When we were preparing our earlier article [1], we thought to look back to see what else had appeared in the Gazette on the subject of polyominoes. A polyomino is a finite collection of cells in the square grid with connected interior - so it is insufficient that cells be connected only corner to corner. Some authors require polyominoes to have a simply connected interior, that is, to be without holes, as was appropriate, for example, in [1] for stack polyominoes. The classic text on polyominoes is Solomon Golomb's engaging book [2], first published in 1965, now supplemented at an even more accessible level by [3].

Nice polynomials are polynomials whose coefficients, roots, and critical points are integers. The earliest papers on nice polynomials [1, 2, 3] give an explicit formula for all nice cubics. All the derivations of this formula use Pythagorean triples.

In 1999 the problem of finding, constructing, and classifying nice polynomials was added to the list of unsolved problems [4] in The American Mathematical Monthly. Other papers soon followed, including [5] and [6], with extending the results of nice polynomials to polynomials with coefficients, roots, and critical points in integral domains D. Such polynomials are called D-nice.

Shortly before the start of my term as MA President, a colleague asked me what my theme was going to be for the year. “Do Presidents have themes?”, I asked. He didn’t know, but thought it might be a good idea. As it happens, a perfect theme was handed to me on a plate. Planning for the 2008 Annual Conference was already underway when I joined Council, and it had already been decided that it was to be called ‘Joined Up Mathematics’.

New discoveries about the Soddy circles of a triangle were published in the Gazette [1] in 1995; they are summarised below. Extensions of wellknown results in the geometry of the triangle to that of the tetrahedron were presented by Zeeman in his 2004 Presidential Address at the Mathematical Association Conference in York [2]. Using the computer software package Cabri 3D, we have discovered new results which extend the Soddy circles of a triangle to Soddy spheres of a special class of tetrahedra. This article is the first of two which present our discoveries, as well as relevant aspects of the established geometry of tetrahedra, together with their proofs.

Is there a three-dimensional equivalent of the Pythagorean theorem? It all depends, of course, what one means by equivalent. There are in fact several distinct results which can with some justice be regarded as threedimensional equivalents of the theorem.

One interpretation of the Pythagorean theorem is as a statement about any rectangle, expressing the square of its diagonal as the sum of the squares of its height and length. This readily extends to the case of a rectangular box (cuboid), for which the square of its long diagonal always equals the sum of the squares of its height, length and depth.

What is the area of the shaded region between the tyre tracks of a moving bicycle such as that depicted in Figure 1 ? If the tracks are specified, and equations for them are known, the area can be calculated using integral calculus. Surprisingly, the area can be obtained more easily without calculus, regardless of the bike’s path, using a dynamic visual approach called the method of sweeping tangents that does not require equations for the curves.

This is the second of two articles presenting new results which extend the Soddy circles of a triangle to Soddy spheres of a special class of tetrahedra. In Part 1 [1], after a brief résumé of the Soddy theorems in two dimensions, we stated the corresponding theorems in three dimensions, discovered partly as the result of computer investigations. These threedimensional Soddy theorems are valid only for a special type of tetrahedron - the four-ball tetrahedron - and we went on to prove some basic results about such tetrahedra.

A friendly engineer recently sent me a version of Figure 1 below, and asked me to explain the connection between it and the well-known Fibonacci Problem (FP) about calculating a population of rabbits. He knew that Figure 1 was related to the ‘Divine Proportion’ or ‘Golden Ratio’ ϕ ( = ½(√5 - l), which also occurs in the solution to FP, and wondered how such different problems could be related by such a number. (He unfortunately regarded ϕ as 0.618 exactly, thus missing a lot of stuff to arouse curiosity.) I knew of various references that I could recommend, but none covered all the things my engineer mentioned, so I constructed the following mathematical development, leaving the relations with biology and architecture to be explained in the books referenced later, see for example [1]. A matrix approach is used here, and may be new to those readers of the Gazette who may be quite familiar with the other material.

In this article a theorem similar to Napoleon’s theorem is established for the Fermat point configuration of a triangle. Areal coordinates are used throughout, with ABC as triangle of reference. For a full account of how to define and use these coordinates see Bradley [1]. Alternative synthetic proofs, generously supplied by the referee, are also included. The presentation of the paper with its emphasis on coordinates was designed in part to show that an algebraic treatment of the Fermat point configuration is possible, as well as the more familiar synthetic or complex number treatments.

