Figure 1 shows how a regular dodecahedron can be dissected into three slices by two planes through the two sets of vertices, each set defining a regular pentagon parallel to the top and bottom faces. A surprising result emerges if we calculate the ratio of the volumes of the three slices. We first prove this result directly and then show it by a dissection argument using simple polyhedral pieces of five types. These pieces can be used to build many polyhedra, including the regular dodecahedron, the regular icosahedron, the great dodecahedron, the small and great stellated dodecahedra and all the Archimedean polyhedra which have icosahedral symmetry.

On 22 September 1636, Fermat wrote to Roberval, [I, FO.II.XIII, p.71], saying that he had found the quadrature of an infinite number of curves. He named in particular the ‘solid parabola’, y = x3, and said that his method was different from Archimedes‘ quadrature of the parabola. He invited Roberval to share his thoughts on this and other matters.

On 11th October 1636, Roberval replied to Fermat [1. FO.II.XIV, p.75], indicating his method for determining the quadrature of the ‘solid parabola’ and extending it to the quadrature of y = x4 and y = x5.

In [1] and [2] we demonstrated how Pascal-like triangles arose from the probabilities associated with the various outcomes of a particular game (see Definition 1 below). It was also shown that they could be considered as generalisations of Pascal's triangle. In this article we show how Fibonacci-like sequences arise from our Pascal-like triangles, and demonstrate the existence of simple relationships between these Fibonacci-like sequences and the Fibonacci sequence itself. In addition we will investigate a generalisation of the binomial coefficients that appears when considering an extended version of the game. We start by describing this game.

A classical exercise in recreational mathematics is to find Pythagorean triples such that the legs are consecutive integers. It is equivalent to solve the Pell equation with k = 2. In this case it provides all the solutions (see [1] for details). But to obtain all the solutions of a Diophantine system in one stroke is rather exceptional. Actually this note will show that the analogous problem of finding four integers A, B, C and D such that

Do the following objects shown on the cover belong together?

We will argue that they do bear a certain kinship, sharing the common gene of cylindrical intersection. In fact, we hope this essay will awaken in the reader the ability to discern several other members of this family in the world around him or her.

Cylinder intersection is used widely in construction. Beautiful examples abound in old world architecture, for instance the famous Florence cathedral in Figure 1.

The beautiful infinite product of radicals

due to Vieta [1] in 1592, is one of the oldest non-iterative analytical expressions for π, In a previous paper [2] the author proved the following two Vieta-like products:

for N even, and

for N odd. Here N is a positive integer, FN and LN are the Fibonacci and Lucas numbers, and is the golden section. (The Fibonacci numbers are F1 = 1, F2 = 1, with the recursion relation , while the Lucas numbers are L1 = 1, L2 = 3 with the same recursion relation )

This work grew out of a simple problem: how could I introduce the basic concepts of calculus to students who were mathematically under-prepared? It has been pointed out by many authors (see [1] and its references) that limits are a major stumbling block for many students beginning their study of calculus. Part of the difficulty with limits is the conceptual confusion between process and value. Some limits can be evaluated by a simple process of substitution, for example,

while others cannot, for example,

I often recommend to my students that they follow their noses when solving mathematical problems. The role of intuition in creative mathematical work is difficult to explain. Still its value should never be underestimated and its development may be fostered by experience, including even unsuccessful trips down dead ends and meandering forays into thickets of apparently untameable complexity. In this article I invite you to corne with me and explore a simple calculus problem using any tools and ideas that seem to offer some promise of moving us along in the right direction. The scenic route embraces Taylor's formula, geometric series expansions, relations in Pascal's triangle, and matrix diagonalisation. Don't forget to bring along a pencil and some scrap paper… and a computer algebra system too, if you like.

The ellipse is well known to students of mathematics at an advanced high school level and at an undergraduate level through its associations with, say, conic sections and their deployment as hardware in dual reflector antenna systems. Perhaps less well known is its appearance on the software side, so to speak, for example in RF (Radio Frequency) engineering in the context of polarisation, where it is known as the polarisation ellipse. It is the purpose here to discuss this manifestation and dwell particularly on the concept of a circumscribing rectangle for this ellipse because such aspects are of interest to the RF engineer who may be involved with measurements to determine its properties. This exercise affords an interesting insight into a very practical application of tertiary level mathematics and whilst it is of course helpful to appreciate the physical concepts involved, a lack of such knowledge does not detract from an appreciation of the underlying sums, which should be within the grasp of the above, interested student.

In this note we will discuss five topics from triangle geometry and occasionally encounter something new. Areal coordinates (presented as Appendix A in [1]) will be used throughout, with the exception of the final section where it is advantageous to employ Cartesians.

The topics are:

(1) examples of collinearities where the distance ratios for important points vary with triangle shape;

(2) a pencil of lines which includes the Fermat, Napoleon and Vecten axes;

(3) two more circles related to the Lester circle;

(4) the family of rectangular hyperbolae which circumscribe the triangle of reference ABC and whose principal member is the Kiepert hyperbola [2];

(5) Cartesian coordinates leading to a solution of the ‘Euler disc problem’ [3].

In [1], Chris Ormell raised a question about the uncountability of the real numbers. Ormell affirmed that only a countable collection of numbers may be defined by a finite number of words, a notion that stems from a letter of Richard (1905), reproduced in [2, p. 142]. This appears to conflict with Cantor's proof (1874) of the uncountability of the reals [3, p. 839]. Poincaré, whom Ormell claims as a witness in his defence, said ‘Now as is well known, Cantor proved that the continuum is not denumerable; this contradicts the proof of Richard. The question therefore arises which of the two proofs is correct. I maintain that they are both correct and that the contradiction is only apparent.’ [3, p. 1072] Poincaré then supplied a new proof that the points on an interval are not countable.