The idea of a recursive function is an important one for students of mathematics and computer science so it is especially interesting to find an elementary though non-trivial example based on traditional mathematics. In problem 68. A (Gazette, March 1984, solutions October 1984) we are given the values of three positive integers bv b7, b3 with the remaining terms of a sequence (bn) obtained recursively by the determinantal condition

The subject matter of my talk is geometry in the sense that I shall deal with such entities as lines, circles, and squares. But the results I aim at bear little resemblance to the theorems of traditional geometry, which describe properties of given configurations. Our present object is quite different; we shall begin by specifying not configurations but processes (for example processes of selection), and then seek to determine how economically or how effectively these processes can be carried out. Questions of this nature are quantitative and the solution, if it is known, is generally expressed by a number. It is for this reason that I speak of 'arithmetical' geometry. The problems of the type I shall discuss are usually simple to state and require very little in the way of mathematical equipment for their understanding. However, their solution often presents formidable difficulties and is in many cases as yet far from complete. Let us now consider in some detail a few of the characteristic problems in this field.

James R. Newman observed that “the most painful thing about mathematics is how far away you are from being able to use it after you have learned it” ([1], Vol. 3, p. 1978). Perhaps there is some sort of natural justice, then, in the fact that one of the most pleasant things about mathematics is that you don’t need to know very much about the subject in order to enjoy the puzzles, pastimes and toys that the study of mathematics has produced.

This article gives instructions for a new way to construct the mathematician’s toy which is called a “rotating ring”. I publish it in hopes that it may give you some pleasure and also provide a means by which you can entice your students to study the more serious aspects of solid geometry.

Formulae for Σl=1n ik when k is a positive integer are well known, and many methods for finding them have been described. This article gives an account of an approximate, but remarkably accurate, formula which I have found for the case k =1/2.

That the advantage of using vector analysis in mathematical physics has met with so little recognition by British applied mathematicians must appear rather strange to those of other countries. There are probably not more than half a dozen among us who habitually use vector methods and notation, at any rate in published work. The fact is still more surprising when we recall the work of Maxwell and Heaviside, seeing that the former gave a sort of authority to the curl and divergence, while the latter had so much to do with the systematic development of the vectorial calculus.

Perhaps the chief reason why vector analysis has not come into more general favour with us is that our leaders in applied mathematics have not felt the need of it. It has often been remarked, and perhaps with some degree of truth, that nothing can be accomplished by vector methods that cannot also be done by Cartesian analysis ; and therefore, it is argued, the change is unnecessary and useless. If we had to deal only with minds of special mathematical ability and analytical insight, this conclusion might be accepted. But with the average student so much of his attention is occupied in dealing with the complex array of symbols of partial differentiation to which he is often led in Cartesian analysis, that he is unable to grasp the inner meaning of the work. It is difficult for him in many cases even to see exactly what is expressed in the formulae obtained, involving as they do the three components of a vector quantity in combinations not easy to visualise. And even if the student succeeds in following the argument it is often almost impossible for him to remember either the train of reasoning or the result arrived at. He does not perhaps see why his equations should be differentiated partially with respect to x, y, z and added, or with respect to z, y and subtracted.

The first part of this article covers some well-known ground in preparation for the second part on the Archimedean honeycomb duals. Mathematically speaking some of this second part is, so far as I know, no more than conjecture: I do not myself know of a proper systematic approach to such honeycombs.

As far as possible I have used names for the polyhedra from standard references (e.g. [1]). But references vary, and I have not found all of them described. In such cases I have tried to choose names which are appropriate and technically correct.

A great deal of attention has recently been given to the problem of revising both school and university syllabuses of pure mathematics. Though applied mathematics has not escaped attention, it has been discussed less intensively. It is therefore of interest that in his inaugural lecture, Professor Bonnor subjected the London University syllabuses in this subject to searching criticism on the ground that it gives a possibly unnecessarily thorough training in classical theoretical physics. He noted that, excepting Imperial College, most of the subject matter contained in the London B.Sc. Special Examination contains ideas which are eighty years old, or older.