The Geometric Vein

Geometry has been defined as that part of mathematics which makes appeal to the sense of sight; but this definition is thrown in doubt by the existence of great geometers who were blind or nearly so, such as Leonhard Euler. Sometimes it seems that geometric methods in analysis, so-called, consist in having recourse to notions outside those apparently relevant, so that geometry must be the joining of unlike strands; but then what shall we say of the importance of axiomatic programmes in geometry, where reference to notions outside a restricted repertory is banned? Whatever its definition, geometry clearly has been more than the sum of its results, more than the consequences of some few axiom sets. It has been a major current in mathematics, with a distinctive approach and a distinctive spirit.

Tilings of the plane in which each tile is a closed topological disk, or some other simple kind of set, have been extensively studied from many points of view; see [18]. Of special interest are tilings whose tiles are regular polygons; there are many such tilings and they occur frequently in practical applications. In particular, the three regular tilings have been known since ancient times and the uniform (or Archimedean) tilings have been known since Kepler’s pioneering work in the seventeenth century [19].

H. S. M. Coxeter’s work on regular and uniform polytopes is, perhaps, his best-known contribution to geometry. By central projection one can relate each of these polytopes to a tiling on a sphere, and the symmetry properties of the polytopes then lead naturally to various transitivity properties of the corresponding tilings. The main purpose of this paper is to classify all tilings on the 2-sphere with these transitivity properties (and not just those obtained from three-dimensional polytopes). Our results are exhibited in Tables 3 and 4. Here we enumerate all “types” of tilings whose symmetry groups are transitive on the tiles (isohedral tilings), on the edges (isotoxal tilings), or on the vertices (isogonal tilings). The word “type” is used here in the sense of “homeomeric type” for details of which we refer the reader to recent literature on the subjects of patterns and plane tilings, especially [18] and [20].

The motivation for the mathematics presented here should really be viewed as originating with the practitioners of the weaver’s craft. The catalyst that resulted in this particular effort, however, was some recent work of Branko Grünbaum and G. C. Shephard [4,5]. They have carefully analyzed certain geometric objects which represent an idealization of woven fabrics in the plane and their investigations lead, among other things, to remarkable theorems concerning the number and nature of the different kinds of what they call “isonemal”1 fabrics in the plane. They have posed many open problems. The models described and pictured here (see Plates A-E, following page 120) were the result of my investigating one of their problems. The resulting models were a joy to discover and are truly beautiful to behold, but as so frequently happens in mathematics, as the existence of the answer to the original question was unveiled other similar questions seemed to spring forth. And herein lies the major difficulty involved with presenting such embryonic material. It is tempting (and, of course, desirable in the long run) to attack the problem with a great deal of mathematical rigor and preciseness (a) because it will certainly yield to that kind of discussion and (b) because there are beautiful and psychologically satisfying results. I will choose not to do that here because I believe that it is beneficial for the reader to observe first some of the natural beauty and surprise that is felt when viewing these models for the first time (unencumbered by technical detail). My second reason is that I wish, right now, to write an article—not a book.

We say that the convex body (compact convex set with nonempty interior) K tilesd-dimensional Euclidean space Ed (by translation) if there is some family T of translation vectors, such that (i) K covers Ed, and (ii) if t _i ∈ T (i = 1,2) with t₁ ≠ t₂, then K + t₁ and K + t₂ have disjoint interiors; that is, K is simultaneously a covering and packing of Ed. We call K a tiling of Ed (by translation), and call K and its translates in Ktiles. A particularly important case is when T is a lattice (discrete additive subgroup of Ed), when we call K a lattice tiling.

The aim of this paper is to illuminate the connection between the geometry and the arithmetic of the radix representations of the complex numbers and other algebraic number fields. We indicate how these representations yield a variety of naturally defined fractal curves and surfaces of higher dimensions.

