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.
- 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
- 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