H. S. M. Coxeter

Covering space with equal spheres

In a recent paper Rogers [13] has discussed packings of equal spheres in n -dimensional space and has shown that the density of such a packing cannot exceed a certain ratio σ n . In this paper, we discuss coverings of space with equal spheres and, by using a method which is in some respects dual to that used by Rogers, we show that the density of such a covering must always be at least

World-structure and non-Euclidean honeycombs

Milne (1934) described a one-dimensional system of discrete particles in uniform relative motion such that the aspect of the whole system is the same from each particle. The purpose of the present paper is to construct analogous systems in two and three dimensions. If the uniformly moving observers regraduate their clocks so as to describe each other as relatively stationary, the private Euclidean spaces of the Special Theory of Relativity become public hyperbolic space. This point of view leads to a discussion of uniform honeycombs in hyperbolic space, four of which were discovered by Schlegel (1883, p. 444). One of the new honeycombs, called {4, 4, 3}, has for its vertices the points whose four co-ordinates are proportional to the integral solutions of the Diophantine equation t2 - x2 - y2 - z2 = 1. As a by-product, a simple set of generators and generating relations are obtained for the group of all integral Lorentz transformations (Schild 1949, p. 39). Another by-product is the enumeration of those groups generated by reflexions in hyperbolic space whose fundamental regions are tetrahedra of finite volume. The work culminates in the discovery of a point-distribution whose mesh is seven times as close as that of {4, 4, 3}, though apparently still far too coarse to be of direct cosmological significance. It follows that some irregularity in the distribution of the extragalactic nebulae is almost certainly geometrically inevitable.

Some families of finite groups having two generators and two relations

A class of groups Fa,b,c having two generators and two relations is studied. This class is of interest for two reasons. It is relevant to the study of trivalent O-symmetric graphs and also adds infinite families to the known class of finite groups with two-generator two-relation presentations. Adding a third relation to the presentation for Fa’b’c gives a homomorphic image Ha’b’c. In this paper the orders of the groups Ha’b’c are determined, also the orders of certain subclasses of the groups Fa’b’c. Finally a conjecture is given concerning the normal subgroup G such that Ha,b,c≅Fa,b,c/G.

Symmetrical Combinations of Three or Four Hollow Triangles

A hollow triangle is the planar region bounded by two homothetic and concentric equilateral triangles, that is, a flat triangular “ring.” The edges of the inner triangle are half as long as those of the outer triangle. We will look at two striking constructions in three-space made from such figures.

One-Dimensional Projectivities

We shall find that the one-dimensional projectivity considered in Chapter 4 has two different analogues in two dimensions: one relating points to points and lines to lines, the other relating points to lines and lines to points. The former kind is a collineation, the latter a correlation. Although the general theory is due to von Staudt, * and the names collineation and correlation to Mobius (1827), some special collineations were used much earlier, e.g. by Newton and La Hire.† Moreover, the classical transformations of the Euclidean plane, viz. translations, rotations, reflexions, and dilatations, all provide instances of collineations. Poncelet considered the relation between the central projections of a plane figure onto another plane from two different centres. He called this special collineation a homology. In §5·2 we shall give a purely two-dimensional account of it. Poncelet also considered a special correlation: the polarity induced by a conic. In §5·5, following von Staudt again, we obtain the same transformation without using a conic. We then find that several famous properties of conies are really properties of polarities (which are simply correlations of period two).

Projectivities on a Conic

This chapter deals with those properties of a non-degenerate conic which may be most readily derived by means of the notion that the points on the conic form a range, resembling in many ways the points on a line. Pascal’s theorem is the most famous instance; but its original proof must have been different. The idea of projectivity on a conic is due to Bellavitis (1838). We shall see that the construction for such a projectivity is simpler than for a projectivity on a line. In fact, some authors, such as Holgate, rearrange the material so as to treat ranges on a conic before ranges on a line. Involutions are especially easy to deal with, for the joins of pairs of corresponding points are concurrent, as we shall see in § 7.5.

A Comparison of Various Kinds of Geometry

The ordinary geometry taught in school, dealing with circles, angles, parallel lines, similar triangles and so on, is called Euclidean geometry because it was first collected into a systematic account by the Greek geometer Euclid, who lived about 300 B.C. His treatise, The Elements, is one of the most famous books in the world; probably the Bible is its only rival in the number of copies made and the number of languages into which it has been translated. With a few unimportant changes it is still suitable for the instruction of the young.

Order and Continuity

The order of arrangement of lines in a pencil, like that of points on a circle, is cyclic; we cannot say of three that one is between the other two, but we can say of four that two separate the other two. The correspondence between a pencil and its section enables us to carry over this cyclic order from pencils to ranges. If A and B separate C and D, we write AB//CD. (The idea of a point C lying between A and B belongs to affine geometry and may be interpreted as meaning that AB//CD, where D is the point at infinity on AB.)

The Use of Coordinates

In Chapter 11 we saw how a system of coordinates is inherent in synthetic geometry. In the present chapter we shall reverse the process, building up the analytic geometry from first principles, and deriving the theorems (including the axioms) from properties of numbers. We shall find that the analytic method enables us to solve some problems more easily. On the other hand, it would be a grave mistake to abandon the synthetic method, which is far more stimulating to one’s geometrical ingenuity.

The Introduction of Coordinates

In Chapter 10 we discussed many properties of the real projective line, but there remain certain questions that would be difficult, if not impossible, to answer without using the concept of a coordinate or abscissa. For instance, how can you be sure that a harmonic net does not exhaust all the points on the line?