Regular polyhedra. Duality and interrelations

Regular polyhedra, also called Platonic solids, are 3D shapes where all faces are congruent regular polygons and the same number of faces meet at each vertex. There are exactly five such solids. Duality is a relationship between polyhedra: for each regular polyhedron, you can construct a dual polyhedron by swapping faces and vertices. The dual of a regular polyhedron is also regular, and the process can be reversed.

Important properties

• There are five regular polyhedra: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

• Duality pairs: cube and octahedron are duals, dodecahedron and icosahedron are duals, and the tetrahedron is self-dual.

• Duality means that the number of faces and vertices are swapped in the dual.

• The edges of a polyhedron and its dual correspond to each other.