Platonic Solids

Polygons and Polyhedra

Time and again, the study of geometric solids - polyhedra - proves to be very satisfying for both the experts and the amateurs in the field of Euclidian Geometry. Polyhedra have attractive shapes and the relations between them never fail to surprise and amuse a curious investigator.

Before we go further let's introduce some vocabulary:

Polygon is a two-dimensional shape bounded by line segments. A polygon is regular if all its edges and angles are equal.

A polyhedron is a three-dimensional solid made up of polygons. A polyhedron is regular if all its faces are congruent regular polygons that meet at equal angles.
There is an infinite number of regular polygons. So, it may come as a surprise to discover that there are only five regular polyhedra. These five very special polyhedra are known as the Platonic Solids.


	Five Platonic Solids The simplest possible regular solid is called the Tetrahedron. It is constructed by joining equilateral triangles such that three triangles meet at each vertex of the solid.
	To make an Octahedron use the equilateral triangles again, but now put four of them around each vertex.
	Putting five equilateral triangles around each vertex yields the Icosahedron, If you try to put six equilateral triangles around each vertex you will discover (sooner or later) that they lay flat in a plane. No solid can be formed. You have made a triangular tiling instead. So the Icosahedron is the last possible solid made up of triangles.

	Next try squares and create the familiar Cube by putting three squares in each corner. However, four squares in each corner lay flat again - now you have made a square tiling. So the cube is the first and last solid made up of squares. No others are possible.

	Three pentagons joining at each vertex yield the Dodecahedron. Here the process ends. Three hexagons lay flat again forming hexagonal tiling.
	There are no more regular polyhedra. Just five are possible! Five Platonic Solids. All the above constructions, simple and conceptually accessible to elementary school children, constitute a profound geometry lesson used in the Geometro programs.
	Semi-regular polyhedra For further amusement and education you might start exploring the solids formed by using two different kinds of polygons instead of one.