page_title: Polyhedron
section_title: Symmetry groups
hierarchical_section_title: Polyhedron / Symmetries / Symmetry groups
caption_attribution_description: English: KaleidoTile, Topology and Geometry Software, Jeff Weeks Spherical domains for tetrahedral symmetry
context_page_description: In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron.
  A convex polyhedron is the convex hull of finitely many points, not all on the same plane.
  Cubes and pyramids are examples of convex polyhedra.
  A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions.
context_section_description: Many of the symmetries or point groups in three dimensions are named after polyhedra having the associated symmetry. These include:
  T – chiral tetrahedral symmetry; the rotation group for a regular tetrahedron; order 12.
  Td – full tetrahedral symmetry; the symmetry group for a regular tetrahedron; order 24.
  Tₕ – pyritohedral symmetry; the symmetry of a pyritohedron; order 24.
  O – chiral octahedral symmetry;the rotation group of the cube and octahedron; order 24.
  Oₕ – full octahedral symmetry; the symmetry group of the cube and octahedron; order 48.
  I – chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron; order 60.
  Iₕ – full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron; order 120.
  Cₙᵥ – n-fold pyramidal symmetry
  Dₙₕ – n-fold prismatic symmetry
  Dₙᵥ – n-fold antiprismatic symmetry.
  Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. Examples include the snub cuboctahedron and snub icosidodecahedron.
caption: Full icosahedral symmetry divides the sphere into 120 triangular domains.
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