# FELLESKOLLOKVIUM FREDAG 28. APRIL 2006

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Douglas Rogers:

"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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