As demonstrated in [1], it is possible to generalise the familiar 3-dimensional cube to n dimensions for each n > 0. We present here a fascinating link between a geometrical aspect of these 'n-dimensional hypercubes' and the normal distribution.

In what follows we shall identify with n-dimensional space, and use n-D to denote ‘;n-dimensional’. For the sake of mathematical convenience we initially consider n-D hypercubes of edge length 1 unit whose 2n vertices are situated in at all possible n-tuples (c1, c2, ... , cn), where ck is equal either to 0 or 1, . We denote these mathematical objects by Cn, n = 0,1,2,3, .... Our results, however, can easily be generalised to hypercubes of any edge length, and situated anywhere in .

According to various sources (e.g. [1, p. 102]), the terminology of the power of a point with respect to a circle is due to Steiner. His definition appears in most classical and contemporary geometry textbooks (to mention just a few references, see [2, 3, 4, 5]). The concept of the power of a point has been revisited not only in advanced Euclidean geometry, but also in computational geometry and other areas of mathematics.

In the current literature there are two different definitions of the power of a point with respect to a circle, which we study in detail in section 2. In the first half of the twentieth century there have been published several attempts to generalise the concept of power of the point to real algebraic curves.

Are there solutions of the equation x2 + 1 = 0 ? Carl Fredrich Gauss (1777–1855) conjectured that there was a solution and that it was the square root of - 1 . But since the squares of all real numbers, positive or negative, are positive, Gauss introduced a fanciful idea. His solution to this equation was , which he named i. He integrated i with the real numbers to form a set known as , the complex numbers, where each element in that set was of the form a + bi, where a, . Gauss illustrated this on a graph, the horizontal axis became the real axis and represented the real coefficient, while the vertical axis became the imaginary axis and represented the imaginary coefficient.

The nine points for which the nine-point circle is named are the vertices of three triangles: the medial, orthic, and Euler triangles of a reference triangle ABC. All nine of the vertices lie on the circle, which Dan Pedoe [1] proclaims ‘the first really exciting one to appear in any course on elementary geometry’. An online summary of properties of the circle, which is also called the Euler circle, is given at MathWorld [2].

Readers might at times have wondered at some of the reflector shapes to be seen at airports and satellite communication centres around the world. Typically, such shapes occur when a cone of energy emanates from a focal point to illuminate an offset portion of a paraboloidal reflector (a parabola of revolution). The purpose of this note is to show how such shapes can be determined mathematically using the tools of trigonometry and geometry established at school level. The cross-section of the energy cone is usually circular but in some instances it is elliptic and it is this case that is the focus of the present article.

We first present a general property of nice polynomials, which leads to an important modification of the concept of equivalence classes of nice polynomials. We then establish several properties of nice symmetric and antisymmetric polynomials with an odd number of distinct roots. We give a complete description of all nice symmetric and antisymmetric polynomials with three distinct roots. We then apply our results to nice symmetric sextics and octics. As an important application, all the properties we have established have dramatically increased the speed of a computer search for examples, allowing us to discover the first examples of nice symmetric and antisymmetric polynomials with five distinct roots and the first two examples of nice polynomials with six distinct roots.

Pascal’s triangle is the most famous of all number arrays - full of patterns and surprises. One surprise is the fact that lurking amongst these binomial coefficients are the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers which arose in the Hindu studies of arrangements and selections, together with the Fibonacci numbers from medieval Italy. New identities continue to be discovered, so much so that their publication frequently excites no one but the discoverer.

Euler’s Φ(n) is the number of positive integers n with no factor in common with n. We wish to calculate

This problem arises in connection with the Farey series Fn, i.e. all the fractions from 0/1 to 1/1 (inclusive) with denominator n in their lowest terms, in order of size, so that for example

The parabolic orbit is rarely found in nature although the orbits of some comets have been observed to be very close to parabolic. The parabola is of interest mathematically because it represents the boundary between the open and closed orbit forms. An object moving along a parabolic path is on a oneway trip to infinity never being able to retrace the same orbit again. The velocity of such an object is the escape velocity and its total energy is zero.