## Archive for May, 2019

### Symplectic Geometry

For many decades, a search has been under way to find a theory of everything, that accounts for all the fundamental physical forces, including gravity. The dictum “physics is geometry” is a guiding principle of modern theoretical physics. Einstein’s General Theory of Relativity, which emerged just one hundred years ago, is a crowning example of this synergy. He showed how matter distorts the geometry of space and this geometry determines the motion of matter. The central idea is encapsulated in an epigram of John A Wheeler:

$\displaystyle \mbox{Matter tells space how to curve. Space tells matter how to move.}$

### Chase and Escape: Pursuit Problems

From cheetahs chasing gazelles, through coastguards saving shipwrecked sailors, to missiles launched at enemy aircraft, strategies of pursuit and evasion play a role in many areas of life (and death). From pre-historic times we have been solving such pursuit problems. The survival of our early ancestors depended on their ability to acquire food. This involved chasing and killing animals, and success depended on an understanding of relative speeds and optimal pursuit paths.

### The Rise and Rise of Women in Mathematics

Sonya Kovalevskya (1850-1891)

The influential collection of biographical essays by Eric Temple Bell, Men of Mathematics, was published in 1937. It covered the lives of about forty mathematicians, from ancient times to the beginning of the twentieth century. The book inspired many boys to become mathematicians. However, it seems unlikely that it inspired many girls: the only woman to get more than a passing mention was Sofia Kovalevskaya, a brilliant Russian mathematician and the first woman to obtain a doctorate in mathematics [TM163 or search for “thatsmaths” at irishtimes.com].

### Bouncing Billiard Balls Produce Pi

There are many ways of evaluating ${\pi}$, the ratio of the circumference of a circle to its diameter. We review several historical methods and describe a recently-discovered and completely original and ingenious method.

### Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph

The Greeks regarded the heavens as the epitome of perfection. All flaws and blemishes were confined to the terrestrial domain. Since the circle is perfect in its infinite symmetry, it was concluded by Aristotle that the Sun and planets move in circles around the Earth. Later, the astronomer Ptolemy accounted for deviations by means of additional circles, or epicycles. He stuck with the circular model [TM162 or search for “thatsmaths” at irishtimes.com].

Left: Elliptic orbit with velocity vectors. Right: Hodograph, with all velocity vectors plotted from a single point.