School Launch

Twelve Pentagons

Build a ball out of pentagons and hexagons, with three faces meeting at every corner. Add as many hexagons as you like: there are always exactly 12 pentagons. Spin the ball and find out why.

Drag to spin. On a keyboard, focus the ball and use the arrow keys.

Choose a ball

Try:

Colors

Spin slowly

12pentagons P

0hexagons H

12faces F

30edges E

20corners V

1T = m²+mn+n²size step

V − E + F = 20 − 30 + 12 = 2

Why always 12?

Drag to turn the shape. Hidden edges are dashed.

Pick a shape

Count them

Faces (flat sides)	?
Edges (where two faces meet)	?
Corners (vertices)	?

Count all three, then try: faces + corners − edges. Do it for every shape. What do you notice?

Euler’s formula V − E + F = 2 holds for every convex polyhedron (and any solid without holes).

Student guide

1. The ball

Every ball here is made of pentagons (5 sides) and hexagons (6 sides), and three faces meet at every corner. A soccer ball has 12 pentagons and 20 hexagons. Bigger balls have more hexagons, but count the pentagons: always 12.

Let a ball have P pentagons and H hexagons with three faces at every corner. Counting edge ends gives 2E = 5P + 6H; counting corners gives 3V = 5P + 6H. Substituting into V − E + F = 2 with F = P + H, the H terms cancel and P/6 = 2. So P = 12, whatever H is. (H can be 0 or any whole number except 1.)

Goldberg balls

Michael Goldberg described these shapes in 1937. GP(m,n) is made by walking m steps, turning 60°, and walking n steps between neighboring pentagons. With T = m² + mn + n², it has 10(T−1) hexagons, 30T edges and 20T corners. GP(1,0) is the dodecahedron and GP(1,1) the soccer ball. When m and n differ and neither is 0, like GP(2,1), the ball comes in a left-handed and a right-handed form.

Bigger balls are made by stepping farther between pentagons. Look how the 12 pentagons stay spread out, like the corners of the dodecahedron.

2. Euler’s 2

For a cube: 6 faces + 8 corners − 12 edges = 2. Try a pyramid, a prism or any shape without a hole. You always get 2!

Leonhard Euler described the relation in a 1750 letter to Christian Goldbach. René Descartes had found an equivalent fact about corner angles a century before. A flat sheet of hexagons fits perfectly, because three 120° corners make 360°. To close a ball, 720° must be removed in total. With ideal regular faces, each pentagon corner is 108° instead of 120°, so it removes 12°; a pentagon has 5 corners, so it removes 60°, and 720° ÷ 60° = 12 pentagons.

Print and fold

Ideas to try on paper: a cube net (a cross of six squares), a square pyramid net (a square with four triangles), and a dodecahedron net (two “flowers” of six pentagons). Count faces, edges and corners on what you fold.

Wonder

From soccer balls to carbon molecules to some virus shells, the same rule of 12 turns up. “O LORD, how manifold are thy works! in wisdom hast thou made them all” (Psalm 104:24, KJV).

Practice

Use the tool to answer. Your teacher has the answer key.

Count the faces, edges and corners of a cube. Add faces and corners, then subtract edges.
Do the same for a square pyramid and a triangle prism.
How many pentagons and how many hexagons are on a soccer ball? How many panels in all?
Can you cover a ball with only hexagons? Use the ball tab to explain.
For GP(3,0), find T, H, F, E and V, then check V − E + F.
A ball has 12 pentagons and 30 hexagons with three faces at each corner. Find E and V. Which Goldberg ball is it?
Show the algebra that turns V − E + F = 2 into P = 12 when every corner has three faces.
The C₆₀ molecule has a carbon atom at each corner of the soccer-ball shape. How many bonds (edges) and how many rings (faces) does it have?
For GP(2,1), find T and the number of hexagons. Why does it come in two mirror-image forms?
A ball has 20 corners and 12 faces. How many hexagons does it have, and what is its name?