On a square billiard table with corners (± 1, ± 1), the path of a ball is easily seen to be periodic if and only if it begins with a line $$ Xx + Yy = N, $$ where X and Y are integers and |N| < |X| + |Y| [16, p. 82]. Ignoring a trivial case, we shall assume XY ≠ 0. We lose no generality by taking these integers to be positive and relatively prime. After any number of bounces, the path is still of the form $$ \pm Xx \pm Yy = N \pm 2k, $$ where k is an integer. Among these paths for various values of k, those that come closest to the origin are of the form $$ \pm Xx \pm Yy = N \pm = N', $$ where 0 ⩽ N′ ⩽ 1. The distance of such a path from the origin is $$ N'/\sqrt {{X^2} + {Y^2}} . $$

The previous paper [4] on the subject of the finite Fourier series (f.F.s.) dealt with some known and some new applications to problems of elementary geometry. In the present second paper we apply it to a beautiful theorem of Jesse Douglas [3] on skew pentagons in space. It is shown here that Douglas’s theorem amounts to the graphical harmonic analysis of skew pentagons and that it is also the source of striking outdoor sculptures. This last opinion is shared by two great art experts, Allan and Marjorie McNab, whom I wish to thank for their encouragement.

It is not generally known among archeologists that there is a universal, cross-cultural classification scheme for the repeated patterns occurring in such diverse media as textiles, pottery, basketry, wall decoration, and the art of M. C. Escher. In this paper we introduce this scheme by means of a “flowchart” which reduces the analysis of any particular pattern to a sequence of simple questions (mostly answered “yes” or “no”). We then apply it to the analysis of the decorated pipes excavated from the K2 site of the Kramo quarter of Begho (Ghana) in January-March, 1979, under the direction of Professor Merrick Posnansky. In the sequel we refer to this site as Begho K2.

The note that follows is based essentially on some investigations which I undertook in about 1930 [4,5], in response to Coxeter’s earliest researches [1] on the pure Archimedean polytopes (PA) _n in n dimensions, later fitted into his more general notation [2] as (n − 4)_2,1 (3 ⩽ n ⩽ 9). It had been remarked that the 27 vertices of (PA)₆ correspond in an invariant manner to the 27 lines on a general cubic surface; and in the discussions that followed amongst the group of students that surrounded H. F. Baker, it soon emerged that there was a similar correspondence between (PA) _n (n = 3,4,5) and the lines on the del Pezzo surface of order 9 − n, between (PA)₇ and the bitangents of a general plane quartic curve, and between (PA)₈ and the tritangent planes of a certain twisted sextic curve. The theory I propose now to outline provides a systematic explanation of all these correspondences, as well as others that were remarked later.

The construction of a cubature formula of strength t for the unit sphere Ω _d in ℝd amounts to finding finite sets X₁,..., X _N ⊂ Ω _d and coefficients a₁,..., a _N ∈ ℝ such that 1.1$$ |{\Omega _d}{|^{ - 1}}\int_{{\Omega _d}}^{} {f(\xi )d\omega } (\xi ) = \sum\limits_{i = 1}^N {{a_i}|{X_i}{|^{ - 1}}} \sum\limits_{x \in {X_\iota }} {f(x),} $$ for all functions f represented on Ω _d by polynomials of degree ⩽ t; cf. [16], [15], [11]. Sobolev [14,15] introduced group theory into the construction of cubature formulae by considering orbits X _ι under a finite subgroup G of the orthogonal group O(d). Thus spherical polytopes and root systems (cf. Coxeter [3]) enter the discussion. There are further relations to Coxeter’s work, since the obstruction to higher strength for a cubature formula is caused essentially by the existence of certain invariants. For finite groups generated by reflections, the theory of exponents and invariants goes back to Coxeter [4].

