Solid Faces

3. The Buckyball Asteroid

Solid Faces

The shapes pictured on your space capsule's screen are examples of geometric figures known as solids (see "The Buckyball Asteroid"). Solids have three dimensions: length, width, and height. Many solid objects, from pyramids and dice to baseballs and cereal boxes, have shapes that can be described in simple geometric terms.

Dice serve as examples of the solid geometric shape known as a cube.

A solid formed by polygons that enclose a single region of space is called a polyhedron. Five of the six shapes you saw on the screen had surfaces made up identical, regular polygons, which meet at each corner, or vertex, in exactly the same way.

A regular tetrahedron has four faces, each one an equilateral triangle. Here's what a tetrahedron might look like if it were cut open and unfolded into a flat shape.

A regular tetrahedron, as seen whole (left) and cut open and unfolded (right), has four faces, each one an equilateral triangle.

A cube, or regular hexahedron, has six square faces (below).

A regular octahedron has a surface consisting of eight equilateral triangles (below).

A regular dodecahedron is made up of twelve regular pentagons. If you were to cut it into two equal parts, each part would resemble a flower having five pentagon-shaped petals around a central pentagon (below).

A regular icosahedron has twenty flat surfaces, each one an equilateral triangle (below).

More than two thousand years ago, the Greek mathematician Euclid proved that these five objects are the only ones that can be constructed from a single type of regular polygon. Known as the Platonic solids, they are named after the Greek philosopher Plato, who lived around 350 B.C.

The Greeks were not the first to study these shapes, however. There is evidence that people in China and in the British Isles knew about them long before Plato's time.

Plato believed that the world was made up of tiny particles consisting of four elements: fire, air, water, and earth. Each particle had the shape of a regular polyhedron.

Fire, the lightest and sharpest of the elements, was a tetrahedron. Earth, as the most stable element, consisted of cubes. Water, as the most mobile, was an icosahedron, the regular solid most likely to roll easily. Air was an octahedron, and the doecahedron represented the entire universe.

TRY IT!
Discover an amazing relationship among the vertices, edges, and faces of polyhedra.

You will need:

• sheet of paper, pencil, ruler
• cube-shaped object, such as a game die, a child's building block, or a box
• any other examples of polyhedra that you can find, such as dodecahedral, octahedral, or tetrahedral dice (used in certain games) or objects like prisms
• soccer ball

What to do:

1. Divide your sheet of paper into four columns.
2. Label the head of each column, in order,  Name of Shape; Faces (F); Vertices (V); Edges (E).
3. Under "Name of Shape," write CUBE.
4. Count the number of faces on the cube and record your total under "F."
5. Count the number of vertices on the cube and record your total under "V."
6. Count the number of edges on the cube and record your total under"E."
7. Record the same information for each polyhedron you have available. For a soccer ball with a surface pattern of hexagons and pentagons, think of each face as a flat rather than a curved surface.
8. Look for patterns in the table to find a relationship among the number of faces, vertices, and edges. Hint: Try adding or subtracting various combinations of F, V, and E for each polyhedron.

The relationship you are looking for was discovered by Leonhard Euler, an eighteenth-century Swiss mathematician. The rule applies to many different kinds of polyhedra. Use it, for example, to calculate how many edges a solid with eight faces and twelve vertices must have.

Answers:

The relationship is F + V − E = 2.

If F = 8 and V = 12, then F + V = 20, so E = 18. That means a polyhedron with 8 faces and 12 vertices has 18 edges. It is called a hexagonal prism.

NEXT: The Amazing Buckyball