Catalan Solids

Dual polyhedra of the Archimedean solids - face-regular convex polyhedra

Rhombic Forms

Rhombic Dodecahedron

Dual to cuboctahedron - 12 congruent rhombi

Rhombic Triacontahedron

Dual to icosidodecahedron - 30 congruent rhombi

Triakis Forms

Triakis Tetrahedron

Dual to truncated tetrahedron - 12 isosceles triangles

Triakis Octahedron

Dual to truncated cube - 24 isosceles triangles

Triakis Icosahedron

Dual to truncated dodecahedron - 60 isosceles triangles

Tetrakis and Pentakis Forms

Tetrakis Hexahedron

Dual to truncated octahedron - 24 isosceles triangles

Pentakis Dodecahedron

Dual to truncated icosahedron - 60 isosceles triangles

Hexakis Forms

Hexakis Octahedron

Dual to truncated cuboctahedron - 48 scalene triangles

Disdyakis Triacontahedron

Dual to truncated icosidodecahedron - 120 scalene triangles

Deltoidal Forms

Deltoidal Icositetrahedron

Dual to rhombicuboctahedron - 24 deltoidal quadrilaterals

Deltoidal Hexecontahedron

Dual to rhombicosidodecahedron - 60 deltoidal quadrilaterals

Pentagonal Icositetrahedron

Dual to snub cube - 24 irregular pentagons

About Catalan Solids

The Catalan solids are the dual polyhedra of the Archimedean solids and are characterized by congruent (but not regular) faces:

Crystallography - Natural crystal forms
Mineralogy - Rock structures
Architecture - Complex facades

Mathematics - Duality theory
Art - Sculptures
3D Design - Parametric models

Duality and Properties

Dual Relationship

Each Catalan solid is dual
to an Archimedean solid

Face Regularity

All faces are congruent
but not regular

Vertex Types

Different vertex types
Not vertex-transitive

Completeness

Exactly 13 Catalan solids
Correspond to 13 Archimedean

Duality: In dualization, vertices become faces and faces become vertices. The number of edges remains the same.

Quick Reference

13 Solids

Catalan Solids

Dual

to Archimedean

Congruent

Faces

V ↔ F

Duality

E = E

Edges Equal

Historical Context

Eugène Catalan (1814-1894): Belgian mathematician who systematically studied these dual polyhedra.

Kepler (1619): Already described some of the Catalan solids in "Harmonices Mundi".

Modern Research: Applications in crystallography and parametric design.