One property of a (convex) polytope in ℝn is that the vertex set defines the actual subdivision into edges, triangles, etc. The cells (dimension n − 1) are the intersections of the convex hull of the vertices with its bounding hyperplanes. The cells intersect in (n − 2)-dimensional elements, and so on. All these are finite. But for a polytope in ℂn convexity is not available; there is some latitude as to the various elements (now subspaces), subject to suitable conditions on their incidences. For example the fractional polytope $$ \frac{1}{3}\gamma _3^3 $$ and generalized cross polytope $$ \beta _3^3 $$ [10] agree as to vertices and “edges,” but the first has 18 “triangles” whereas the second has 27.

A generalized quadrangle (GQ) of order (s,t) is a point-line incidence geometry S = (P,L,I) with pointset P, lineset L and incidence relation I satisfying the following: (1)Two points are incident with at most one line in common.(2)If x ∈ P, L ∈ L, and xIL (i.e. x is not incident with L),there is a unique pair (y,M)∈ P × L for which xIMIyIL.(3)Each point (respectively, line) is incident with 1 + t lines (respectively, 1 + s points).

A loxodromic sequence of tangent spheres in n-space is an infinite sequence of (n − 1)-spheres having the property that every n + 2 consecutive members are mutually tangent. When considering mutually tangent spheres we’ll always suppose they have distinct points of contact. Given any ordered set of n + 2 mutually tangent (n − 1)-spheres, we can invert into n congruent (n — 1)-spheres sandwiched between two parallel hyperplanes, and hence (since the centres of these n are the vertices of a regular simplex) they are all inversively equivalent. Furthermore, any ordered set of n + 1 mutually tangent (n − 1)-spheres (C₀, C₁,..., C _n can be completed to a set of n + 2 spheres in exactly two ways. Hence the spheres belong to just one sequence 1$$ .{\rm{ }}.{\rm{ }}.{\rm{ }},{C_{ - 1}},{C_0},{C_1},{C_2},\;.{\rm{ }}.{\rm{ }}. $$ with the property that every n + 2 consecutive members are mutually tangent.

What is elementary geometry, and when did it originate? The first of these questions—the content of elementary geometry—is not at all simple, and a clear-cut answer is not possible. The most natural answer for present purposes would be the following: “Elementary geometry is the collection of those geometric concepts and theorems taken up in secondary school, together with immediate consequences of these theorems.” However, in spite of the seeming simplicity of this answer, it raises at once a host of objections. The appeal to the word “geometric” in the definition is in itself hard to interpret, since the question “what is geometry?” also admits no clear-cut answer (on that, more below); but in any case, the rapid rate of change in school curricula in all countries of the world, currently seeming to reach its maximum, would oblige us if we adopted that definition to accept the existence of indefinitely many elementary geometries. The concept would have to change not merely from country to country, but for each given country also from year to year if not even from school to school. In addition, such a definition clearly refers only to the content of the school subject “elementary geometry,” while we are here asking about the content of the corresponding science—or, since the word “science” here may seem pompous, about the corresponding direction of scientific thought.

Let me start with extremum properties of the regular solids, pointing out how fertile a remark of Professor Coxeter turned out to be in this field. The researches started thirty years ago by Coxeter’s remark are still in progress.

In this note we discuss four problems. The first three remain totally intractable; the fourth has recently yielded some interesting results that are as yet incompletely understood.

For a convex body K ⊂ Ed let V(K) be its volume and G(K) = card(K ⋂ ℤd) [G0(K) = card(int K ⋂ ℤd)] be the lattice point number of K [int K]. If K is central symmetric (K = — K), then by Minkowski’s fundamental theorem 1$$ {G^0}(K) = 1 \Rightarrow V(K) \leqslant {{\text{2}}^d}. $$

It has been shown by Fujiwara [4] and Bol [2] that if K is a planar convex body which is not a disk, then it is possible to find a congruent copy K′ of K such that K and K′ have more than two points in common on their boundaries. This result was used by Yanagihara [9] to show that if K is a 3-dimensional convex body with the property that for any congruent copy K′ of K, the boundaries of K and K′ intersect in a planar curve (assuming they do, in fact, meet but do not coincide), then K is a ball.

