Activity 14
Polyhedra

A polyhedron is composed of polygons connected at their edges so as to enclose space. There are many different solid shapes that are polyhedra. Examine the shapes below and count the number of faces, edges, and vertices for each. Place your results in the table below and look for a relationship between the number of faces, edges, and vertices. Describe this relationship in a formula using the letters F, E, and V.

Below are drawings of a pyramid with a base that is an n-sided polygon, and a prism with an identical base. Write the number of faces, vertices, and edges for each of these solids.

Regular Polyhedra

A regular polyhedron is a polyhedron that is composed of congruent regular polygons with the same vertex code everywhere. In any regular n-sided polygon, the angle at each vertex is given by [180(n - 2)]/n.

The smallest polygon angle possible is 60° for equilateral triangles. The maximum number of equilateral triangles that can be placed with edges connected is 5, since 6 would make a flat surface. Since the minimum number of polygons that can meet at a vertex is 3, the possibilities would be either 3, 4, or 5 triangles at each vertex. The largest polygon angle possible is 108° for a regular pentagon. It might be possible to connect 3 pentagons at each vertex. The only other polygon angle possible would be 90° for a square. This might allow 3 squares at each vertex.

The possible codes for regular polyhedra are given in the table below. The only way to determine if such polyhedra exist is to try to construct them. Use tabbed regular polygon cutouts to try to construct regular polyhedra with the given codes. When you are successful with one, assign it a name based on the number of faces. For example, a polyhedron with 10 faces would be called a decahedron. Also count the number of vertices and edges and enter these in the table.

Dual of a Regular Polyhedron

One regular polyhedron is the dual of another if the polyhedra have the same number of edges but the number of faces of one equals the number of vertices of the other. Find the dual of each regular polyhedron and write it in the table below.

Archimedean Polyhedra

A semi-regular polyhedron is a polyhedron that is composed of several different regular polygons with the same vertex code everywhere. The possibilities for regular polygons that meet at a vertex are numerous, but only a few can actually be constructed. Rather than examine all of the possibilities by adding angles of polygons, we will take a different approach. We will modify the existing regular polyhedra using two basic techniques: truncating, that is, cutting about one-third of the way from each vertex; and cutting one-half of the way from each vertex. Both methods chop a pyramid from each original vertex, leaving a polygon in its place, and leave a polygon in the center of each original face.

The modification process can be done by drawing on clear plastic models or by actually cutting styrofoam models with a hot-wire cutter. If these materials are not available, the modifications can be shown by drawing on pictures of the regular polyhedra. Construction kits are available for all of the semi-regular polyhedra.

Modify the regular polyhedra using both methods and place your results in the table below.

Two additional Archimedean Polyhedra are formed by drawing diagonals on the square faces of the rhombicuboctahedron and the rhombicosidodecahedron, and then distort ing the right triangles formed into equilateral triangles. The results are called the snub cube and the snub dodecahedron.

The final remaining Archimedean Polyhedron is formed by removing the "roof" from the rhombicuboctahedron, rotating it by 45°, and replacing it. This polyhedron has the same vertex code as the rhombicuboctahedron, but is a different solid. It is called the pseudorhombicuboctahedron.

Fill in the table below for the remaining seven Archimedean Polyhedra.