Geometry

To make a Christmas gift for her brother Leon, Alex asked all her friends to save empty cereal boxes. She collected about a dozen boxes.

She cut the boxes open, which gave her several big sheets of thin cardboard.

Then she carefully traced the templates for a regular triangle, square, pentagon, and hexagon, as shown below.

Click here to download the polygon templates

She drew the dark outline of each polygon with a ballpoint pen, pressing hard to score the cardboard so the tabs would bend easily.

She cut out shapes until her fingers felt bruised: 20 each of the pentagon and hexagon, 40 each of the triangle and square.

Alex bought a bag of small rubber bands for holding the tabs together. Each rubber band can hold two tabs, forming an edge of the polyhedron. So, for instance, it takes six squares and twelve rubber bands to make a cube.

Finally, she stuffed the whole kit in a plastic zipper bag, along with the following instructions.

Polyhedra Have “Many Faces”

Poly means many, and hedron means face, so a polyhedron is a 3-D shape with many faces.

The plural of polyhedron is polyhedra, thanks to the ancient Greeks, who didn’t know that the proper way to make a plural was to use the letter s.

Each corner of a polyhedron is called a vertex, and to make it more confusing, the plural of vertex is vertices.

Regular Polyhedra

Regular polyhedra have exactly the same faces and corners all around. If one side is a square, then all the sides will be squares. And if three squares meet to make one vertex, then all the other vertices will be made of three squares, just like that first one.

There are only five possible regular polyhedra. Can you figure out why?

Here are the five regular polyhedra, also called the Platonic solids. Try to build each of them with your construction kit.

Tetrahedron: three equilateral triangles meeting at each vertex.

Hexahedron: three squares meeting at each vertex. Do you know its common name?

Octahedron: four triangles at each vertex.

Icosahedron: five triangles at each vertex.

Dodecahedron: three pentagons per vertex.

You can find pictures of these online, but it’s more challenging to build them without peeking at the finished product. Just repeat the vertex pattern at every corner until the polygons connect together to make a complete 3-D shape.

Semi-Regular Polyhedra

Semi-regular polyhedra have each face a regular polygon, although not all the same. Each corner is still the same all around. These are often called the Archimedean polyhedra.

For example, on the cuboctahedron, every vertex consists of a square-triangle-square-triangle combination.

Here are a few semi-regular polyhedra you might try to build, described by the faces in the order they meet at each corner:

Icosidodecahedron: triangle, pentagon, triangle, pentagon.

Truncated octahedron: square, hexagon, hexagon.

Truncated icosahedron: pentagon, hexagon, hexagon. Where have you seen this?

Rhombicuboctahedron: triangle, square, square, square.

Rhombicosidodecahedron: triangle, square, pentagon, square.

Now, make up some original polyhedra of your own. What will you name them?

To Be Continued…

Read all the posts from the December 2000/January 2001 issue of my Mathematical Adventures of Alexandria Jones newsletter.

• Alexandria Jones and the Magic Christmas Cards

CREDITS: “50/52 Weeks of Teddy – Merry Christmas” photo by Austin Kirk via Flickr (CC BY 2.0).