Alex's Adventures in Numberland - Alex Bellos [37]
Regular polygons.
One of Euclid’s pursuits was to investigate the three-dimensional shapes that can be made from joining identical regular polygons together. Only five shapes fit the bill: the tetrahedron, the cube, the octahedron, the icosahedron and the dodecahedron, the quintet known as the Platonic solids since Plato wrote about them in the Timaeus. He equated them with the four elements of the universe plus the heavenly space that surrounds them all. The tetrahedron was fire; the cube, earth; the octahedron, air; the icosahedron, water; and the dodecahedron, the encompassing dome. The Platonic solids are particularly interesting because they are perfectly symmetrical. Twist them, roll them, invert them or flip them and they always stay the same.
The Platonic solids.
In the thirteenth and final book of The Elements, Euclid proved why there are only five Platonic solids by working out all the possible solid objects that can be made from regular polygons, starting with the equilateral triangle, and then moving on to squares, pentagons, hexagons and so on. The diagram overleaf shows how he reached his conclusion. To make a solid object from polygons you must alwys have a point where three sides meet, a corner, or what’s called a vertex. When you join three equilateral triangles at a vertex, for example, you get a tetrahedron (A). When you join four, you get a pyramid (B). A pyramid is not a Platonic solid because not all the sides are the same, but by sticking an inverted pyramid on the bottom you get an octahedron, which is. Join five equilateral triangles together and you have the first part of an icosahedron (C). But join six and you get…a flat piece of paper (D). You cannot make a solid angle with six equilateral triangles, so there are no other ways to create a different Platonic solid made up of them. Continuing this procedure with squares, it is evident that there is only one way to join three squares at a corner (E). This will end up as a cube. Join four squares and you get…a flat piece of paper (F). No more Platonic solids here. Similarly, three pentagons together give a solid angle, which becomes the dodecahedron (G). It is impossible to join four pentagons. Three hexagons meeting at the same point lie flat alongside each other (H), so it is impossible to make a solid object out of them. There are no more Platonic solids since it is impossible to join three regular polygons of more than six sides at a vertex.
Using Euclid’s method, many mathematicians after him have asked new questions and made new discoveries. For example, in 1471 the German mathematician Regiomontanus wrote a letter to a friend in which he posed the following problem: ‘At what point on the ground does a perpendicularly suspended rod appear largest?’ This has been rephrased as the ‘statue problem’. Imagine there is a statue on a plinth in front of you. When you are too close you have to crick your neck and you have a very narrow angle to look up at it. When you are far away, you have to strain your eyes and, again, see it through a very narrow angle. Where, then, is the best place to view it?
Proof that there are only five Platonic solids.
Consider a side-on view of the situation, as in the diagram opposite. We want to find the point on the dotted line, which represents eye level, such that the angle from the point to the statue is greatest. The solution follows from the third book of The Elements, on circles. The angle is largest when a circle that goes through the top and bottom of the statue rests on the dotted line.
The problem is equivalent to that faced by rugby players wanting to know the best distance from the tryline to kick a conversion. If you are too near the opposing tryline the angle