PLATONIC SOLIDS Audrey Johnson

Characteristics of Platonic Solids zThey are regular polyhedra zA polyhedron is a three dimensional figure composed of polygons zThere are exactly five platonic solids zAll the faces are the same regular polygon zThe same number of polygons meet at each vertex

To Be a Platonic Solid… zAt least three faces must meet zthe sum of the interior angles of the sides meeting at each vertex must be less than 360 degrees yFor example, the tetrahedron is made up of equilateral triangles which consists of three 60 degree angles y3 equilateral triangles meet at each vertex so the sum of the interior angles is 180 degrees which is less than 360 degrees

More Examples: zOctahedron ymade up of four equilateral triangles y4*60=240 < 360 degrees zIcosahedron ymade up of five equilateral triangles y5*60=300 < 360 degrees zAs a result there cannot be 6 equilateral triangles since 6*60=360. If this was so the triangles would form a single-planed figure and not a solid

zThe cube: yMade up of three squares y3*90=270 < 360 zAs a result, if four squares met at a vertex then the interior angles would equal 360 and would form a plane and not a solid

Unique Numbers zTetrahedron 4 faces 6 edges 4 vertices zCube 6 faces 12 edges 8 vertices zOctahedron 8 faces 12 edges 6 vertices zDodecahedron 12 faces 30 edges 20 vertices zIcosahedron 20 faces 30 edges 12 vertices

Unique Relationship zDuality yThe cube and octahedron are duals yThe icosahedron and dodecahedron are duals yThe tetrahedron is a dual to itself zThis means that one can be created by connecting the midpoints of the faces of the other

In Real Life zThe five platonic solids are ideal models of crystal patterns that occur throughout the world of minerals in numerous variations zTo the Greeks, these solids symbolized fire, earth, air, spirit, and water