Of all the changes that have taken place in the mathematical curriculum, both in schools and universities, nothing is more striking than the decline in the central role of geometry. Euclidean geometry, together with the allied subject of projective geometry, has been dethroned and in some places almost banished from the scene. While educational reform was certainly needed there is always the danger that the pendulum may swing too far the other way and that insufficient attention may be paid to geometry in its various forms. Much of the difficulty here centres round the elusive nature of the subject: What is geometry? I would like to examine this question in a very general way in the hope that this may clarify the educational reasons for teaching geometry, and for deciding what is appropriate material at different levels.

A puzzle which introduces simple ideas relating to deduction within a mathematical system in a concrete and problem-oriented way is the following:

In a recent article (this Gazette, 65 (1981), 87-92) Dr Anderson raised the question of the derivation of the well-known formula for the sum of integer squares, in contradistinction to its mere verification by induction. Some time ago, in response to a request from a colleague, I derived each of the well-known formulas

by arguments based upon the decomposition of rectangles. As a complement to Dr Anderson’s article, these may be of interest; especially, in view of the derivation, to those requiring the formulas for elementary integration from first principles.

Take a look at the ‘string’ (or finite sequence)

6,2,1,0,0,0,1,0,0,0.

Count the occurrences of the numbers:

Notice that the bottom row is the string we started with! In fact the above string is the unique answer to the popular question (posed, for example, in a recent edition of the Trans World Airlines magazine, The Ambassador, August 1978) to find a string

The ‘digital root’ of 73219 is 4, since 7 + 3 + 2 +1 + 9 = 22 and 2 + 2 = 4. This idea features in some popular puzzles (and in a recent Gazette note 65.26). Section 1 of this article concerns some basic properties of digital roots, while Section 2 gives a characterisation of functions like the ‘digital root’ function. The former provides interesting exercises in modular arithmetic; the latter in elementary ring theory.

The purpose of this article is threefold. First to suggest other approaches to the ‘Glass rod problem’ (Gazette note 65.6, March 1981). Second, to show the unexpected connection between that and the article ‘Triangle of triangles’ by H. B. Griffiths on page 10 of the same issue and with the article on ‘Random triangles’ by T. Easingwood in the December 1981 edition. And third, to explore further problems which arise from a consideration of the above.

In the June 1981 Gazette, Ian Anderson discussed several ways of finding the sum of the first n squares:

His methods are mostly algebraic and I wondered if there were more geometric ways of obtaining the formula. After some time, I have found several methods which I present here. These methods are based on counting balls in piles and are suitable for exploratory activity at school level.

The natural numbers are the elements of the set N = {1,2,3,…}, the integers are the elements of the set Z = {0,± 1,±2,…} and the rational numbers are the elements of the set Q = {p/q:pεZ, qε6 N}. A real number is called algebraic if it is the root of a polynomial equation

P(x) = anxn + an−1xn−1 +…+ a1x + a0 = 0

In making a study recently of “whiplash injury” in vehicle impacts, I came upon a fact which surprised me, and further investigation led to some pleasing results involving only elementary mathematics, together with the design of a model which might have a future as an “executive toy” related to Newton’s cradle.

We are looking for all triangles with sides of integer lengths a, b, and c say, with an angle of 60° opposite the side of length a.

One of the problems of the first introduction to the calculus and the subsequent mental imagery developed by the student is that the functions involved are usually given by simple formulae such as f(x) = xn and the derivatives calculated by formulae crunching: f′(x) = nxn−1. The fundamental ideas of the calculus and any relational understanding recede into the background. Getting out of the strait-jacket and considering more general functions at some stage is rarely considered. When it is, it is usually performed in the context of university analysis where pictures are banned because they are claimed to mislead the intuition.

The Voyager missions to Saturn in November 1980 and August 1981 produced a wealth of new information and many spectacular pictures. Naturally attention focused on the physical structure of the planet and its major satellites and on the intricate fine structure of its rings. In the course of both missions many less spectacular observations were made, including the study of a number of small satellites. One of these tiny moons, Dione B (also known as S12), whose discovery from Earth was confirmed by Voyager 1, is the subject of this article.

The Greeks were the first real mathematicians. Littlewood expressed this thought in the following terms; they were not clever schoolboys or scholarship candidates, but Fellows of another College. The sentiment is universally accepted by mathematicians, no matter how clever we consider the ancient Egyptians, nor how under-rated the achievements of the Babylonians (nor indeed how inscrutable the works of the ancient Chinese). But if the Greeks were the first in the field, who was the first of the Greeks?

The lack of any obvious pattern amongst the Bernoulli numbers is one of the shocks of analysis which subsequent familiarity with the many beautiful and simple means of deriving them does not altogether assuage. Historically, they first arose in connection with the sums of the rth powers of the first n integers

In this article we describe how a group of third year mathematics students of the University of Rome who were following a mathematics education option explored some of the properties of hyperbolic plane geometry before preparing a unit on non-Euclidean geometry to be used with sixth-formers at the Liceo Virgilio, Rome. We recall here that the Mathematical Association considered the possibility of such work in its 1923 report on geometry [1]. Nowadays the report is associated with the three ‘stages’ of geometry teaching, A, B and C, but in fact it proposed two further stages. The fifth stage included non-Euclidean geometry both ‘for a few gifted specialists’ and also because it might interest ‘able boys who are not mathematical specialists’.

If you were to ask a random sample of students from your school if they liked mathematics, what do you suppose their responses would be? Almost certainly most of them would respond negatively and some would do so with considerable vigour. But what of the others? Some students do like mathematics and we might well ask why this is so? Do you suppose it is because these students realise that knowledge of mathematics is a necessary survival skill in today’s increasingly technological society? Surely the answer is no. Do you suppose it is because they realise that mathematics has many useful real life applications or that lack of knowledge of mathematics will effectively prevent them for entering college in most disciplines? Again the answer is almost surely no. Certainly mathematics is relevant and we should teach our students that this is so and we should teach them useful and appropriate applications as well. At the same time, it is my experience that student attitudes towards mathematics are much like they are towards exercise—in general, they do not exercise because it contributes to good health, but because exercising is usually done in the form of a game or a competition that they find attractive in and of itself. Similarly, what turns students on to the study of mathematics and provides the needed motivation for learning are the remarkable patterns and surprising results with which our subject is so replete. Successful teachers realise this fact and utilise it to maintain a high level of interest and enthusiasm which enables them at the same time to teach ideas which, while useful and necessary, may be more prosaic.