Popular expositions of the new subject of catastrophe theory have tended to concentrate on ‘soft’ applications, but, as I hope to show, it is perfectly possible to make a serious, important, and mathematically more demanding application intelligible to sixth formers with a minimum of differential calculus. The application is in that part of engineering science which deals with the stability of structures. Catastrophe theory produces nothing really novel in this field, but, by assuming Thorn’s theorem, which will be explained below, it is possible to make the whole subject accessible to those for whom Taylor’s theorem for a number of variables, and the associated concepts, are too difficult. Even for engineers already familiar with the results, catastrophe theory offers an attractive unity within the subject and can point to analogies in related fields.

Many school textbooks state, but few of them prove, that the determinant of a 2 × 2 matrix is the area scale factor of its transformation; whilst a proof of the corresponding result for 3 × 3 matrices is usually delayed until the scalar triple product is available as a technique. Teachers of SMP, or of other ‘modern’ courses containing some development of transformation geometry, may be interested in the following approach to the topic of determinants.

When I began teaching secondary mathematics after several years in industry, I hadn’t been trained as a teacher. Having been away from school mathematics for so long, I was sometimes nonplussed by pupils’ questions.

One day I was showing a mixed-ability first-year class how to obtain the decimal expansion of a vulgar fraction. After explaining the how and, I hoped, the why of the method I did several examples on the blackboard and concluded by telling them that the expansion would either terminate or recur (or words to that effect). At once a small boy demanded to be told how I knew this. This floored me: I hadn’t got a ready answer and I had to promise to think about it.

A tiling of the plane is a family of sets, called tiles, that cover the plane without gaps or overlaps. Usually we are concerned with tilings whose tiles are of a small number of different shapes; familiar examples are the regular and uniform tilings (see, for example, [1]).

Although the mathematical theory of tiling is very old, it still contains a rich supply of interesting and challenging problems. The purpose of this article is to describe some of these. In many cases the mathematical aspect is enhanced by the aesthetic appeal of the resulting tilings.

Among the definitions at the beginning of Book I in Euclid’s Elements [1] there are several that pick out special kinds of triangles and quadrilaterals. In his commentary [2] on Book I, Proclus observes that Euclid classifies triangles in two ways: firstly ‘by sides’ into equilateral, isosceles and scalene triangles; and secondly ‘by angles’ into right-angled, obtuse-angled and acute-angled triangles. With regard to quadrilaterals, Proclus ([2], p. 134) attributes to Posidonius the classification scheme on p. 39, which is to be found in Heath’s edition of the Elements ([1], p. 189). Thus the ancient classification of triangles and quadrilaterals produces three (or six) species of triangles and seven species of quadrilaterals.

In its simplest form the theory of Markov chains states that, if the (i,j) element of matrix M is the probability pij of a one-step transition from state i to state j, then the (ij) element of the matrix Mk gives the probability of transition from state i to state j in k steps. Since the probabilities of transition to the different states add to 1, the sum of the elements in any row of M is necessarily equal to 1. (Readers are warned that in some elementary treatments the alternative convention is adopted of writing pij in the (j,i) position, so that the column sums are 1, in order that the probability vectors should be column matrices, which are more familiar in school mathematics.)

In the Gazette for June 1975 (59, No. 408, 98–106) Colin Tripp showed how the familiar problem illustrated in Fig. 1 (in the isosceles triangle, for certain given values of the angles marked a, b, c, to find θ) could be tackled for most of the possible triples (a, b,c) revealed by a computer investigation.

It is well known to ‘curve-stitchers’ that, if equally spaced points around a circle are numbered 1, 2, 3,..., n (and then repeated cyclically), then the chords joining 1 to 2, 2 to 4, 3 to 6, ..., n to 2n envelop a cardioid. (See [1], p. 41, §16, where the cardioid is described as “the caustic of a circle with respect to a point on its circumference”.) Correspondence with E. H. Lockwood has led to a rather unexpected result from this envelope.

1. It is well known that if p is a prime number not equal to 2, then exactly half the non-zero residues of p are quadratic residues (that is, squares in the arithmetic mod p). What happens if the modulus m is not prime? We can look at this problem in two ways: we can insist that our residues are relatively prime to m, or we can consider all the residues.

The property of “φ sequences” which D. A. Barwick describes in Note 60.31 was proved by reference to the theory of Farey sequences. When the author first showed me the sequences I set about trying to obtain a proof from first principles. This note is the result of that investigation.

The usual basic topological concepts, as seen by students for the first time in a course such as analysis, often seem unintuitive. The following procedure may be more digestible.

Virtually everyone can recognise a boundary point of a set A in a space X. It is merely a point of X all of whose neighbourhoods contain both a point belonging to A and a point not belonging to A. The boundary bA of A consists of the boundary points of A.