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 in those with tetrahedral, octahedral and icosahedral symmetry. For both octahedral and icosahedral symmetry there are six forms. The only Catalan solid with genuine tetrahedral symmetry is the triakis tetrahedron (dual of the truncated tetrahedron). Rhombic dodecahedron and tetrakis hexahedron have octahedral symmetry, but they can be colored to have only tetrahedral symmetry. Rectification and snub also exist with tetrahedral symmetry, but they are Platonic instead of Archimedean, so their duals are Platonic instead of Catalan. (They are shown with brown background in the table below.)

Tetrahedral symmetry
Archimedean
(Platonic)
Polyhedron 4-4.png Polyhedron truncated 4a max.png Polyhedron truncated 4b max.png Polyhedron small rhombi 4-4 max.png Polyhedron great rhombi 4-4 max.png Polyhedron snub 4-4 left max.png
Catalan
(Platonic)
Polyhedron 4-4 dual blue.png Polyhedron truncated 4a dual max.png Polyhedron truncated 4b dual max.png Polyhedron small rhombi 4-4 dual max.png Polyhedron great rhombi 4-4 dual max.png Polyhedron snub 4-4 left 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
Face angles (°) Dihedral angle (°) 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
112.885
33.557
33.557
129.521 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
70.529
109.471
70.529
109.471
120 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
117.201
31.400
31.400
147.350 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
83.621
48.190
48.190
143.130 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
81.579
81.579
81.579
115.263
138.118 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
87.202
55.025
37.773
155.082 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
114.812
114.812
114.812
114.812
80.752
136.309 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
63.435
116.565
63.435
116.565
144 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
119.039
30.480
30.480
160.613 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
68.619
55.691
55.691
156.719 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
86.974
67.783
86.974
118.269
154.121 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
88.992
58.238
32.770
164.888 120 180 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
118.137
118.137
118.137
118.137
67.454
153.179 60 150 92 I

Geometry[edit]

All dihedral angles of a Catalan solid are equal. Denoting their value by , and denoting the face angle at the vertices where faces meet by , we have

.

This can be used to compute and , , ... , from , ... only.

Triangular faces[edit]

Of the 13 Catalan solids, 7 have triangular faces. These are of the form Vp.q.r, where p, q and r take their values among 3, 4, 5, 6, 8 and 10. The angles , and can be computed in the following way. Put , , and put

.

Then

.

For and the expressions are similar of course. The dihedral angle can be computed from

.

Applying this, for example, to the disdyakis triacontahedron (, and , hence , and , where is the golden ratio) gives and .

Quadrilateral faces[edit]

Of the 13 Catalan solids, 4 have quadrilateral faces. These are of the form Vp.q.p.r, where p, q and r take their values among 3, 4, and 5. The angle can be computed by the following formula:

.

From this, , and the dihedral angle can be easily computed. The faces are kites, or, if , rhombi. Applying this, for example, to the deltoidal icositetrahedron (, and ), we get .

Pentagonal faces[edit]

Of the 13 Catalan solids, 2 have pentagonal faces. These are of the form Vp.p.p.p.q, where p=3, and q=4 or 5. The angle can be computed by solving a degree three equation:

.

Application to other solids[edit]

All of the formulae of this section apply to the Platonic solids, and bipyramids and trapezohedra with equal dihedral angles as well, because they can be derived from the constant dihedral angle property only. For the pentagonal trapezohedron, for example, with faces V3.3.5.3, we get , or . This is not surprising: it is possible to cut off both apexes in such a way as to obtain a regular dodecahedron.

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]