Let L be a simple closed, rectifiable, plane curve of perimeter 4p. Using a continuity argument, one can prove that there exist on L four consecutive points, A, A′, B, and B′ which divide the perimeter in four equal parts while the segments AB and A′B′ are orthogonal. In March 1956, according to my recollection, Peter Ungar of CIMS (then the NYU Institute for Mathematics and Mechanics) conjectured, while studying properties of quasiconformal mappings, that: If L is convex, then$$ \overline {AB} \; + \;\overline {A\prime B\prime } \; \ge \;{\rm{2}}p, $$with equality iff L is a rectangle.

In this paper an algebraic method will be developed to deal with geometry problems of apparently varied nature. We use as examples the following three theorems.

Surfaces whose prime (i.e. hyperplane) sections are hyperelliptic were studied and classified by Castelnuovo (2). If the sections have genus p, no surface can have order greater than 4p + 4, and any of lesser order is a projection of a normal surface Ф in a projective space S of 3p + 5 dimensions. There is a pencil of conies, none of them singular, on Ф; through each point of Ф passes one of the conies and their planes generate a threefold V of order 3p + 3 (2, § as the paper was later republished with different pagination, it may be advisable to refer to it by sections).

In line with the elementary-geometric character of this paper, our primary concern is with circular transformations in the (Euclidean) plane (thought of differently for different types of transformations—see, for example, [1]). The (easy) extension of the various constructions to n-dimensional space is discussed at the end of the paper. Our aim is to show that it is possible to develop entirely analogous elementary theories of the circular point transformations of Steiner and Möbius [8], the circular axial transformations of Laguerre [5], and the circular contact transformations of Lie [6,7J.

This paper on the geometry of cycles (oriented circles and lines) consists of a new look at some old ideas, using mainly synthetic methods. The figures are used for the communication of essentially simple geometrical ideas, and algebraic calculations are kept to a minimum.

Let me begin by describing one of the gems of classical mathematics which first stirred my own enthusiasm for inversive geometry. It illustrates the elegance of the subject and provides a point of interest which we shall glimpse again in the closing chapters of this account.

There are two different geometric transformations that go by the name of “inversion.” One is inversion in a point, also called “central” inversion, and the other is inversion in a circle or a sphere, which is the basis of inversive geometry. Other than the fact that both of these transformations are of period two, they seem to have little in common except for the name. However, when we extend the concept of inversion in a circle to include inversion in imaginary circles, we find that inversion in an ordinary or ideal point of hyperbolic space can be identified with inversion in an imaginary or real circle at infinity, thus uniting the two meanings of the term. Such a correspondence is possible because the group of isometries of a hyperbolic space of two or more dimensions is isomorphic to the group of circle-preserving transformations of an inversive space of one dimension less. This isomorphism, noted in 1905 by Liebmann [13, p. 54] and subsequently by many others, has been dealt with extensively in two papers by Coxeter [6; 10J.

In order to obtain information about a mapping, it is often advantageous to factorize it into mappings of a special nature. We shall announce a number of results dealing with the factorization of affinities and projectivities into axial affinities and central collineations, respectively. These mappings are as simple as possible, since they leave all points of a hyperplane fixed. We shall distinguish different types of axial affinities such as shears, affine reflections, and affine hyperreflections, and of central collineations such as elations, projective reflections, and projective hyperreflections. We shall be interested in factorizations with a minimal number of factors for each mapping and also in finding upper bounds for the number of factors needed to express all mappings in a certain group.

Let σ be an isometry in a Euclidean vector space V = Vn(ℝ,f) of dimension n, the bilinear form f representing the ordinary inner product.

Deep connections between the classical theory of simple Lie groups and the theory of finite simple groups, discovered by C. Chevalley [1], lead to the construction of finite geometries analogous to geometries of classical Lie groups.

