For young readers
About ages 8 to 12. Read it together, or on your own.
At the very end of his book, Euclid builds shapes you can hold. A regular solid is a solid whose faces are all the same regular shape, with the same number meeting at every corner. A cube is one: six square faces, three at every corner.
How many regular solids are there? You might guess there are as many as you like. Euclid proved there are exactly five, and showed how to build each one.
Try this
Why only five? Cut out paper triangles with three equal sides. Tape 3 around a point and fold them up: a corner of a pyramid. 4 works too (an octahedron), and 5 (an icosahedron). Try 6: they lie flat! Six triangles make 360°, a full turn, so there is nothing to fold. Now try squares (3 works, 4 lies flat) and pentagons (3 works). Hexagons? Three of them already lie flat. That is Euclid's argument.Euclid also measured shapes. He proved that a cone holds exactly a third of a cylinder with the same base and height. You can test it with a paper cone, a paper tube and some rice.
For teens and adults
The full lesson.
Who, when and why
Books XI to XIII finish the Elements (Euclid, Alexandria, c. 300 BC) by moving into three dimensions. Book XI sets up solid geometry: lines and planes in space, solid angles, and boxes. Book XII measures curved and pointed solids. Book XIII constructs the five regular solids inside a sphere and proves there are no others.
Why end there? The five regular solids were famous: Plato's dialogue Timaeus (c. 360 BC) had matched four of them to earth, air, fire and water, and the fifth to the whole cosmos. Euclid's own interest is mathematical. Book XIII uses almost everything before it: proportion, the golden section ("extreme and mean ratio", from Book VI), and the irrational lines of Book X. Many readers see it as the summit the whole work climbs toward, though that is a reading, not something Euclid says.
Measuring by exhaustion
How do you prove a pyramid is a third of a prism without calculus? Book XII uses the method of exhaustion, credited to Eudoxus. Fill the shape with simpler pieces you can measure, then show that the leftover can be made smaller than any amount you name. Then prove, by a double argument, that the answer can be neither more nor less than claimed. Archimedes pushed the method much further.
Key idea
Any cone is a third part of the cylinder which has the same base with it and equal height.
Worked example: A cone a third of a cylinder
- The cylinder holds base × height = $30 \times 12 = 360$ cubic centimeters.
- By XII.10 the cone is a third of that: $360 \div 3 = 120$.
Answer: 120 cubic centimeters
Worked example: Spheres by their diameters
- By XII.18 spheres are to one another as the cubes of their diameters.
- $3^3 = 27$.
Answer: 27 times as much
Exactly five
After the last construction, Euclid adds a short argument that no other regular solid exists. At each corner at least three faces must meet, and the angles at the corner must add up to less than four right angles (360°), or the faces lie flat. Equilateral triangles have 60° corners, so 3, 4 or 5 of them fit (180°, 240°, 300°). Squares have 90°, so only 3 (270°). Regular pentagons have 108°, so only 3 (324°). Hexagons have 120°, and three already make 360°. Nothing larger fits at all.
I say next that no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another. For a solid angle cannot be constructed with two triangles, or indeed planes. With three triangles the angle of the pyramid is constructed, with four the angle of the octahedron, and with five the angle of the icosahedron; but a solid angle cannot be formed by six equilateral and equiangular triangles placed together at one point.
| Solid | Faces | Meeting at each corner | Faces | Edges | Corners |
|---|---|---|---|---|---|
| Tetrahedron (pyramid) | triangles | 3 | 4 | 6 | 4 |
| Octahedron | triangles | 4 | 8 | 12 | 6 |
| Icosahedron | triangles | 5 | 20 | 30 | 12 |
| Cube | squares | 3 | 6 | 12 | 8 |
| Dodecahedron | pentagons | 3 | 12 | 30 | 20 |
Common mistake
What later science kept, and what it changed
- Kept: the volume results of Book XII and the five-solid theorem, both exactly right.
- Changed: Kepler, early in his career, tried to fit the planets' orbits between nested regular solids (1596). The idea was wrong, but the search led him to his real laws of planetary motion.
- Found in nature: some viruses and many minerals have the symmetry of these solids; table salt forms cubes, and many virus shells are icosahedral.
Going deeper
Context, the big questions, and how later books answered.
From exhaustion to calculus
Exhaustion proofs are rigorous but slow: you must already know the answer, then prove that nothing else is possible. Archimedes, Cavalieri, Fermat and finally Newton and Leibniz looked for a method that finds the answer, not just checks it. The integral of calculus is the descendant of Book XII.
Counting corners
Euler's formula, $V - E + F = 2$, holds for every solid without holes, regular or not. It was one of the first results of topology, the geometry of shapes that can be stretched. Euclid's careful counts of faces and edges contain it, though he never wrote it down.
Big questions
- Why might the fact that there are exactly five regular solids, no more, have seemed to Plato and Kepler to say something about the universe itself?
- Is a beautiful idea more likely to be true? Kepler's nested solids were beautiful and wrong; his ellipses seemed less beautiful and were right.
- What is the difference between filling a shape with smaller and smaller pieces forever, and actually finishing?
Talk about it together
- Build the five solids from paper or with toothpicks and clay. Which is hardest? Count the faces, edges and corners of each.
- Why can't six equilateral triangles meet at the corner of a solid?
- Dice for games come in all five shapes. Why would a game maker want a regular solid for a die?
- Plato thought the five solids were the shapes of the elements. Kepler thought they spaced the planets. Why do people look for simple patterns in nature, and when does that go wrong?
- A cone holds a third of a cylinder. How could you test that at home?