PLATONIC SOLIDS

Audrey Johnson

Characteristics of Platonic Solids

They are regular polyhedraA polyhedron is a three dimensional

figure composed of polygonsThere are exactly five platonic solidsAll the faces are the same regular

polygonThe same number of polygons meet

at each vertex

To Be a Platonic Solid…

At least three faces must meetthe sum of the interior angles of the sides

meeting at each vertex must be less than 360 degrees For example, the tetrahedron is made up of

equilateral triangles which consists of three 60 degree angles

3 equilateral triangles meet at each vertex so the sum of the interior angles is 180 degrees which is less than 360 degrees

More Examples:

Octahedron made up of four equilateral triangles 4*60=240 < 360 degrees

Icosahedron made up of five equilateral triangles 5*60=300 < 360 degrees

As 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

The cube: Made up of three squares 3*90=270 < 360

As 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

Tetrahedron 4 faces 6 edges 4 vertices

Cube 6 faces 12 edges 8 vertices

Octahedron 8 faces 12 edges 6 vertices

Dodecahedron 12 faces 30 edges 20 vertices

Icosahedron 20 faces 30 edges 12 vertices

Unique Relationship

Duality The cube and octahedron are duals The icosahedron and dodecahedron are

duals The tetrahedron is a dual to itself

This means that one can be created by connecting the midpoints of the faces of the other

In Real Life

The five platonic solids are ideal models of crystal patterns that occur throughout the world of minerals in numerous variations

To the Greeks, these solids symbolized fire, earth, air, spirit, and water