SYMmetryplus No. 61 - Autumn 2016

14 SYMmetryplus 61 Autumn 2016 Matters might have rested there but for the birth of the electronic computer towards the end of World War II. Computer calculations were further enhanced by the discovery of more powerful algorithms. Just after the turn of the new millennium a computation of  exceeded a trillion digits. Now it is possible to call up a particular digit of  expressed in hexadecimal (base 16) or binary form! The irrationality of  , (  cannot be expressed as p q where p , q are positive integers), was proved by Johann Lambert in 1767. Its transcendence (  cannot be the root of a polynomial equation with integer coefficients), was established by Carl Lindemann in 1882. So why continue to compute increasingly long strings of digits of  ? Programming is no trivial matter and testing the reliability of the technology using different algorithms and computational methods is as good an explanation as any. Graham Hoare EULER ON A DOUGHNUT Regular readers of SYmmetry plus will know Euler’s polyhedron formula: vertices + faces = edges + 2 v + f = e + 2 and will also know that it works for graphs which are not formed by the edges of a polyhedron, for example a map showing county boundaries. It is important to include the region around the edge because the map should really be drawn on a topological sphere (as it is for a polyhedron). We are going to write the formula 2 = v + f – e because it is the ‘ 2 ’ we are interested in. To make sure of the rules governing our maps we will start with the simplest possible graph and change it: 1 2 3 4 v + f – e: 1 + 1 – 0 2 + 1 – 1 1 + 2 – 1 2 + 2 – 2 An edge must have a vertex at each end. In 3 we have bent the edge around and joined the ends so that the 2 vertices become 1. (An edge must have a vertex on it.) 4 is rather odd but it is a valid graph. There is a very important requirement on a face which these pictures do not show: we must be able to shrink to a point any closed curve (loop) we draw on it. We will call this the ‘shrinking loop’ rule because we will need to check that it is obeyed in every figure we draw. If you draw loops around 1 and 2, how can you shrink them? Answer: Around the back. (Remember you are looking at a sphere.) JOHANN HEINRICH LAMBERT (1728 – 1777) WAS A SWISS POLYMATH WHO DEVISED THEOREMS BASED ON CONIC SECTIONS THAT MADE THE CALCULATION OF ORBITS OF COMETS SIMPLER. CARL LOUIS FERDINAND VON LINDEMANN (1852 – 1939) WAS A GERMAN MATHEMATICIAN.