Area-preserving diffeomorphisms

This chapter is devoted to two-dimensional symplectomorphisms, which are just area- and orientation-preserving diffeomorphisms. The chapter includes an exposition of Birkhoff’s proof of Poincaré’s last geometric theorem, which asserts that an area-preserving twist map of the annulus must have at least 4 two distinct fixed points. It also discusses some applications of these ideas to billiard problems.

A Porism Concerning Cyclic Quadrilaterals

We present a geometric theorem on a porism about cyclic quadrilaterals, namely, the existence of an infinite number of cyclic quadrilaterals through four fixed collinear points once one exists. Also, a technique of proving such properties with the use of pseudounitary traceless matrices is presented. A similar property holds for general quadrics as well as for the circle.

A curious misattribution: the early history of ‘Simson's line’

When William Wynne Willson died in the summer of 2010 he left behind an incomplete text covering a number of geometrical topics, including the so-called ‘Simson line’. Geoff Wain, who was completing and editing the text with a view to possible publication, sent me a copy. One chapter, entitled ‘Whose line is it anyway?’ contains the following paragraph.‘In 1799 the Scottish mathematician William Wallace … published a geometric theorem. Nowadays this result is known as ‘Simson's line’. It is not clear why it should have this name as Robert Simson, another Scottish mathematician, had died aged 80 in 1768, the year in which Wallace was born. Although Simson published a considerable body of mathematical writings … there seems to be no trace of this theorem among them.