Early in the classical period of Greek mathematics there arose a crisis over the meaning of “proportion”, or alternatively of “area”. This crisis, which was resolved by Eudoxus, the pupil of Plato, about the middle of the fourth century B.C., led to the introduction of axioms, or postulates, as “self-evident truths”, necessary for the proof of theorems, equally true but presumably less self-evident. Among these axioms was included the famous Fifth Postulate of Euclid, equivalent to saying (see below for its rather complicated wording) that “through a given point (the point T in Fig. 1) there is at most one (straight) line parallel to a given line (the line MK)”. But this postulate had all the earmarks of a theorem; it looks true yet not altogether self-evident, and its converse, to the effect that there is at least one such line, is an easily provable theorem.

I want to start by explaining some of the terms I shall use in this lecture. First, there is pure mathematics. Anyone who has done Ordinary Level Mathematics has some idea of what pure mathematics is about. Degree, or postgraduate pure mathematics for that matter, is—to quote A. N. Whitehead—“the science concerned with the logical deduction of consequences from general premisses”. The humbler algebra and geometry of school mathematics is concerned with exactly this, or should be. And in both cases, it is all done, at least in principle, for the fun of the thing with no application to the real world in mind.

The regular decagon will be dissected into other regular polygons and into Greek and Latin crosses by the methods outlined in my previous article (Math. Gaz. 45, 1961, p. 94).

The first step is to find tessellations. In one of those that are used in this note, shown in 1 and 2, the decagon is cut into three pieces, and in the other only two. A further tessellation element, not used here but worth noting, is shown in 5.