Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:
Geometric properties[edit]
Angles[edit]
There are a number of angles associated with each Platonic solid. The dihedral angle is the interior angle between any two face planes. The dihedral angle, θ, of the solid {p,q} is given by the formula
This is sometimes more conveniently expressed in terms of the tangent by
The quantity h (called the Coxeter number) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively.
The angular deficiency at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2π. The defect, δ, at any vertex of the Platonic solids {p,q} is
By a theorem of Descartes, this is equal to 4π divided by the number of vertices (i.e. the total defect at all vertices is 4π).
The three-dimensional analog of a plane angle is a solid angle. The solid angle, Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by
This follows from the spherical excess formula for a spherical polygon and the fact that the vertex figure of the polyhedron {p,q} is a regular q-gon.
The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4π steradians) divided by the number of faces. This is equal to the angular deficiency of its dual.
The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in steradians. The constant φ = 1 + √5/2 is the golden ratio.
Related polyhedra and polytopes[edit]
Uniform polyhedra[edit]
There exist four regular polyhedra that are not convex, called Kepler–Poinsot polyhedra. These all have icosahedral symmetry and may be obtained as stellations of the dodecahedron and the icosahedron.