"Bounds Archimedes missed : exercises in geometric extrapolation."
Abstract : Pi is a topic of abiding fascination that engages the interest of all mathematicians, pure and applied alike. We know, or think we know, that it was Archimedes who early calculated pi to considerable accuracy by bounding a circle inside and out by regular polygons. However, this program, with an explicit argument in the case of inscribed polygons, is already contained in Book XII of Euclid's Elements. Closer examination of the works of Euclid and of Archimedes suggests that everything you can do with inscribed and circumscribed polygons together can be done just as well with inscribed polygons alone. Moreover, it seems that the Chinese mathematician Liu Hui, working over seventeen hundred years ago, was able to improve the lower bound on the area of a circle by interpolation using only inscribed polygons. Perhaps even more surprisingly, whereas the combined work of Euclid and Archimedes shows that the difference between areas of circumscribed and inscribed polygons more than halves on doubling the number of sides of these polygons, an argument that would have been accessible to both of them, as well as to Liu Hui, shows that, in fact, it more than quarters. The talk is presented as an exercise in ''mathematics from history'', where we take the mathematics from a given period and see what (more) can be extracted by means of it alone. Thus, when we look back on this material from the later perspective of the calculus, we find that these geometric arguments remarkably powerful, giving results akin to Richardson-Romberg integration - the quartering inequality just mentioned is accurate up to the term in the sixth power of the reciprocal of the number of sides of the largest and smallest polygons. It seems that we - not just Archimedes - might have been missing something.
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