The Rhombicosidodecahedron

(Small Rhombicosidodecahedron)

The Rhombicosidodecahedron, sometimes called the Small Rhombicosidodecahedron to distinguish it from the Great Rhombicosidodecahedron (Truncated Icosidodecahedron), is a visually stunning Archimedean solid. Its surface is a mosaic of three types of regular polygons: triangles, squares, and pentagons.

It can be formed by expanding either a dodecahedron or an icosahedron, pulling the faces outward radially and inserting square faces into the resulting gaps along the original edges, and triangular faces at the original vertices. This expansion highlights its connection to both Platonic solids from the icosahedral symmetry group.

Geometric Properties

• Faces: 62 total (20 triangles, 30 squares, 12 pentagons)
• Edges: 120
• Vertices: 60
• Vertex Configuration: 3.4.5.4 (Each vertex joins one triangle, two squares, and one pentagon, in that order around the vertex)

Its dual polyhedron is the Deltoidal Hexecontahedron. The arrangement of faces ensures that no two faces of the same type share an edge (though squares share edges with other squares).

Symbolism and Interpretation

Metaphysically, the Rhombicosidodecahedron represents a complex but harmonious integration of different fundamental principles. The triangles (fire/spirit), squares (earth/structure), and pentagons (aether/life/quintessence) meet at each vertex, symbolizing points of synthesis where diverse energies or concepts converge.

It embodies balanced complexity and multifacetedness. Unlike the Truncated Icosidodecahedron which features larger polygons (hexagons and decagons), the Rhombicosidodecahedron uses the more fundamental regular polygons (triangle, square, pentagon), suggesting a synthesis closer to the primary building blocks. Meditating on this shape can help in integrating different aspects of the self or understanding how diverse elements can form a stable, harmonious, and beautiful whole. Its near-spherical form connects it to wholeness and comprehensive understanding.