Let P be a finite projective plane of arbitrary odd order n, and let π be a regular polarity of P: that is, a polarity for which there exists an integer s = s(π) such that every line containing two or more absolute points of π contains s + 1 absolute points [11, p. 247J. Baer [1] has shown that the absolute points form an oval when n is odd and nonsquare, and Segre [13] has shown that every oval in a Desarguesian projective plane is a conic.

Let G be a finite, primitive subgroup of GL(V)= GL(n, D), where V is an n-dimensional vector space over the division ring D. Assume that G is generated by “nice” transformations. The problem is then to try to determine (up to GL(V)-conjugacy) all possibilities for G. Of course, this problem is very vague. But it is a classical one, going back 150 years, and yet very much alive today. The purpose of this paper is to discuss both old and new results in this area, and in particular to indicate some of its history. Our emphasis will be on especially geometric situations, rather than on representation-theoretic ones.

An important event in the history of Riemann surface theory was F. Klein’s investigation [6] of the principal congruence subgroup at level seven of the modular group and his subsequent discovery of the famous quartic curve x3y + y3z + z3x = 0. This genus 3 curve was soon put to good use. A. Hurwitz [5] showed that a compact Riemann surface of genus g has no more than 84(g — 1) conformal homeomorphisms and cited Klein’s curve to show that this bound is attainable. It is an easy consequence of Hurwitz’s work that there is a one-to-one correspondence between conformal equivalence classes of compact Riemann surfaces that attain the upper bound and normal subgroups of finite index of $$ \left( {2,3,7} \right)\;{\rm{: = }}\left\langle {\left. {x,y:{x^2}\;{\rm{ = }}{y^3}\;{\rm{ = }}{{(xy)}^7} = \;1} \right\rangle .} \right. $$ Factors of (2,3,7) are therefore called Hurwitz groups. In [2], all such normal subgroups were obtained whose factor is an extension of an Abelian group by PSL₂(1). In other words all Abelian covers of Klein’s surface by compact Riemann surfaces exhibiting Hurwitz’s upper bound were obtained. Here a new infinite family of Hurwitz groups is given whose members act on covers of Klein’s curve. Further, matrix representations of certain groups from [2] are obtained and used to decide precisely when the extension splits.

The object of this paper is the comparison of two notions of (combinatorial) buildings, that of [14] (or [1]), and an earlier version (cf. e.g. [10]), which has lately regained interest through the work of F. Buekenhout on sporadic groups (cf. [2], [3], [16]).

The properties described here were found while investigating relations between the Lie group E₈, Thompson’s simple subgroup of the “Monster” which centralizes an element of order three therein, and the cube root of the modular function j.

Let G be a finite group represented on a real vector space V. We can make G act on the polynomial algebra S(V) on V by g · f(x) = f(g−1x). Classical invariant theory studies the invariant subalgebra $$ S{\left( V \right)^G}\;{\rm{ = }}\mathop \oplus \limits_{j = 0}^\infty \;{S_j}{\left( V \right)^G}. $$ Alternatively, one has the graded, homogeneous ideal I _G , generated by the positive components of S(V)G, and we can form the quotient algebra S _G = S(V)/I _G . For convenience, we call this the coinvariant algebra of G and its elements coinvariants (though this terminology has been used for other purposes).

A finite group defined by n generators and m relations must have m ⩾ n. A finite group is said to have deficiency zero if it has a presentation with n generators and n relations. In 1907 Schur [13] proved important results showing that certain finite groups could not have deficiency zero presentations. Let SL(2, p) denote the group of 2 X 2 matrices of determinant 1 over the field GF(p) p an odd prime, and put PSL(2, p) = SL(2, p)/{±I}. Now PSL(2, p) and SL(2,p) can be generated by two elements, but Schur’s result showed that PSL(2, p) required at least three relations. However, the possibility of a 2-generator 2-relation presentation for SL(2, p) was not excluded.

