Names and Dimensions

• Johannes Kepler named this polyhedron a rhombicosidodecahedron.
• The name rhombicosidodecahedron is short for truncated icosidodecahedral rhombus.
• Kepler also named a rhombic triacontahedron as icosidodecahedral rhombus.
• There are different truncations of a rhombic triacontahedron into a topological rhombicosidodecahedron.
• The rectification of the dual icosidodecahedron is the core of the dual compound.
• The surface area of a rhombicosidodecahedron with edge length 'a' is approximately 59.3059828449a^2.
• The volume of a rhombicosidodecahedron with edge length 'a' is approximately 41.6153237825a^3.

Geometric Relations

• A rhombicosidodecahedron can be obtained by expanding an icosahedron or its dual dodecahedron and patching the resulting square holes.
• Alternatively, a rhombicosidodecahedron can be obtained by expanding five cubes and patching the resulting pentagonal and triangular holes.
• Two clusters of faces of the bilunabirotunda can be aligned with a congruent patch of faces on the rhombicosidodecahedron.
• The rhombicosidodecahedron shares the vertex arrangement with the small stellated truncated dodecahedron and uniform compounds of pentagrammic prisms.
• Zometool kits use expanded rhombicosidodecahedra as connectors for making geodesic domes.

Cartesian Coordinates and Orthogonal Projections

• The Cartesian coordinates for the vertices of a rhombicosidodecahedron with an edge length of 2 centered at the origin are permutations of specific values.
• The circumradius of this rhombicosidodecahedron is the common distance of these points from the origin.
• For unit edge length, the circumradius is halved.
• The rhombicosidodecahedron has six special orthogonal projections.
• These projections include vertex-centered, edge-centered, and face-centered projections.
• The rhombicosidodecahedron can also be represented as a spherical tiling and projected onto the plane via a stereographic projection.
• The stereographic projection preserves angles but not areas or lengths.
• The rhombicosidodecahedron can be represented by various projections in different contexts.

Definition and Characteristics

• Archimedean solid
• Polyhedron with 20 regular triangular faces, 30 square faces, and 12 regular pentagonal faces
• Dual of the small stellated dodecahedron
• Convex and uniform polyhedron
• Named after its rhombic and icosidodecahedral symmetry
• Triangular faces have angles of 60 degrees
• Square faces have angles of 90 degrees
• Pentagonal faces have angles of 108 degrees
• All edges are congruent in length
• Symmetry group is isomorphic to the alternating group A5

Applications and Related Polyhedra

• Used in architecture and design
• Can be found in jewelry and decorative objects
• Serves as a model for mathematical concepts and calculations
• Used in computer graphics and modeling
• Provides inspiration for artistic creations
• Related Polyhedra: Rhombic triacontahedron, Small stellated dodecahedron, Great rhombicosidodecahedron, Great stellated dodecahedron, Small ditrigonal icosidodecahedron

Historical Significance

• First studied by Archimedes in ancient Greece
• Mentioned in Plato's dialogue 'Timaeus'
• Played a role in the development of polyhedra theory
• Has been a subject of mathematical exploration for centuries
• Represents the beauty and elegance of geometric shapes

Rhombicosidodecahedron Data Sources

Reference	URL
Glossary	https://harryandcojewellery.com.au/blogs/glossary/rhombicosidodecahedron
Wikipedia	http://en.wikipedia.org/wiki/Rhombicosidodecahedron
Wikidata	https://www.wikidata.org/wiki/Q672521
Knowledge Graph	https://www.google.com/search?kgmid=/m/01py_z