Polyhedra

A polyhedron is the three-dimensional version of a polygon: it is a chunk of space with flat walls. In other words, it is a three-dimensional figure made by gluing polygons together. The word is Greek in origin, meaning many-seated. The plural is polyhedra. The polygonal sides of a polyhedron are called its faces.

Discussion

Collect some triangles, either the snap-together plastic polydrons or paper triangles. Try gluing them together in various ways to form polyhedra.

1. Fasten three triangles together at a vertex. Complete the figure by adding one more triangle. Notice how there are three triangles at every vertex. This figure is called a tetrahedron because it has four faces (see the table of Greek number prefixes.)
2. Fasten triangles together so there are four at every vertex. How many faces does it have? From the table of prefixes below, deduce its name.

3. Do the same, with five at each vertex.

4. What happens when you fasten triangles six per vertex?
5. What happens when you fasten triangles seven per vertex?

 
Table 1: The first 20 Greek number prefixes

Homework

A regular polygon is a polygon with all its edges equal and all angles equal. A regular polyhedron is one whose faces are regular polygons, all congruent, and having the same number of polygons at each vertex.

For homework, construct models of all possible regular polyhedra, by trying what happens when you fasten together regular polygons with 3, 4, 5, 6, 7, etc sides so the same number come together at each vertex.

Make a table listing the number of faces, vertices, and edges of each.

What should they be called?