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All Platonic solids except the tetrahedron are centrally symmetric, meaning they are preserved under reflection through the origin. The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parentheses (likewise for the number of symmetries).
Picture Name Schläfli symbol Vertex/Face configuration exact dihedral angle (radians) dihedral angle – exact in bold, else approximate (degrees) Platonic solids (regular convex)
All five have C 2 ×S 5 symmetry but can only be realised with half the symmetry, that is C 2 ×A 5 or icosahedral symmetry. [ 9 ] [ 10 ] [ 11 ] They are all topologically equivalent to toroids . Their construction, by arranging n faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane .
The conjugacy classes of full tetrahedral symmetry, T d ≅ S 4, are: identity; 8 × rotation by 120° 3 × rotation by 180° 6 × reflection in a plane through two rotation axes; 6 × rotoreflection by 90° The conjugacy classes of pyritohedral symmetry, T h, include those of T, with the two classes of 4 combined, and each with inversion: identity
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.
This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3] It corresponds to the Euler characteristic of the sphere (i.e. χ = 2 {\displaystyle \ \chi =2\ } ), and applies identically to spherical polyhedra .
Many convex polytopes having some degree of symmetry (for example, all the Platonic solids) can be projected onto the surface of a concentric sphere to produce a spherical polyhedron. However, the reverse process is not always possible; some spherical polyhedra (such as the hosohedra ) have no flat-faced analogue.
A Johnson solid is a polyhedron whose faces are all regular, but which is not uniform. This means the Johnson solids do not include the Archimedean solids, the Catalan solids, the prisms, or the antiprisms. Some of them are constructed involving the removal of the part of a regular icosahedron, a process known as diminishment.