The article Problems, solved and unsolved in the Gazette for last March (60 (411), 1–11) drew reactions from a number of readers. There is not sufficient space to show in full all the proofs sent in, but I will try to indicate the methods used. For statements of the problems themselves I must ask readers to refer to the earlier article.

In a recent session of the Royal Institution Master Classes in Cambridge, the enthusiastic participants explored a computer graphics program, designed and written by Derek and Alan Ball. The program first draws a circle, and, given an integer N, marks N equally spaced points on it, these being cyclically labelled 1,2,…,N. The operator then inputs some rule, say n ↦ 5n + 8, and the program draws, for each n, the chord from the point marked n to the point marked 5n + 8 (modulo N). Before long, the children took N to be large and flower-like envelopes of the chords started to appear (as illustrated below). Here, we show how the equations of these envelopes can be obtained in a simple way with some general parameter k, and that for positive k they are all epicycloids, that is, curves obtained by rolling a circle around a fixed circle, the simplest one being a cardioid. A similar analysis shows that for negative k they are hypocycloids (see for example E. H. Lockwood’s Book of curves, C.U.P. 1961) except when k = 1 when parallel chords occur.

This article started life as an investigation into certain aspects of the Fibonacci numbers, ‘morphed’ seamlessly into the structure of some infinite matrices and finally resolved into a general set of results that link structural aspects of Fibonacci numbers with trigonometric and hyperbolic functions. It is a surprising fact, but while I can find evidence that the link between these areas has been noted in the past, I can find no evidence that the link has been systematically developed.

Although the gradient of a scalar, and the curl and divergence of a vector can be obtained by various elementary methods in terms of curvilinear components and coordinate vectors, the corresponding results for the components of strain and the vector divergence of stress are usually obtained by specializing the formulae for covariant derivatives. The purpose of this note is to give an elementary approach to these latter results in the spirit of the theory of Cartesian tensors. Apart from standard elementary vector analysis the only prerequisite is a geometrical appreciation of the analysis of deformation. For completeness an elementary analytical derivation of the results about gradient, curl and divergence is given first by a method which, whilst certainly not new, does not appear to be very well known.

If you visit the Manchester Museum of Science and Technology take a look at the sculpture in front of the museum. There are two large stainless steel objects, a sphere and an inverted cone. On the ground around the cone are variously shaped patterns of black and brown. However, when the patterns are viewed as reflected from the cone they take on the portraits of four famous scientists connected with the city of Manchester (see figure 1 on the previous page). This is an example of anamorphosis, or anamorphic art. Anamorphosis refers to an image or drawing which is distorted so that it can only be recognized when viewed in a particular way, or using a special device (like the cone in the above example). The word is derived from the Greek anamorphoun - to transform.

Nice polynomials are polynomials whose coefficients, roots, and critical points are integers. The earliest papers on nice polynomials [1, 2, 3] give an explicit formula for all nice cubics. All the derivations of this formula use Pythagorean triples.

In 1999 the problem of finding, constructing, and classifying nice polynomials was added to the list of unsolved problems [4] in The American Mathematical Monthly. Other papers soon followed, including [5] and [6], with extending the results of nice polynomials to polynomials with coefficients, roots, and critical points in integral domains D. Such polynomials are called D-nice.

This article came about as a response to the Supplement to the Gazette Number 514 [1] which consists of an extended version of Sir Christopher Zeeman’s Presidential Address to The Mathematical Association at York in 2004. His intention was to stimulate the teaching of 3-dimensionaI geometry in schools, and his publication will certainly satisfy some long-felt needs. It contains a short section on conies, which seemed relevant to a topic I am already investigating: the level of knowledge of conic geometry potentially available to the Greeks before the time of Apollonius. Professor Zeeman has encouraged me to publish the first stage of this research in the Gazette to provide a background to his discussion.

About four years ago after consulting my copy of Cundy and Rollett [1] and then leaving it on my desk, I noticed that it had fallen open at a diagram of an approximation to Von Koch’s ‘snowflake’ curve. Interest in such curves has increased of late but for most mathematics students they make a fleeting appearance in the analysis course as examples of closed, continuous curves of infinite length and differentiable nowhere.