The definition of the groups $$ {F^{a,b,c}}\;{\rm{ = }}\left\langle {\left. {x,y|{x^2}\;{\rm{ = }}1, x{y^a}x{y^b}x{y^c}\;{\rm{ = 1}}} \right\rangle } \right. $$ was suggested by H. S. M. Coxeter because of their relevance to the search for trivalent 0-symmetric Cayley diagrams, and these groups are studied in [1]. The structure of these groups depends heavily on that of the groups $$ {H^{a,b,c}}\;{\rm{ = }}\left\langle {\left. {x,y|\;{x^2} = 1, {y^{2n}}\;{\rm{ = 1, }}x{y^a}x{y^b}x{y^c}\;{\rm{ = 1}}} \right\rangle ,} \right. $$ where n = a + b + c. The groups Ha,b,c and related groups are studied in [1] and [2]. Perhaps rather surprisingly, the orders of the groups Fa,b,c and Ha,b,c sometimes involve Fibonacci numbers; see for example Theorem 9.1 of [1]. Another connection with Fibonacci numbers emerges from a study of the apparently unrelated class of groups discussed in [3]. In the work of this last paper the groups T(2m) = 〈x,t|xt2xtx2t = 1, xt2m + 1 = t2x2)〉 are shown to have order $$ \mathop {40mf}\nolimits_m^3 \;{\rm{if }}m is odd and \mathop {8mg}\nolimits_m^3 $$ if m is even, where f _m and g _m are Fibonacci and Lucas numbers respectively. (Definitions and useful properties of Fibonacci and Lucas numbers may be found in [3], [5], [8].) To see that the groups T(2m) are in fact closely related to the groups Fa,b,c, put a = xt, b = t to obtain $$ T\left( {2m} \right)\;{\rm{ = }}\left\langle {a,b|{b^{ - 1}}{a^2}b = {a^{ - 2}},aba{b^{ - 2}}a{b^{2m + 1}} = 1} \right\rangle . $$ Then 〈a2〉 is normal in T(2m), and the factor T(2m)/〈a2〉 is isomorphic to F1,−2,2m+ 1.

Contemplation of the convex polyhedra leads to some interesting enumerative problems. How many combinatorially distinct polyhedra are there with n edges? Or, as Kirkman asked, how many with p faces and q vertices?

The purpose of this note is to exhibit examples related to the work of Maciej Syslo [3,5]. The examples concern the nonexistence of Hamilton circuits in some fundamental cycle graphs. Also, some problems concerning fundamental cycle graphs are posed.

For integers n ⩾ 1, k ⩾ 0, and w ⩾ 0, let (n|k) _w denote the number of k-choices 1$$ 1\; \le {\rm{ }}{x_1}\; < \;{x_2}\; < \; \cdot \cdot \cdot \; < {x_k}\; \le \;n $$ from {1,2,3,...,n} satisfying the restrictions 2$$ {x_i}\; - \;{x_{i - 1}}\; - \;{\rm{1 }} \ge {\rm{ }}w for i = 2,3, \ldots , k $$ and $$ n\;{\rm{ + }}{x_1}\; - \;{x_k}\; - \;{\rm{1 }} \ge {\rm{ }}w. $$

Recently Vera Pless, N. J. A. Sloane, and I completed the classification of ternary self-dual codes of length 20 [7]. We produced most of the codes by building on known codes of shorter length (the techniques are outlined and applied to codes of length 16 in a paper of Conway, Pless, and Sloane [1]). However, we used a different method for finding those codes having minimum weight 6. It is based on regarding the words of weight 6 as binary words and then examining the resulting set of binary vectors. The second section of this paper contains a summary of the results used in this approach; a detailed exposition will appear elsewhere [9]. Since the method links binary and ternary codes, the third section presents applications to the two most conspicuous of such linked codes, the ternary extended Golay code of length 12 and the binary extended Golay code of length 24.