Catalan solid

Set of Catalan solids

The Catalan solids are the dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. ^[1] The faces of the Catalan solids correspond by duality to the vertices of Archimedean solids, and vice versa. ^[2]

The solids

The Catalan solids are face-transitive or isohedral meaning that their faces are symmetric to one another, but they are not vertex-transitive because their vertices are not symmetric. Their dual, the Archimedean solids, are vertex-transitive but not face-transitive. Each Catalan solid has constant dihedral angles, meaning the angle between any two adjacent faces is the same. ^[1] Additionally, two Catalan solids, the rhombic dodecahedron and rhombic triacontahedron, are edge-transitive, meaning their edges are symmetric to each other.^{[ citation needed ]} Some Catalan solids were discovered by Johannes Kepler during his study of zonohedra, and Eugene Catalan completed the list of the thirteen solids in 1865. ^[3]

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

In general, each face of a dual uniform polyhedron (including the Catalan solid) can be constructed by using the Dorman Luke construction. ^[4] Some of the Catalan solids can be constructed, starting from the set of Platonic solids, all faces of which are attached by pyramids. These examples are the Kleetope of Platonic solids: triakis tetrahedron, tetrakis hexahedron, triakis octahedron, triakis icosahedron, and pentakis dodecahedron. ^[5]

Two Catalan solids, the pentagonal icositetrahedron and the pentagonal hexecontahedron, are chiral, meaning that these two solids are not their own mirror images. They are dual to the snub cube and snub dodecahedron respectively, which are also chiral.

Eleven of the thirteen Catalan solids are known to have the Rupert property that a copy of the same solid can be passed through a hole in the solid. ^[6]

The thirteen Catalan solids

Name	Image	Faces	Edges	Vertices	Dihedral angle [7]	Point group
triakis tetrahedron		12 isosceles triangles	18	8	129.521°	Td
rhombic dodecahedron		12 rhombi	24	14	120°	Oh
triakis octahedron		24 isosceles triangles	36	14	147.350°	Oh
tetrakis hexahedron		24 isosceles triangles	36	14	143.130°	Oh
deltoidal icositetrahedron		24 kites	48	26	138.118°	Oh
disdyakis dodecahedron		48 scalene triangles	72	26	155.082°	Oh
pentagonal icositetrahedron		24 pentagons	60	38	136.309°	O
rhombic triacontahedron		30 rhombi	60	32	144°	Ih
triakis icosahedron		60 isosceles triangles	90	32	160.613°	Ih
pentakis dodecahedron		60 isosceles triangles	90	32	156.719°	Ih
deltoidal hexecontahedron		60 kites	120	62	154.121°	Ih
disdyakis triacontahedron		120 scalene triangles	180	62	164.888°	Ih
pentagonal hexecontahedron		60 pentagons	150	92	153.179°	I

References

Footnotes

1. Diudea (2018), p.  39.
2. Wenninger (1983), p. 1, Basic notions about stellation and duality.
3. • Diudea (2018), p.  39
   • Heil & Martini (1993), p.  352
4. • Cundy & Rollett (1961), p. 117
   • Wenninger (1983), p. 30
5. • Brigaglia, Palladino & Vaccaro (2018)
   • Çolak & Gelişgen (2015)
6. Fredriksson (2024).
7. Williams (1979).

Works cited

• Brigaglia, Aldo; Palladino, Nicla; Vaccaro, Maria Alessandra (2018), "Historical notes on star geometry in mathematics, art and nature", in Emmer, Michele; Abate, Marco (eds.), Imagine Math 6: Between Culture and Mathematics, Springer International Publishing, pp. 197–211, doi:10.1007/978-3-319-93949-0_17, hdl: 10447/325250 , ISBN   978-3-319-93948-3 .
• Çolak, Zeynep; Gelişgen, Özcan (2015), "New Metrics for Deltoidal Hexacontahedron and Pentakis Dodecahedron", Sakarya University Journal of Science, 19 (3): 353–360, doi:10.16984/saufenbilder.03497 (inactive 1 July 2025)
• 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)

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