron
(Animation)
truncated cuboctahedron
48 72 26 scalene triangle
V4.6.8
Oh
pentagonal icositetrahedron
Pentagonal icositetrahedron Pentagonal icositetrahedron (Ccw)Pentagonal icositetrahedron (Cw)
(Anim.)(Anim.)
snub cube
24 60 38 irregular pentagon
V3.3.3.3.4
O
rhombic triacontahedron
Rhombic triacontahedron Rhombic triacontahedron
(Animation)
icosidodecahedron
30 60 32 rhombus
V3.5.3.5
Ih
triakis icosahedron
Triakis icosahedron Triakis icosahedron
(Animation)
truncated dodecahedron
60 90 32 isosceles triangle
V3.10.10
Ih
pentakis dodecahedron
Pentakis dodecahedron Pentakis dodecahedron
(Animation)
truncated icosahedron
60 90 32 isosceles triangle
V5.6.6
Ih
deltoidal hexecontahedron
(Or trapezoidal hexecontahedron)
Deltoidal hexecontahedron Deltoidal hexecontahedron
(Animation)
rhombicosidodecahedron
60 120 62 kite
V3.4.5.4
Ih
disdyakis triacontahedron
(or hexakis icosahedron)
Disdyakis triacontahedron Disdyakis triacontahedron
(Animation)
truncated icosidodecahedron
120 180 62 scalene triangle
V4.6.10
Ih
pentagonal hexecontahedron
Pentagonal hexecontahedron Pentagonal hexecontahedron (Ccw)Pentagonal hexecontahedron (Cw)
(Anim.)(Anim.)
snub dodecahedron
60 150 92 irregular pentagon
V3.3.3.3.5
I

See also

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References

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  • 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)
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