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]:

I could not resist combining the theme of BCME7 (Mathematical Progressions) with some thoughts on my personal journey and a focus on how mathematics can enable people to progress in so many areas. I like the double meaning tied up in the conference theme. It not only relates to some very interesting mathematics but also shows how mathematics enables progression in many areas. I am going to indulge myself on the way with some glimpses into my own journey through mathematics and the influences on the choices I made. I will also explore such influences more widely, as well as possible implications for teaching, learning and professional development.

The Earth (more precisely, the ‘geoid’ thereof) is known to approximate closely to a slightly oblate spheroid whose unique axis coincides with the Earth's axis of rotation [1,2]. (By ‘spheroid’ is meant is an ellipsoid of revolution, i.e. one with two semi-axes equal; a slightly oblate one has these two semi-axes slightly longer than the unique one.) To the nearest km, the diameter of the ‘geoid’ pole-to-pole is 43 km less than the equatorial diameter of 12756 km. There is a reduction of practical significance (0.527%) in the acceleration of free fall" at sea level between the poles and the equator, and therefore in the weight of objects. Of this, 0.345% derives directly from the rotation of the Earth; the balance of 0.182% results from the purely gravitational effect of the Earth's deviation from sphericity.

The history of cryptography is punctuated by the invention of clever systems to encipher messages and, sometime later, equally clever systems for cryptanalysing the enciphered messages to determine their meaning. Most enciphering schemes of any worth enjoy a relatively lengthy period of prominence before sufficiently determined cryptanalysts undermine their security by figuring out how to attack them. In response, cryptographers devise new and improved schemes and then the cycle repeats. Cryptographers have learned from history that it is dangerous to declare any enciphering scheme unbreakable; at best they are considered to be very secure. But there is one scheme, a scheme that has been around since 1917, that truly is unbreakable. It is the perfect cipher.

Figure 1 shows how a regular dodecahedron can be dissected into three slices by two planes through the two sets of vertices, each set defining a regular pentagon parallel to the top and bottom faces. A surprising result emerges if we calculate the ratio of the volumes of the three slices. We first prove this result directly and then show it by a dissection argument using simple polyhedral pieces of five types. These pieces can be used to build many polyhedra, including the regular dodecahedron, the regular icosahedron, the great dodecahedron, the small and great stellated dodecahedra and all the Archimedean polyhedra which have icosahedral symmetry.

This article is written as a companion piece to the recent account of cryptography in [1]. The aim will be similar—to explain the meaning and principles of coding, give some implementations that are feasible at school and early undergraduate level, and give a flavour of the types of mathematics involved.

2. Cryptography and coding—What's the difference?

Cryptography concerns the transmission of secret or sensitive information in such a way that if intercepted en route by an ‘enemy’, that enemy will be incapable of understanding it and therefore be prevented from using it for any political, financial, military or other advantage.

In an earlier communication to the Gazette [1], the authors in effect showed, in a somewhat complicated manner, how to evaluate the integral One can show in a simpler manner, however, how to evaluate, for integers n and m, a more general integral of the form where n ≥ m, provided that if m = 1, then n is odd.

In addition, the final section to this article shows how to extend the procedure to include integrals for which m does not even have to be an integer, and also how to integrate where such an integral converges.

On 22 September 1636, Fermat wrote to Roberval, [I, FO.II.XIII, p.71], saying that he had found the quadrature of an infinite number of curves. He named in particular the ‘solid parabola’, y = x3, and said that his method was different from Archimedes‘ quadrature of the parabola. He invited Roberval to share his thoughts on this and other matters.

On 11th October 1636, Roberval replied to Fermat [1. FO.II.XIV, p.75], indicating his method for determining the quadrature of the ‘solid parabola’ and extending it to the quadrature of y = x4 and y = x5.

