56 A History ofMathematics
(3, 3) (tetrahedron), (3, 4) (octahedron), (4, 3) (cube), (3, 5) (icosahedron), and (5, 3)
(dodecahedron). Each ofpandqmust be greater than 2; while the angles of a regular
p-sided polygon areπ( 1 −( 2 /p)). (Why? Check examples, and prove the rule.) The sum of
the angles at a vertex is thereforeπ(q−( 2 q/p)). This must be less than 2πfor the result
to be a solid (if it is equal to 2π, the polygons lie flat). From this, with a little algebra,
pq< 2 p+ 2 q,or(p− 2 )(q− 2 )<4. It is now clear that the combinations of(p,q)which
have been given are the only ones.
There is much more on this (covering star-polyhedra, ‘semi-regular’ polyhedra, and
higher dimensions) in Coxeter (1963), an excellent introduction to the subject, with
information on its later (post-Plato) history.- The process cannot stop, because when you invertx, the ‘remainder’ turns out to bex
again. So you cannot (as you would with a ratio of integers, see our example) have a
sequence of ratios with smaller denominators; from which it follows thatxcannot be a
ratio of integers.
Supplementary problem
I have left you one of the most interesting and typical problems in the history of early
Greek mathematics as a ‘research problem’ to think about. This is known asHippocrates’s
quadrature of lunes.It is a clever area calculation, supposed to have been worked out by
Hippocrates of Chios before 400bce(and so, in a sense, our earliest serious result in Greek
mathematics). You can find in Fauvel and Gray; and also some account of the transmission
line, which makes its status rather unreliable (the description comes from a text about 700
years later). Consider- the result;
- how its status is evaluated by various modern historians—accepted without question,
or grudgingly, or set on one side as dubious; - how you would assess the importance of the result, and of its authority;
- the problems about how Hippocrates might have arrived at it, and why.