# **SEE-Math**

## Texas A&M University

# **Platonic Solids**

## Yasskin, Kobiela

The students get to play with the Polydron construction set which consists of triangles, quadrilaterials, pentagons, hexagons and octagons which snap together at the edges and flex to build solids. We discuss the names for the various polygons as well as the distinction between regular and irregular polygons. They discover that there are an infinite number of regular polygons.
Then we discuss the names for the solids the students have built. These include pyramids, prisms, diamonds (bipyramids), drums (antiprisms), tetrahedron, cube (hexahedron), octahedron, dodecahedron and icosahedron as well as stellated and truncated versions of some of these. All of these are polyhedra.

We then discuss the distinction between regular and irregular polyhedra. The instructor asks "How many regular polyhedra are there?" Depending on how many have been constructed so far, the students may say 3, 4, 5 or infinity (since there are infinitely many regular polygons). The instructor challenges them to build some more. Eventually they conjecture that there are only 5, namely tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron, as pictured below.

These 5 polyhedra are called **Platonic Solids** after Plato who first studied them. Finally, the instructor gives a constructive proof that the 5 Platonic solids are, in fact, all of the regular solids.

© Philip B. Yasskin, 2005.

Last updated Jan 22, 2005