The simplest way of picturing a three-dimensional object in a plane is by orthogonal projection; that is, as it were, by taking a photograph from a distance. (Perspective, showing its appearance when viewed from nearby, is more accurate and lifelike but much more troublesome.) For technical drawing, it is customary to use a plan and elevation (and sometimes also side elevation). Quite often there are three perpendicular directions naturally associated with the object, such as those of the sides of a rectangular building. We call these the “left-right”, “front-back” and “down-up” axes, and the plan and elevations associated with these directions are usually easy to draw. Unfortunately, technical drawings of plans and elevations often appear rather artificial, and require training and practice to interpret properly.

The study of unique factorisation is almost as old as mathematics itself. The so-called Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be factorised into a product of primes in only one way, was probably the first major theorem proved. It is interesting to note that Euclid (c. 300 B.C.) did not give the result in this form. He proved (IX 14) that “if a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it” (see [1]).

I am most grateful to the Mathematical Gazette for publishing, in the March 1977 issue, my letter about the problem of finding five positive integers such that the sum of any pair is a perfect square. The response far exceeded my expectations and showed that the problem aroused great interest.

Why is such a nice result rarely, if ever, found in texts? Perhaps because it is usually necessary to use trigonometry to calculate the areas of regular polygons. A regular n-gon inscribed in a unit circle has an area n times that of the isosceles triangle having an edge as base and the centre as apex; that is,

area of regular n-gon = n × (1/2. 1.1. sin (360/n)°) = 1/2n sin (360/n)°.

If we have a system of m linear equations to solve, it is a great simplification to write them in matrix form

Ax = b,

where A is an m x n matrix of coefficients, b is an m-dimensional vector of constants and x is an n-dimensional vector of unknowns.

Some of the group axioms are easy to check. The entries within the table are those outside, so the set is closed under the operation. The first row and column inside the table copy the row and column at the top and side, so 1 is a two-sided identity. A 1 appears in each row and each column of the table, and overall the 1 s appear symmetrically in the table, so each element has a two-sided inverse. The table is a latin square, so that, for given a and b, the equations ax = b and ya = b have unique solutions for x and y.

The following problems concern a rigid four-legged table whose feet may be considered as coplanar points, on a floor to be treated as rigid, conveniently smooth and large, but not flat.

1. (a) If the feet form a square, show that there is a position where all feet are on the floor.

2. (b) As (a), but with isosceles trapezium in place of square.

Of course (b) includes (a), but the diminished symmetry calls for a subtler argument.

1. (c) If the feet do not form a cyclic quadrilateral, show that there may be no position with all feet on the floor.

Most elementary texts on group theory define factor groups. Most beginners find factor groups difficult. The present exposition is based on the notion that it may be helpful to have a look at factor groups in a context slightly wider than that of “factoring out normal subgroups”.

In this article I outline the natural interplay between some simple collections of partial orders, graphs and sequences. The work is elementary and self-contained; I hope that, in broad outline, it provides an interesting tutorial topic.

The first published description of the geoboard appears to have been given as an incidental illustration to Denis Diderot’s Letter on the blind (1774).

The savants of the eighteenth century were interested in blindness as a problem for speculative philosophy. If the mind is a tabula rasa at birth, how will it develop if the sovereign sense is absent? Diderot held that the blind man would not only form different physical concepts, but different aesthetic and even different moral values. In the end he would inevitably come to a different set of religious beliefs and hence, Diderot argued, the concept of God was relative to our senses.

Among the various logically equivalent sets of axioms for boolean algebra ([1] discusses four of them), that referred to in the title of this article is popular because of its symmetry (self-duality) and convenience in application. The textbook presentation is essentially as follows; the qualifiers ‘left’, ‘right’ are often omitted, although the symbolic version indicates that they are intended.