The use of the Greek letter π to denote the ratio of the circumference of a circle to its diameter appears to have been popularised by Euler following his adoption of the symbol in 1737 and later in his many popular textbooks. The original use of the symbol for this purpose was in fact by the Welsh mathematician William Jones who used it his book Synopsis Palmariorum Matheseos published in 1706. This book was described in 1824, over a hundred years later as follows ‘This work has long been esteemed as a compendious, but comprehensive summary of Mathematical Science’.

An appreciation of the geometry underlying algebraic techniques invariably enhances understanding, and this is particularly true with regard to polynomials.

With visualisation as our theme, this article considers the cubic equation and describes how the French mathematicians François Viète (1540–1603) and René Descartes (1596–1650) related the ‘three-real-roots’ case (casus irreducibilis) to circle geometry. In particular, attention is focused on a previously undescribed aspect, namely, how the lengths of the chords constructed by Viète and Descartes in this setting relate geometrically to the curve of the cubic itself.

Family history research has become a very popular hobby during the last few years due to the ready access to records of births and marriages. Many people have rigorously traced their ancestors back to 1700 and earlier, corresponding to some 12 generations. The principal motivation of most amateur genealogists is to learn about the lives of their forebears and they trust these ‘legal’ records without question and do not usually acknowledge that cuckoldry may have occurred and that their true ancestors may not have been the ones recorded.

In [1, Chapter 3, Section 2], we collected together results we had previously obtained relating to the question of which positive integers m were Lucasian, that is, factors of some Lucas number Ln. We pointed out that the behaviors of odd and even numbers m were quite different. Thus, for example, 2 and 4 are both Lucasian but 8 is not; for the sequence of residue classes mod 8 of the Lucas numbers, n ⩾ 0, reads

and thus does not contain the residue class 0*. On the other hand, it is a striking fact that, if the odd number s is Lucasian, then so are all of its positive powers.

One of the bedrock theorems of mathematics is the statement that a real polynomial of degree n has at most n real zeros. Probably the best-known proof is the algebraic one, by factorisation. But there is also a pleasant analytic proof, by deduction from Rolle’s theorem.

A slightly different question is how many positive zeros a polynomial has. Here the basic result is known as ‘Descartes′ rule of signs’. It says that the number of positive zeros is no more than the number of sign changes in the sequence of coefficients. Descartes included it in his treatise La Géométrie which appeared in 1637. It can be proved by a method based on factorisation, but, again, just as easily by deduction from Rolle’s theorem.

A ‘magic’ hexagon has rows of numbers in three directions that add to the same total. It is possible to construct such a hexagon from a honeycomb array of hexagons and from an array of triangles (Figure 1). There is known to be only one magic hexagon formed from hexagons. However, numerous magic arrangements are possible for the hexagon of triangles and these arrangements have many additional interesting properties. In this article we give the reasons why such configurations are possible, we look at the number of arrangements possible when there are 2 triangles on each side of the hexagon and we show that in general the arrangements are balanced in many ways – including that they are physically balanced, a property we call ‘magic moments’.

Many problems in topology are easy to state, but some are extremely difficult to solve. And even if they are easy, they may have surprising solutions. For example, take a tyre inner tube, that is, a toroidal surface with a hole (for the valve!). Can one turn it inside out? The answer is ‘yes’ but if one actually carries out this procedure the result is entirely unexpected.

Facetting, or faceting, is the process of removing parts of a polyhedron, without creating any new vertices. It has received far less attention than the process of stellation, which is its exact reciprocal. Facetting is perhaps less easy to visualise clearly, and the results may not have the same immediate appeal. But once broached, it has a fascination all its own. Also, to understand the rules of stellation properly, it can be illuminating to study the reciprocal rules of facetting. Now that suitable software is available (see Further investigations), the study of facetting is gaining momentum.