Catalan Solid — Definition, Formula & Examples
A Catalan solid is a convex polyhedron that is the dual of an Archimedean solid. There are exactly 13 Catalan solids, and each one has identical faces that are not regular polygons but are all the same shape and size.
A Catalan solid is a face-transitive convex polyhedron whose dual is an Archimedean solid. Each Catalan solid has congruent but non-regular polygon faces, and its vertices (rather than its faces) may meet in different configurations. The 13 Catalan solids were described by Belgian mathematician Eugène Catalan in 1865.
How It Works
To understand a Catalan solid, start with an Archimedean solid and construct its dual by swapping the roles of faces and vertices. Each face of the Archimedean solid becomes a vertex of the Catalan solid, and each vertex of the Archimedean solid becomes a face. Because Archimedean solids are vertex-transitive (every vertex looks the same), Catalan solids are face-transitive (every face looks the same). However, while Archimedean solids use regular polygons of different types, Catalan solid faces are all congruent non-regular polygons.
Worked Example
Problem: The rhombic dodecahedron is the Catalan solid dual to the cuboctahedron. The cuboctahedron has 12 vertices and 14 faces. Find the number of vertices, faces, and edges of the rhombic dodecahedron, and verify Euler's formula.
Step 1: In a dual, the number of faces and vertices swap. The cuboctahedron has 14 faces and 12 vertices, so the rhombic dodecahedron has 14 vertices and 12 faces.
Step 2: Use Euler's formula to find the number of edges.
Step 3: Verify: the rhombic dodecahedron has 12 congruent rhombus faces, 24 edges, and 14 vertices, consistent with known data.
Answer: The rhombic dodecahedron has 14 vertices, 24 edges, and 12 faces, satisfying Euler's formula .
Why It Matters
Catalan solids appear in crystallography, where mineral structures often take the shape of rhombic dodecahedra or pentagonal icositetrahedra. Understanding duality between Archimedean and Catalan solids deepens your grasp of symmetry, a central theme in both geometry courses and materials science.
Common Mistakes
Mistake: Assuming Catalan solids have regular polygon faces like Platonic or Archimedean solids.
Correction: Catalan solid faces are congruent (all the same shape) but are not regular polygons. For example, the rhombic dodecahedron has rhombus faces, not squares.