In 1979 Philip W. Kuchel published a paper [1] in the Mathematical Gazette on using curved mirrors as a means of demonstrating the transformation known as inversion in a circle. He called the mirrors ‘anamorphoscopes’ since he came to the idea as a special case of the conical mirror anamorphosis which was a popular optical toy from the seventeenth century onwards [2]. In this paper we revisit his ideas with current technology and provide some extensions to Kuchel's derivation.

Three of the oldest and most celebrated formulae for π are:

The first is Vieta's product of nested radicals from 1592 [1]. The second is Wallis's product of rational numbers [2] from 1656 and the third is Lord Brouncker's continued fraction [3,2], also from 1656. (In the remainder of the paper, for continued fractions we will use the more convenient notation

In [1] there is a rather nice story regarding the coming into being of the hook-length formula. The year was 1953, and the Canadian mathematician Gilbert Robinson was visiting a fellow mathematician, James Frame, at Michigan State University. One of their discussions concerned the work of Ralph Staal [2], an ex-student of Robinson, and this led to Frame conjecturing the formula. Apparently, Robinson was not at all convinced initially that the formula could be as simple as the one Frame was proposing. He was, however, eventually won over, and the combined efforts of these two mathematicians soon elicited a proof.

Computer programs for dynamic geometry provide a very effective tool for motivating students. It is my experience that a certain effort in preparing an adequate applet and spending some time on a careful presentation can result in a great change of the atmosphere in the classroom. On the other hand, these programs also open new perspectives in geometry exploration and can provoke new interesting research questions related to the already known topics. The possibility of experimenting and the chance of visual examination of the results add new dimensions to research and can also be appealing to non-mathematical audiences.

In the paper [1], Euler was examining properties of the conic sections that could be shared by more general curves. Most of the paper is concerned with ‘oblique-angle diameters’, a concept that seems to have been familiar to his readers in the eighteenth century, but has been ignored today. In this paper we will explain this concept and, led by Euler, develop some of its consequences.

Morley's theorem states that the points of intersection of adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle, known as Morley's triangle. See Figure 1.

Concise proofs of the theorem are given in recent papers [1, 2]. A good picture of important previous work can be obtained by looking at [3, p. 1999], and examining references cited there.

I was prompted to write this article after two of my students came to me (independently) with what was essentially the same query. This concerned the graphical solutions of a particular scenario occurring in Game Theory. In order to state and then fully to understand their question we need to make a little foray into this area of mathematics, which appears as one of the topics in the Decision 2 module of the OCR A level (see [1], for example).