Catalan solid

Set of Catalan solids
The rhombic dodecahedron's construction, the dual polyhedron of a cuboctahedron, by Dorman Luke construction

The Catalan solids are the dual polyhedron of Archimedean solids, a set of thirteen polyhedrons with highly symmetric forms semiregular polyhedrons in which two or more polygonal of their faces are met at a vertex.[1] A polyhedron can have a dual by corresponding vertices to the faces of the other polyhedron, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.[2] One way to construct the Catalan solids is by using the method of Dorman Luke construction.[3]

These solids are face-transitive or isohedral because their faces are transitive to one another, but they are not vertex-transitive because their vertices are not transitive to one another. Their dual, the Archimedean solids, are vertex-transitive but not face-transitive. Each has constant dihedral angles, meaning the angle between any two adjacent faces is the same.[1] Additionally, both Catalan solids rhombic dodecahedron and rhombic triacontahedron are edge-transitive, meaning there is an isometry between any two edges preserving the symmetry of the whole.[citation needed] These solids were also already discovered by Johannes Kepler during the study of zonohedrons, until Eugene Catalan first completed the list of the thirteen solids in 1865.[4]

The pentagonal icositetrahedron and the pentagonal hexecontahedron are chiral because they are dual to the snub cube and snub dodecahedron respectively, which are chiral; that is, these two solids are not their own mirror images.

Eleven of the thirteen Catalan solids are known to have the Rupert property (a copy of the same solid can be passed through a hole in the solid).[5]

The thirteen Catalan solids
Name Image Faces Edges Vertices Dihedral angle[6] Point group
triakis tetrahedron Triakis tetrahedron 12 isosceles triangles 18 8 129.521° Td
rhombic dodecahedron Rhombic dodecahedron 12 rhombi 24 14 120° Oh
triakis octahedron Triakis octahedron 24 isosceles triangles 36 14 147.350° Oh
tetrakis hexahedron Tetrakis hexahedron 24 isosceles triangles 36 14 143.130° Oh
deltoidal icositetrahedron Deltoidal icositetrahedron 24 kites 48 26 138.118° Oh
disdyakis dodecahedron Disdyakis dodecahedron 48 scalene triangles 72 26 155.082° Oh
pentagonal icositetrahedron Pentagonal icositetrahedron (Ccw) 24 pentagons 60 38 136.309° O
rhombic triacontahedron Rhombic triacontahedron 30 rhombi 60 32 144° Ih
triakis icosahedron Triakis icosahedron 60 isosceles triangles 90 32 160.613° Ih
pentakis dodecahedron Pentakis dodecahedron 60 isosceles triangles 90 32 156.719° Ih
deltoidal hexecontahedron Deltoidal hexecontahedron 60 kites 120 62 154.121° Ih
disdyakis triacontahedron Disdyakis triacontahedron 120 scalene triangles 180 62 164.888° Ih
pentagonal hexecontahedron Pentagonal hexecontahedron (Ccw) 60 pentagons 150 92 153.179° I

References

Footnotes

  1. ^ a b Diudea (2018), p. 39.
  2. ^ Wenninger (1983), p. 1, Basic notions about stellation and duality.
  3. ^
  4. ^
  5. ^ Fredriksson (2024).
  6. ^ Williams (1979).

Works cited

  • Cundy, H. Martyn; Rollett, A. P. (1961), Mathematical Models (2nd ed.), Oxford: Clarendon Press, MR 0124167.
  • Diudea, M. V. (2018), Multi-shell Polyhedral Clusters, Carbon Materials: Chemistry and Physics, vol. 10, Springer, doi:10.1007/978-3-319-64123-2, ISBN 978-3-319-64123-2.
  • Fredriksson, Albin (2024), "Optimizing for the Rupert property", The American Mathematical Monthly, 131 (3): 255–261, arXiv:2210.00601, doi:10.1080/00029890.2023.2285200.
  • 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.
  • Heil, E.; Martini, H. (1993), "Special convex bodies", in Gruber, P. M.; Wills, J. M. (eds.), Handbook of Convex Geometry, North Holland, ISBN 978-0-08-093439-6
  • Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
  • 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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