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].

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.

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 .

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.

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