John Rigby

Kaleidoscopes : selected writings of H.S.M. Coxeter

Partial table of contents: The Nine Regular Solids. The Regular Sponges, or Skew Polyhedra. Two Aspects of the Regular 24-Cell. The Densities of the Regular Polytopes I. The Densities of the Regular Polytopes II. The Densities of the Regular Polytopes III. A Challenging Definite Integral. Groups Whose Fundamental Regions Are Simplexes. Discrete Groups Generated by Reflections. Finite Groups Generated by Reflections, and Their Subgroups Generated by Reflections. Orthogonal Trees. The Product of the Generators of a Finite Group Generated by Reflections. Extreme Forms. Regular and Semi-Regular Polytopes I. Regular and Semi-Regular Polytopes II. Regular and Semi-Regular Polytopes III. Factor Groups of the Braid Group. Finite Groups Generated by Unitary Reflections. Index.

Tritangent centres, Pascal's theorem and Thebault's problem

Thébaults problem was posed in 1938, but a solution did not appear until 1983. Since then other solutions of the original problem and of various generalisations have been given. In this paper we discuss the generalisations (including some new results) and various background theorems on tritangent centres, making extensive use of Pascals theorem.

(The Bride's Chair)2

This story begins in Nick Lords workshop session at the Joined Up Mathematics conference at Keele in April 2008. He had decided to talk about Figure 1 and the results concerning it featured in [1]. He was also having an interesting e-mail exchange with Douglas Rogers, who pointed out that iterating such constructions twice produced a triangle homothetic to the original. The exchange focused on the analogous iterated Vecten configuration which John Mason recently discussed in [2]; but Nick could not see a similarly quick argument for the results in [1], which he thus set as a ‘homework problem’ for the workshop participants.

The Soddy spheres of a 4-ball tetrahedron: Part 1

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.

Sums of squares of edge-lengths of cyclic polygons

Problem 87.I in a recent Gazette concerns cyclic quadrilaterals. Let A , B , C and D be distinct points on a circle with radius r . Show that When does equality occur? We shall investigate generalisations of the problem by considering cyclic polygons and also points on a sphere, by distinguishing between sides and diagonals, and by looking at minima as well as maxima. The term ‘edge’ in the title is intended to encompass both sides and diagonals. It is interesting to see the wide variety of methods needed to solve the generalised problems.

86.53 A Proof of Nickalls' Theorem on Tangents and Foci of a Conic

We use Nickalls’ notation 6 _ XY Z to denote the directed angle, measured modulo 180, from the line Y X to the line Y Z; this angle is positive or negative according as the direction of rotation from Y X to Y Z is anticlockwise or clockwise. Also we introduce further notation, non-standard but useful. Choose an initial line (which will later be the x-axis when we come to introduce coordinates). The directed angle between this initial line and the line PQ will be denoted by (PQ). Then clearly

Perfect Precise Colourings of Triangular Tilings

The regular tiling 3,n is a tiling made up of equilateral triangles, with n triangles meeting at each vertex; it occurs on the sphere, in the Euclidean plane, or in the hyperbolic plane, according as n 6. If this tiling can be coloured with n colours in such a way that each colour occurs once at each vertex, such a colouring is said to be precise. It is easy to find a precise colouring when n = 4, and precise colourings when n = 5 and 6 are shown in Fig. 1. In a problem in the American Mathematical Monthly in 1993 [1], Raphael Robinson asked whether a precise colouring exists when n = 7. We shall show that precise colourings exist whenever n < 3, that these colourings can be chirally perfect, and that they can be fully perfect when n is even and n ≠ 8.

ApoUonius: Conies Books V to VII; The Arabic translation of the lost Greek original in the version of the Banu Musa , edited with translation and commentary by G. J. Toomer. Pp 888. DM 238. 1990. ISBN 3-540-97216-1 (Springer)

Contents: Introduction: The Ancient Mathematical Background. Descriptions of Manuscripts. Editorial Principles. Conics: Text and Translation.- Notes on the Text and Translation.- Appendices.- Figures.- Bibliography and Indexes.