The Platonic Solids

A polygon is a two-dimensional shape bounded by straight line segments. A polygon is said to be regular if the edges are of equal length and meet at equal angles. A polygon is convex if the line connecting any two points inside the polygon is itself entirely inside the polygon.

A square is an example of a regular polygon. In general, a rectangle is not regular because not all of its edges are of the same length. Similarly, a rhombus is not regular because not all of its interior angles are equal.

A polyhedron is a three-dimensional figure bounded by polygons. Extending the notion of regularity to polyhedrons, we will say that a polyhedron is regular if the faces are congruent regular polygons that meet at equal angles. Likewise, a polyhedron is convex if the line connecting any two points inside the polyhedron is itself entirely interior to the polyhedron.

Even though a convex regular polygon is a very special kind, it is easy to see that there are an infinite number of them. It may seem surprising, then, to learn that there are only five convex regular polyhedrons! These 5 very special polyhedrons have become known collectively as the Platonic Solids.

Since all faces are identical regular polygons and the vertices are identical, stating the number of sides of a face and the number of faces that meet at each vertex fully specifies one of these solids.

Let P be the number of sides to each face, and let Q be the number of faces that meet at each vertex. The ordered pair {P,Q} then identifies the polyhedron.

The simplest polygon is the triangle, so P is at least 3, and the number of faces at each vertex, Q, is also at least 3. Therefore the simplest possible solid is {3,3}, and in fact it exists; it is called the Tetrahedron.

Tetrahedron {3,3}

Keep the equilateral triangle for the face, but now put four of them around each vertex. This yields the Octahedron, having eight faces and 6 vertices.

Octahedron {3,4}

Putting five equilateral triangles around each vertex yields the Icosahedron, which has 20 faces and 12 vertices. Six such triangles lay flat in a plane, so this is the last possible solid with P=3.

Icosahedron {3,5}

We next try square faces, and succeed in creating the familiar Cube. However, four squares again lay flat, so there are no others possible with P=4.

Cube {4,3}

Three pentagons joining at each vertex yields the Dodecahedron.

Dodecahedron {5,3}

Here the process ends, as three hexagons lay flat, so there can be no other polyhedra that satisfy the Platonic criteria. Thus, we see that five are possible, and in fact they all exist!

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