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.

Expressions like

are known as continued radicals and they occur frequently in the mathematical literature, if only because of their orderly structure and their close links with continued fractions. But it’s worth bringing them out again here because they can make very interesting classroom exercises on sequences, and they lead to unusual ways of expressing π and the golden ratio, etc.

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 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.

The stereographic projection of the sphere is a perspective projection from a point on the surface onto the opposite diametral plane. Analytical geometry provides a unified method of treating the stereographic projection and problems in spherical trigonometry. The basic result is the equation for the projection of a general circle (eq. 1). This is applied to the polar, equatorial and oblique stereographic projections, and a simple proof of Cayley’s theorem is obtained. Next, the cosine formula for a right-angled spherical triangle and the relation between angles measured along different great circles are derived. Finally, the construction used on astrolabes for determining planetary time is shown to be the stereographic projection of a problem on the sphere, which is solved.

The Mathematical Association was established in 1871 as the Association for the Improvement of Geometrical Teaching (AIGT). Gradually through the nineteenth century, the AIGT broadened its range of mathematical concerns beyond those of an essentially anti-Euclid pressure group. Its forms of professional support developed to include meetings, presentation of papers and publications, and its own journal, The Mathematical Gazette, was launched in 1894 [1]. By this time the AIGT had also started to build a small collection of books and periodicals, which was initially boosted by the donation of mainly geometrical texts from its first President, T. A. Hirst (1830-1892). By 1897 the AIGTcouid comfortably change its name to the Mathematical Association (MA), advertised as ‘an association of teachers and students of elementary mathematics’ [2].

Mathematicians are fascinated by relatios and have worked our extensive theories about them. The treatments tend to be abstract and sometimes the basic ideas are lost in the abstraction. Here we investigate some common ground between mathematical relations and down-to-earth genealogy, the study of family relations.

Family relations have been studied by mathematicians, perhaps none more playfully than Thomas P. Kirkman (the nineteenth century expert in combinatorial mathematics) in his little puzzle rhyme he sent to the Educational Times [1, p.117]: