Most school mathematics courses nowadays include a study of symmetry in the euclidean sense—isometries in 2- or 3-dimensional euclidean space which map a figure to itself. This article explores the theme of symmetry in a more general, topological sense. It could form the basis of a short self-contained unit in topology with a worthwhile end-point, which teachers might care to try out with students at sixth form level or perhaps in a college of education course.

“The total work done by the external forces acting on a rigid body is equal to the change in the kinetic energy of the body.” O.K. ? Well, it all depends on definitions.

The work done by a constant force is usually (in nearly all textbooks ?) defined as the product of the magnitude of the force and the displacement of its point of application, this displacement being measured in the direction of the force. Consider then a sheet of paper on which a student has drawn in pencil a straight line which he is now about to erase. He causes the eraser as it moves along to exert a constant force on it, and he exerts another constant force with his other hand to keep the paper still. The desk (supposed smooth) exerts constant normal forces. Of all these forces on the paper the first force does some work, according to the definition, while the others do none. The total work done by them is positive, yet there is no change in kinetic energy of the paper.

The content of this article seems not only to have appeal for, but also to be well within the problem-solving capabilities of, many students of elementary geometry. No new results are cited (see [1]) but the concepts are presented differently, providing a viable and concrete means of explaining elementary partitions of space.

Recently I came across L. A. Graham’s delightful book of mathematical puzzles [1] for the first time and found there the problem I shall give in a moment. It occurred to me to try to solve it in a more general form.

I was able to find an unexpectedly neat and satisfying answer, using a famous result of E. Helly. The solution provides an excellent example of the type of use to which Helly’s theorem can be put, and a striking example of its power.

Transformation geometry is now taught in many schools, and complex numbers are frequently used to describe the geometric transformations. For example, a translation is represented by (where b is a complex number), a rotation by a magnification by and a reflection in the x-axis by

The set of all transformations of the form forms a group and consists of precisely those transformations which preserve distance. Among the subgroups of this group there are the groups which give rise to the well known 17 distinct ‘wallpaper patterns’. These can be described in an elementary and geometric way and are (quite rightly) a popular topic in transformation geometry.

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.

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.