Thirteen Catalan Polyhedra
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Theorem
There exist exactly $13$ distinct Catalan polyhedra:
- Triakis tetrahedron
- Triakis octahedron
- Disdyakis dodecahedron
- Tetrakis hexahedron
- Triakis icosahedron
- Disdyakis triacontahedron
- Pentakis dodecahedron
- Rhombic dodecahedron
- Rhombic triacontahedron
- Deltoidal icositetrahedron
- Deltoidal hexecontahedron
- Pentagonal icositetrahedron
- Pentagonal hexecontahedron
Proof
By definition, the Catalan polyhedra are the dual polyhedra of the Archimedean polyhedra.
There are $13$ Archimedean polyhedra, and so there are $13$ Catalan polyhedra.
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Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $13$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$