This essay demonstrates an application of prime numbers to the development of a calculator program that solves sudoku puzzles. Among the positive integers, the primes—numbers with exactly two divisors, the numbers themselves and 1—are central to our thinking about numbers. They give us a basis for factoring and divisibility and they contribute to the solution of many problems in the mathematical field of number theory. More important for the purposes of this paper, they provide a way of representing numbers uniquely by prime factors. For example, 6221592 = 23 × 32 × 13 × 172 × 23, any other factorisation differing only in the order of factors.

The three Pauli matrices are normally given [1] as the 2 × 2 matrices:

where ‘i’ is the usual complex number imaginary unit.

These matrices obey the relations a2 = I = b2 = c2(where I is the 2 × 2 identity matrix), as well as the anticommutation relations:

Within the quantities ia,ib and ic,i is a scalar multiplier of the 2 × 2 Pauli matrices and, of course, commutes with each of a, b, c.

In [1] and [2] we demonstrated how Pascal-like triangles arose from the probabilities associated with the various outcomes of a particular game (see Definition 1 below). It was also shown that they could be considered as generalisations of Pascal's triangle. In this article we show how Fibonacci-like sequences arise from our Pascal-like triangles, and demonstrate the existence of simple relationships between these Fibonacci-like sequences and the Fibonacci sequence itself. In addition we will investigate a generalisation of the binomial coefficients that appears when considering an extended version of the game. We start by describing this game.

A classical exercise in recreational mathematics is to find Pythagorean triples such that the legs are consecutive integers. It is equivalent to solve the Pell equation with k = 2. In this case it provides all the solutions (see [1] for details). But to obtain all the solutions of a Diophantine system in one stroke is rather exceptional. Actually this note will show that the analogous problem of finding four integers A, B, C and D such that

In the year 1656 John Wallis published his Arithmetica Infinitorum, [1], in which he displayed many ideas that were to lead to the integral calculus of Newton. In this work we find the celebrated infinite product of Wallis which gives π,

Earlier in 1593, Vieta [2] found another infinite product which gives π

But, since Wallis does not mention it, we suppose that he was unaware of it. (Remarkably, these two seemingly different products are special cases of a more general formula [3].)

From which position on a rugby pitch should a player choose to kick in order to convert a try? The problem is to find the optimal distance from the try-line conforming to a geometrical constraint: Law 13 of the Rugby Union code stipulates, ‘after a try has been scored, the scoring team has the right to take a place kick or drop kick at goal, on a line through the place where the try was scored’.

The mathematical problem is not new and has been treated in the Gazette on several occasions [1, 2, 3], and in each case the modelling has been in two dimensions, taking a planar view of the pitch. Figure 1 represents this ‘planar model’.

This story begins in Nick Lord's workshop session at the Joined Up Mathematics conference at Keele in April 2008. He had decided to talk about Figure 1 and the results concerning it featured in [1]. He was also having an interesting e-mail exchange with Douglas Rogers, who pointed out that iterating such constructions twice produced a triangle homothetic to the original. The exchange focused on the analogous iterated Vecten configuration which John Mason recently discussed in [2]; but Nick could not see a similarly quick argument for the results in [1], which he thus set as a ‘homework problem’ for the workshop participants.

Do the following objects shown on the cover belong together?

We will argue that they do bear a certain kinship, sharing the common gene of cylindrical intersection. In fact, we hope this essay will awaken in the reader the ability to discern several other members of this family in the world around him or her.

Cylinder intersection is used widely in construction. Beautiful examples abound in old world architecture, for instance the famous Florence cathedral in Figure 1.

Linear algebra has many fruitful connections with geometry. This article develops one such connection: the relationship between a 2 × 2 matrix and an associated circle which we call the eigencircle.

This connection was first investigated in a previous paper of ours [1], but the present paper is self-contained, and in fact introduces eigencircles in a different way. Here we discuss some surfaces containing the eigencircle which also have a number of interesting properties and connections with the associated matrix.