CTK Insights

24 Apr

Euclidean Constructions vs Ruler & Compass

The ancient Greeks are credited with the development of the notion of proof that squarely placed their mathematics on the foundation of reason rather than experience. Why they focused on geometric constructions with ruler and compass is anybody's guess. It might be that the ancients thought it is most entertaining to use a minimal set of tools, or, perhaps, because in practice these two instruments are the most handy. Whatever the reason, Euclid in his Elements did not consider any other constructions but only those that could be performed with straightedge and compass. So much so that quite often such construction are known as Euclidean. For example, [Conway and Guy, p. 192] call certain numbers Euclidean that otherwise would be called constructible. However, there is a slight difference between the constructions as articulated by Euclid in the Elements and the geometric methods that in the 19th century led to the solution - albeit negative - of the three problems of antiquity, angle trisection in particular.

The distinction is subtle. The 19th century theory starts with selection of a unit interval - the step omitted by Euclid. Starting with the unit interval the modern theory builds constructible points, lines, circles, and angles - all using exclusively straightedge and compass, as Euclid would. The novelty of fixing a scale may seem like a trifle but it was this trifle that led to algebraization of the theory and the introduction of the notion of constructible numbers - lengths of constructible segments relative to the unit interval.

For example, in Elements VI.12 Euclid constructs \sqrt{pq} for the two given segments of lengths p and q, however arbitrary. It would have never occurred to him to construct \sqrt{p} for a given number p. The modern theory applies Euclidean construction but only to constructible p and q, so as to obtain that \sqrt{pq} is also constructible. In particular, p is constructible only if that is true of \sqrt{p}.

Straightedge and compass could be used with any data, constructible or not. But, as usual, output is only as good as input. Euclidean construction produce nonconstructible objects if applied to nonconstructible data.

This leads to curiosities like being able to obtain \sqrt{\pi} with straightedge and compass without being able to construct \pi itself. (There are of course neusis constructions, to start with.)

There are also nonconstructible but trisectable angles. 3\pi /7 serves an example. If we have managed in any way to obtain that angle, we can then construct its supplementary 4\pi /7 and the difference between the two \pi /7 which is exactly one third of the given angle 3\pi /7. On the other hand, angle 3\pi /7 is not constructible, for otherwise the regular heptagon - the seven sided polygon - would be constructible while we know that it is not.

Reference

  1. J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, NY, 1996.

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