The process of cutting squares from rectangles described by D. G. Ball in the June 1974 Gazette, and its connection with continued fractions given by Professor Goodstein in Mathematical note 3359 (December 1974), has an interesting application to electrical circuits.

After constructing Platonic solids, Archimedean solids and some of the Kepler-Poinsot polyhedra, I became interested in zonohedra, like the rhombic dodecahedron and triacontahedron shown in Fig. 1. These

models were new to me and most of what I knew about them at that time is summed up by H. S. M. Coxeter in his book Regular polytopes as follows:

It has been the experience of one of the authors of this article that students in senior secondary and early tertiary courses tend to regard functions defined on split domains and concepts such as differentiability as somewhat pathological and of interest principally as traps for students in pure mathematics.

The number π (as all—or very nearly all—the world doth know) signifies the ratio of the circumference of a circle to its diameter and is also equal to the ratio of the area to the square of the radius. This suggests two generalisations of π which apply to plane figures of any shape:

The aim of coding theory is not to deceive, but to make information impervious to errors in transmission. When messages are being transmitted along wires, the likelihood of mistakes in transmission is small. When messages are being transmitted by radio, lightning can distort reception. When messages such as television pictures are being transmitted across interstellar space, there is a substantial possibility, because of meteorites, that the message received will differ from the message transmitted.

During and after World War II, operational research teams needed new, computer-adaptable methods for optimization problems in industry, engineering, economics and the armed forces. The calculus methods developed in the sixteenth century to solve problems in astronomy, physics and geometry, even when they were applicable, were not always computer-adaptable. Among these new operational research methods is geometric programming, an optimization technique which in its beginning utilized the theorems in this article. The word “geometric” comes from its use of the geometric mean as well as geometrical concepts such as orthogonality and normalization.

Many readers will be familiar with the puzzle illustrated in Fig. 1, in which the triangle ABC is isosceles and angles a, b, c are given. If a, b, c are integers when expressed in degrees then it might appear, at first sight, that so must angle θ be. But this is not the case. The reader will quickly find that θ is not accessible by simple ‘angle chasing’. By this I mean marking in the

angles obtained by making the angles in any triangle sum to 180° and by making the exterior angle of any triangle equal the sum of the interior opposites. The following question arises: Given a, b, c can angle θ be obtained by pure geometry? My conjecture is that the answer is yes in those cases when angles a, b, c and θ are all multiples of the same angle. However, I am far from proving this, and have obtained geometrical derivations only for a limited number of cases, as will be seen.

In Mathematical note 3356 (Gazette No. 405, October 1974) George Berzsenyi produces new groups from old by redefining multiplication in an appropriate fashion. Though the group axioms are easily checked relative to the new multiplication, the structures of the new groups are not immediately clear. This is a common situation when groups are presented by their multiplication tables which, in the present author’s view, are of little use. They do not give a clear picture of how the groups ‘work’, as anyone who has tried to verify the associative law from a table knows to his cost. Indeed, a multiplication table of a group is usually constructed only after its structure has been worked out by some other means. Take for example the non-commutative group of order 6. Much more informative than a multiplication table are its abstract presentation in terms of generators and defining relations, and its concrete realisations as the group of all permutations on three symbols and the group of isometries of the plane that map an equilateral triangle onto itself.

The slogan which forms the title of this note was coined in the 1960s by the opponents of large scale university expansion. It was maintained that an increase in student numbers could only lower academic standards. The purpose of this note is to show that more can mean better or worse according to the criterion used. The theorems proved in the main part of the paper give formal statements of intuitively obvious results, but the amount of misunderstanding generated on this topic suggests that such an exposition is worthwhile. The conclusion of the note must be that comparisons of academic achievement, based on varying proportions admitted to higher education, should be interpreted with great care.

In 1951 received a letter from Prof. H. Davenport, F.R.S., with some information about the factors of 12n ± 1. The results he gave were interesting and not obvious; whether they were novel I do not know. It occurs to me (whether strictly novel or not) they might well be worth publishing, as of general interest and as a tribute to a great and kindly man. However, since Prof. Davenport’s letter was not intended for publication as written, I paraphrase it here with explanation of the background.

In the June 1974 edition of the Mathematical Gazette Dr. C. W. Puritz gave an interesting description of a school-room exercise on the calculation of π and some associated numerical error analysis. This note suggests some further relevant material which his pupils might also find interesting and instructive.