Polyhedra

Introduction

If you cut six identical squares out of cardboard, you can tape them together to make a cube, which is an example of a polyhedron. A polyhedron is a solid figure in which each side is a flat surface.

Each square is a face of the cube. A line segment where two faces meet is called an edge. A point where three or more faces meet is called a vertex. The faces of a polyhedron are polygons.

In the cube, how many faces meet at each vertex?

Triangular Face Exercise

Cut a few equilateral triangles out of cardboard. Make them all the same size. Keep the cardboard handy, you may need more. Recall: an equilateral triangle is a triangle in which all three sides are the same length and the angles are 60 degrees each.

Can you make a polyhedron in which three of these triangles meet at each vertex? Try it.

How many faces does this polyhedron have?

This polyhedron is a tetrahedron.

Regular Polyhedra

The cube and the tetrahedron are examples of Regular Polyhedra, also called Platonic Solids.

A polyhedron is called regular if the faces are congruent (same size and shape) regular polygons and the same number of faces meet at each vertex.

1. Can you make any more regular polyhedra using equilateral triangles?

2. Try making a solid in which four triangles meet at each vertex.

3. How about five triangles at each vertex?

4. What happens if you try six triangles at each vertex?

5. What are the possibilities with square faces?

6. What about pentagons?

7. What happens if you try hexagons?

More on Regular Polyhedra

Find it on the Web.

Kaleidotile, which you can find at the Geometry Center, is an interactive Macintosh program to explore these shapes and truncated polyhedra.

Dave Chasey maintains a site of photographs of Henry Chasey's Polyhedra Models. These are acrylic models of many different polyhedra, from the Platonic solids, to Archimedean, Stellated and more. Tom Gettys' Polyhedra site has computer generated images of platonic solids, archimedean solids, and more. Plus tips for building your own models.


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