Catalan solid
In
The Catalan solids are all
Additionally, two of the Catalan solids are
Just as
Two of the Catalan solids are
Eleven of the 13 Catalan solids have the
List of Catalan solids and their duals
Conway name | Archimedean dual | Face
polygon |
Orthogonal wireframes |
Pictures | Face angles (°) | Dihedral angle (°) | Midradius[2] | Faces | Edges | Vert | Sym. |
---|---|---|---|---|---|---|---|---|---|---|---|
triakis tetrahedron "kT" |
truncated tetrahedron | Isosceles V3.6.6 |
112.885 33.557 33.557 |
129.521 | 1.0607 | 12 | 18 | 8 | Td | ||
rhombic dodecahedron "jC" |
cuboctahedron | Rhombus V3.4.3.4 |
70.529 109.471 70.529 109.471 |
120 | 0.8660 | 12 | 24 | 14 | Oh | ||
triakis octahedron "kO" |
truncated cube | Isosceles V3.8.8 |
117.201 31.400 31.400 |
147.350 | 1.7071 | 24 | 36 | 14 | Oh | ||
tetrakis hexahedron "kC" |
truncated octahedron | Isosceles V4.6.6 |
83.621 48.190 48.190 |
143.130 | 1.5000 | 24 | 36 | 14 | Oh | ||
deltoidal icositetrahedron "oC" |
rhombicuboctahedron | Kite V3.4.4.4 |
81.579 81.579 81.579 115.263 |
138.118 | 1.3066 | 24 | 48 | 26 | Oh | ||
disdyakis dodecahedron "mC" |
truncated cuboctahedron | V4.6.8 |
87.202 55.025 37.773 |
155.082 | 2.2630 | 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 | 1.2472 | 24 | 60 | 38 | O | ||
rhombic triacontahedron "jD" |
icosidodecahedron | Rhombus V3.5.3.5 |
63.435 116.565 63.435 116.565 |
144 | 1.5388 | 30 | 60 | 32 | Ih | ||
triakis icosahedron "kI" |
truncated dodecahedron | Isosceles V3.10.10 |
119.039 30.480 30.480 |
160.613 | 2.9271 | 60 | 90 | 32 | Ih | ||
pentakis dodecahedron "kD" |
truncated icosahedron | Isosceles V5.6.6 |
68.619 55.691 55.691 |
156.719 | 2.4271 | 60 | 90 | 32 | Ih | ||
deltoidal hexecontahedron "oD" |
rhombicosidodecahedron | Kite V3.4.5.4 |
86.974 67.783 86.974 118.269 |
154.121 | 2.1763 | 60 | 120 | 62 | Ih | ||
disdyakis triacontahedron "mD" |
truncated icosidodecahedron | V4.6.10 |
88.992 58.238 32.770 |
164.888 | 3.7694 | 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 | 2.0971 | 60 | 150 | 92 | I |
ordered by size |
---|
The Catalan solids' midradius in descending order:
All faces of Catalan solids, same scale as above: |
Symmetry
The Catalan solids, along with their dual
Archimedean (Platonic) |
||||||
---|---|---|---|---|---|---|
Catalan (Platonic) |
Archimedean | ||||||
---|---|---|---|---|---|---|
Catalan |
Archimedean | ||||||
---|---|---|---|---|---|---|
Catalan |
Geometry
All
- .
This can be used to compute and , , ... , from , ... only.
Triangular faces
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
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 ,
Pentagonal faces
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 properties
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
- ,
- ,
- ,
- .
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 .
Construction
The face of any Catalan polyhedron may be obtained from the
Application to other solids
All of the formulae of this section apply to the
See also
- List of uniform tilingsShows dual uniform polygonal tilings similar to the Catalan solids
- Conway polyhedron notation A notational construction process
- Archimedean solid
- Johnson solid
Notes
- ^ Weisstein, Eric W. "Archimedean Solid". mathworld.wolfram.com. Retrieved 2022-07-02.
- ^ Cundy & Rollett (1961), p. 117; Wenninger (1983), p. 30.
References
- Eugène CatalanMémoire sur la Théorie des Polyèdres. J. l'École Polytechnique (Paris) 41, 1-71, 1865.
- MR 0124167.
- Gailiunas, P.; Sharp, J. (2005), "Duality of polyhedra", International Journal of Mathematical Education in Science and Technology, 36 (6): 617–642, S2CID 120818796.
- Alan Holden Shapes, Space, and Symmetry. New York: Dover, 1991.
- MR 0730208(The thirteen semiregular convex polyhedra and their duals)
- 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
- 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