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 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.

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.

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 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].)

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.

The December 1978 issue of the Mathematical Gazette [1] contains an elegant and humorous contribution from Anthony Hughes. His Note gives a recipe for determining the optimal spot from which to make a conversion attempt in rugby. Others then elaborated on Hughes' idea; see [2] and [3] in particular. In 1996 Isaksen [4] rediscovered Hughes' results while investigating the kicking of extra points in American gridiron. There are also a number of popularisations and summaries, in which the above results are presented, and in instances rediscovered. The ones of which we are aware are listed in the references; see [5, 6, 7, 8, 9 and 10].