Choosing seven mathematical theorems or ideas to take to a desert island is more difficult than I first thought. In an earlier article (with Colin Fletcher) I discussed other people’s choices. Now comes the time to bite the bullet and make my own. I was reminded that the choices do not have to be ‘great’ or ‘state-of-the-art’ or involve any crystal ball gazing. They do not have to be among the hundreds of fundamental theorems that abound in mathematics all they are bound to be is personal.

There has been a long history in the approximation of ellipses by circular arcs in order to simplify their construction and manipulation. Such approximation was of use for a wide variety of applications, in fields such as mathematics (generating figures), astronomy (analysing orbits), art (marking out large oval frames for ceiling painting), architecture (building masonry arches, floor plans, etc), and, more recently, the conversion of fonts from a general conic specification to circular arcs. Documented evidence goes as far back as the Italian Renaissance when various schemes were published by the architect Sebastiano Serlio in the sixteenth century. More contentiously, it has been argued that fifteen centuries previously the Romans used such approximations when designing and building their amphitheatres.

A recurring topic of discussion is the perceived difficulty, or awkwardness, of establishing the fundamental rules of vector algebra by rigorous geometrical arguments (see, for example, [1]). The following exposition is an attempt to show that, for those acquainted with the rudiments of linear algebra, the essentially simple theory of vector algebra can be clarified by presentation in the context of quaternion algebra, whose elegance and usefulness may still be somewhat undervalued. We make no explicit use of trigonometry, but of course some underlying geometrical ideas are mentioned.

Given chords OA1A2, OB1B2 and OC1C2 of a conic with segment lengths ai = OAi, bi = OBi, ci = OCi (for i = 1, 2) this article will establish a metrical relationship between ai,bi,ci, and the sines of the angular directions of the chords.

We suggest a way to make traditional geometry more interesting for students. Investigating the connection between classical geometry and modern chaos theory seems to be a suitable way of motivating them. Excursions into complex number iterations and interesting Mandelbrot sets offer many methods of discovering and exploring.

In H. Zeitler considers the complex function The classic case n = 2 was investigated by Mandelbrot. We are discussing a more general case where the exponent n is a rational number.

David Pagni drew attention to a result which is ascribed by Dickson [2, p. 286] to Liouville (1857), that the sum of the cubes of the number of divisors of each of the divisors of an integer, is equal to the square of their sum. For example, the divisors of 6 are 1, 2, 3, and 6, which have 1, 2, 2, and 4 divisors respectively, and

Pagni observed, as have others, including Mason et al. [4, p. 179], that the reason Liouville’s result works is because the numbers generated are element-by-element products of sequences of the form {1, 2, ..., t}, which are well known to have the sum of their cubes equal to the square of their sum.

Cyclotomic polynomials, (xp - l)/(x-1), have (p-1) zeros exp(2πiq/p) (q = 1, 2, ... ,p - 1). We consider related polynomials whose zeros are given by cos(qπ/p) (q = 1, 2, ... ,p - 1) and indicate their relation to polynomials Pn (ζ) that satisfy the ‘double angle’ condition Pn(2ζ2 - l) = Pn(ζ)Pn(-ζ). The double angle condition is generalised to a chain of double angle conditions Qi(2ζ2 - 1) = Qi+1(ζ)Qi+1(-ζ), i = 1, 2, ...,n(Q1 ≡ Qn+1) For purposes of illustration, we will focus on P4(ζ), demonstrating how construction of the fourth order polynomial subject to the double angle condition yields an equation for cos . We also discuss more general features of the polynomials Pn (ζ). Furthermore, we illustrate how a simple geometrical construction can be used to derive a polynomial for cos .

The dice game ‘Jolly Roger’ may be played with 5 regular dice or 5 special ‘Jolly Roger’ dice. A ‘Jolly Roger die is a regular die in which the 1 spot is replaced by a skull and crossbones (i.e. replaced by a ‘Jolly Roger’). Play cycles through the players 5 times so that each player has 5 turns in the game. Each turn consists of 3 rolls of the 5 dice. On a particular roll of the 5 dice a player (pirate) can score points (capture cargo) only when some ‘Jolly Rogers’ appear face up. According to Koplow Games (1993):