# Studying Polyhedra

Polyhedra in the Classroom

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#### What is a polyhedron?

A polyhedron is a three-dimensional solid whose faces are polygons joined at their edges. A polyhedron is said to be regular if its faces are made up of regular polygons. A regular polygon is a polygon with sides of equal length placed symmetrically around a common center.

The word polyhedron is derived from the Greek poly (many) and the Indo-European hedron (seat). Five regular polyhedra comprise the convex Platonic solids:

 1. Click on the five different buttons in the applet (F=4, F=6, F=8, F=12, F=20). Can you name the five regular polyhedra? 2. Look at the top of the chart. How many faces does each polyhedron have? How many vertices? 3. Stop the rotation. Look at each of the polyhedra. What polygon do you see on the faces of the     cube?     tetrahedron?     octahedron?     icosahedron?     dodecahedron? 4. Instead of using the stop button you can also try dragging the mouse on the figures to rotate them. 5. Click on the pull down menu with choices color model, light model, metal model, wire model and select wire model and view the cube. Can you count the edges? the vertices? Now change to the other four polyhedra and count the edges and vertices for all of them. Are there any patterns?

applet authored by José Luis Abreu

#### Why study polyhedra?

Sometimes in mathematics we study ideas that are abstract and difficult to see, but polyhedra can be seen! Crystals are real world examples of polyhedra. The salt you sprinkle on your food is a crystal in the shape of a cube.

Another interesting polyhedral shape is the truncated icosahedron. It is not a regular polyhedron, but it is extremely important in the world of science. Carbon-60, also known as the buckyball, takes the shape of a truncated icosahedron.

There's lots to learn about crystals and buckyballs.

My Crazy Dream - Roseanna Vassar
Essay - Jason Marvin

#### Other sites with information about regular polyhedra

Crystallographic Polyhedra
Java applets of a variety of polyhedra including a buckyball.

Platonic Solids
Information and graphics of Platonic solids by Tom Gettys.

Platonic Solids
A color chart from the MacTutor History of Mathematics archive.

#### Other sites with information about Greece and Greek mathematicians

Greek Mathematics and it's Modern Heirs
Information and photo images about Greek mathematicians and their work.

Plato
A biography of Plato from the MacTutor History of Mathematics site.