Catalan solid

Summary

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 solids above (dark) shown together with their duals (light). The visible parts of the Catalan solids are regular pyramids.

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.

List of Catalan solids and their dualsEdit

Name
Conway name
Archimedean Dual Pictures Orthogonal
wireframes
Face
polygon
Face angles (°) Dihedral angle (°) Faces Edges Vert Sym.
triakis tetrahedron
"kT"
truncated tetrahedron        Isosceles
 
V3.6.6
112.885
33.557
33.557
129.521 12 18 8 Td
rhombic dodecahedron
"jC"
cuboctahedron         Rhombus
 
V3.4.3.4
70.529
109.471
70.529
109.471
120 12 24 14 Oh
triakis octahedron
"kO"
truncated cube        Isosceles
 
V3.8.8
117.201
31.400
31.400
147.350 24 36 14 Oh
tetrakis hexahedron
"kC"
truncated octahedron        Isosceles
 
V4.6.6
83.621
48.190
48.190
143.130 24 36 14 Oh
deltoidal icositetrahedron
"oC"
rhombicuboctahedron        Kite
 
V3.4.4.4
81.579
81.579
81.579
115.263
138.118 24 48 26 Oh
disdyakis dodecahedron
"mC"
truncated cuboctahedron        Scalene
 
V4.6.8
87.202
55.025
37.773
155.082 48 72 26 Oh
pentagonal icositetrahedron
"gC"
snub cube        Pentagon
 
V3.3.3.3.4
114.812
114.812
114.812
114.812
80.752
136.309 24 60 38 O
rhombic triacontahedron
"jD"
icosidodecahedron        Rhombus
 
V3.5.3.5
63.435
116.565
63.435
116.565
144 30 60 32 Ih
triakis icosahedron
"kI"
truncated dodecahedron        Isosceles
 
V3.10.10
119.039
30.480
30.480
160.613 60 90 32 Ih
pentakis dodecahedron
"kD"
truncated icosahedron        Isosceles
 
V5.6.6
68.619
55.691
55.691
156.719 60 90 32 Ih
deltoidal hexecontahedron
"oD"
rhombicosidodecahedron        Kite
 
V3.4.5.4
86.974
67.783
86.974
118.269
154.121 60 120 62 Ih
disdyakis triacontahedron
"mD"
truncated icosidodecahedron        Scalene
 
V4.6.10
88.992
58.238
32.770
164.888 120 180 62 Ih
pentagonal hexecontahedron
"gD"
snub dodecahedron        Pentagon
 
V3.3.3.3.5
118.137
118.137
118.137
118.137
67.454
153.179 60 150 92 I

SymmetryEdit

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). The 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)
           
Catalan
(Platonic)
           
Octahedral symmetry
Archimedean            
Catalan            
Icosahedral symmetry
Archimedean            
Catalan            

GeometryEdit

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 facesEdit

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 facesEdit

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. Alternatively, put  ,  ,  . Then   and   can be found by applying the formulas for the triangular case. The angle   can be computed similarly of course. The faces are kites, or, if  , rhombi. Applying this, for example, to the deltoidal icositetrahedron ( ,   and  ), we get  .

Pentagonal facesEdit

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:

 .

Metric propertiesEdit

For a Catalan solid   let   be the dual with respect to the midsphere of  . Then   is an Archimedean solid with the same midsphere. Denote the length of the edges of   by  . Let   be the inradius of the faces of  ,   the midradius of   and  ,   the inradius of  , and   the circumradius of  . Then these quantities can be expressed in   and the dihedral angle   as follows:

 ,
 ,
 ,
 .

These quantities are related by  ,   and  .

As an example, let   be a cuboctahedron with edge length  . Then   is a rhombic dodecahedron. Applying the formula for quadrilateral faces with   and   gives  , hence  ,  ,  ,  .

All vertices of   of type   lie on a sphere with radius   given by

 ,

and similarly for  .

Dually, there is a sphere which touches all faces of   which are regular  -gons (and similarly for  ) in their center. The radius   of this sphere is given by

 .

These two radii are related by  . Continuing the above example:   and  , which gives  ,  ,   and  .

If   is a vertex of   of type  ,   an edge of   starting at  , and   the point where the edge   touches the midsphere of  , denote the distance   by  . Then the edges of   joining vertices of type   and type   have length  . These quantities can be computed by

 ,

and similarly for  . Continuing the above example:  ,  ,  ,  , so the edges of the rhombic dodecahedron have length  .

The dihedral angles  between  -gonal and  -gonal faces of   satisfy

 .

Finishing the rhombic dodecahedron example, the dihedral angle   of the cuboctahedron is given by  .

ConstructionEdit

The face of any Catalan polyhedron may be obtained from the vertex figure of the dual Archimedean solid using the Dorman Luke construction.[1]

Application to other solidsEdit

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 alsoEdit

NotesEdit

ReferencesEdit

  • Eugène Catalan Mémoire sur la Théorie des Polyèdres. J. l'École Polytechnique (Paris) 41, 1-71, 1865.
  • Cundy, H. Martyn; Rollett, A. P. (1961), Mathematical Models (2nd ed.), Oxford: Clarendon Press, MR 0124167.
  • Gailiunas, P.; Sharp, J. (2005), "Duality of polyhedra", International Journal of Mathematical Education in Science and Technology, 36 (6): 617–642, doi:10.1080/00207390500064049, S2CID 120818796.
  • 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 linksEdit

  • Weisstein, Eric W. "Catalan Solids". MathWorld.
  • Weisstein, Eric W. "Isohedron". MathWorld.
  • Catalan Solids – at Visual Polyhedra
  • Archimedean duals – at Virtual Reality Polyhedra
  • Interactive Catalan Solid in Java
  • Download link for Catalan's original 1865 publication – with beautiful figures, PDF format