Abbott’s Flatland (1884) [1] is the story of the encounter between a two-dimensional creature, a Square, and a visitor from space, a Sphere. The Sphere tries to convince the Square of the reality of the third dimension, but with little success at first. Finally, after several long conversations and a few other worldly demonstrations, the Sphere succeeds in making the Square see the light (or the height?), with rather unfortunate consequences for the poor Square. Flatland became a big hit in the years and decades after it came out, and continues to enjoy a robust reputation today. Much of the fallout generated by Flatland in the more than hundred years since it was written is documented in the annotated version of the book brought out by Stewart [2].

In a September 2005 note in The Mathematics Teacher, Gordon [1] gave a variation on the method of completing the square that fraction-phobic students are likely to find easier, along with the comment: ‘This variation is so obvious it must have been discovered many times over, but I have never seen it in print and thought it would be useful to disseminate it more widely.’ The method Gordon [1] gave is to multiply by 4a, and then add b2, to both sides of ax2 + bx = −c to produce 4a2x2 + 4abx + b2 = b2 −4ac. The left side of this last equation can be factorised as (2ax + b)2, and now we have a pure quadratic whose solution is straightforward.

Let Q be a point on the circumcircle of triangle ABC. The reflections of Q in the three triangle sides are known to be collinear, and the line thus defined contains the orthocentre H. This fact can form the basis of a proof of the existence of the Simson line, or alternatively can be deduced from the existence of the Simson line by enlarging the Simson line by a factor of 2 from the centre Q.

In an article full of concrete examples, [1], Malcolm Perella asked what finite groups can be realised as groups of rational functions. When I first thought about this question, I imagined that most of the answers would be easy to locate in standard literature. Some of them are. But several have defied all my attempts to unearth them. I expect that they are hiding somewhere (as we said in our family when precious toys went astray), but they are certainly not in the books where I expected to find them! The present article attempts to answer Perella’s question by referring to appropriate sources where my search has succeeded, and supplying my own answers in other cases. Examples of the groups that arise are given in a series of exercises for readers.

This article sets out to explore some of the connections between two seemingly distinct mathematical objects: trigonometric functions and the integer sequences composed of the Fibonacci and Lucas numbers. It establishes that elements of Fibonacci/Lucas sequences obey identities that are closely related to traditional trigonometric identities. It then exploits this relationship by converting existing trigonometric results into corresponding Fibonacci/Lucas results. Along the way it uses mathematical tools that are not usually associated with either of these objects.

The most basic problem with differential equations is that of being given an equation that can be put into the form

and one pair of solution values x = x0, y = y0, from which to find either an algebraic form of its solution or the numerical value of y corresponding to a particular value of x. This can most conveniently be interpreted as finding, out of the infinity of solution curves to (1.1), an equation for the unique curve that passes through the point (x0, y0) or calculating the coordinates of one particular point on that curve.

This article came about as a response to the Supplement to the Gazette Number 514 [1] which consists of an extended version of Sir Christopher Zeeman’s Presidential Address to The Mathematical Association at York in 2004. His intention was to stimulate the teaching of 3-dimensionaI geometry in schools, and his publication will certainly satisfy some long-felt needs. It contains a short section on conies, which seemed relevant to a topic I am already investigating: the level of knowledge of conic geometry potentially available to the Greeks before the time of Apollonius. Professor Zeeman has encouraged me to publish the first stage of this research in the Gazette to provide a background to his discussion.