Beauty, Form and Function: An Exploration of Symmetry Asset No. 20 PART II Plane (2D) and Space (3D) Symmetry Lecture II-6 The Platonic Solids

Objectives By the end of this lecture, you will be able to: see that tessellation in three dimensions creates polyhedra label polyhedra according to Schäfli notation recognize the 5 Platonic solids describe 1 Archimedean solid

Polyhedra are closed figures with polygonal faces. Convex polyhedra have all dihedral angles (angles between faces) less than 180o when viewed from the outside. Regular polyhedra have all vertices related by symmetry and all faces congruent. A cube (43) is a regular polyhedron with: 8 vertices 12 edges 6 faces 1 4 2 3 squares (4-gons) around each vertex 5 3 6

The Tetrahedron and Octahedron - Platonic Solids 1 2 3 4 A tetrahedron (33) has: 4 vertices 6 edges 4 faces 1 2 3 4 5 6 7 8 A octahedron (34) has: 6 vertices 12 edges 8 faces The five convex regular polyhedrons - tetrahedron, octahedron, cube, icosahedron, dodecahedron - known collectively as Platonic Solids.

The Icosahedron and Dodecahedron A dodecahedton (53) has: 20 vertices 30 edges 12 faces 1 2 4 6 3 5 7 8 9 10 11 1 12 2 3 4 5 8 10 6 7 9 11 12 14 13 An icosahedron (35) has: 12 vertices 30 edges 20 faces 15 16 17 18 19 20

Properties of Platonic Solids 33 34 43 53 35 All the Platonic Solids have vertices on the surface of a sphere and are constructed with a single type of polygon - triangle, square, pentagon. Singular Plural tetrahedron* tetrahedra octahedron octahedra cube (hexahedron) cubes (hexahedra) icosahedron icosahedra dodecahedron dodecahedra Greek tetra 4 octa 8 hexa 6 icosa 20 dodeca 12 Triangle Square Pentagon

Cuboctahedron - An Archimedean Solid Polyhedra with equivalent (i.e. symmetry related) vertices but more than one kind of regular polygonal face are the semi-regular or Archimedean Solids. An important Archimedean solid in crystallography is the cuboctahedron. 1 2 3 4 5 7 10 6 9 A cuboctahedron (3.4.3.4) has: 12 vertices 24 edges 14 faces 8 11 12 13 14 The Euler Rule of Platonic and Archimedean Solids states that V-E+F = 2 which relates the number of vertices (V), edges (E) and faces (F).

Summary There are 5 regular polyhedra known as the Platonic solids - tetrahedron, octahedron, cube, icosahedron, dodecahedron Regular polyhedra are composed of one type of polygon and every vertex is identical Semi-regular polyhedra are composed to two or more polygons with every vertex regular The cuboctahedron is an Archimedean solid