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Article: Rhombicosidodecahedron


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
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