The isomorphism problem for Coxeter groups by B. Muhlherr Coxeter theory: The cognitive aspects by A. V. Borovik From Galois and Lie to Tits buildings by M. Ronan The Coxeter ā¦
The isomorphism problem for Coxeter groups by B. Muhlherr Coxeter theory: The cognitive aspects by A. V. Borovik From Galois and Lie to Tits buildings by M. Ronan The Coxeter element and the branching law for the finite subgroups of $SU(2)$ by B. Kostant Hyperbolic Coxeter groups and space forms by R. Kellerhals Regular and chiral polytopes in low dimensions by P. McMullen and E. Schulte Polytopes, honeycombs, groups and graphs by B. Monson and A. I. Weiss Equivelar polyhedra by J. M. Wills Combinatorics of sections of polytopes and Coxeter groups in Lobachevsky spaces by A. Khovanskii Donald and the golden rhombohedra by M. Senechal Configurations of points and lines by B. Grunbaum Meditations on Ceva's theorem by J. Richter-Gebert Coxeter and the artists: Two-way inspiration by D. Schattschneider The visual mind: Art, mathematics and cinema by M. Emmer.
Spherical and elliptic geometry. As we saw in Ā§1.7, a convenient model for the elliptic plane can be obtained by abstractly identifying every pair of antipodal points on an ordinary ā¦
Spherical and elliptic geometry. As we saw in Ā§1.7, a convenient model for the elliptic plane can be obtained by abstractly identifying every pair of antipodal points on an ordinary sphere. The reflections and rotations which we shall define in Ā§Ā§6.2 and 6.3 are represented on the sphere by reflections in diametral planes and rotations about diameters. In elliptic plane geometry, every reflection is a rotation. This rather startling result is a consequence of the fact that the product of the reflection in any diametral plane and the rotation through Ļ about the perpendicular diameter is the central inversion (or point-reflection in the centre of the sphere), which interchanges antipodal points and so corresponds to the identity in elliptic geometry. In any orientable space (Ā§2.5), a reflection reverses sense, whereas a rotation preserves sense. Thus the above remarks are closely associated with the non-orientability of the real projective plane (in which elliptic geometry operates). On a sphere, corresponding rotations about antipodal points have opposite senses; so the identification of such points abolishes the distinction of sense.
Euclid. Geometry, as we see from its name, began as a practical science of measurement. As such, it was used in Egypt about 2000 B.C. Thence it was brought to ā¦
Euclid. Geometry, as we see from its name, began as a practical science of measurement. As such, it was used in Egypt about 2000 B.C. Thence it was brought to Greece by Thales (640-546 B.C.), who began the process of abstraction by which positions and straight edges are idealized into points and lines. Much progress was made by Pythagoras and his disciples. Among others, Hippocrates attempted a logical presentation in the form of a chain of propositions based on a few definitions and assumptions. This was greatly improved by Euclid (about 300 B.C.), whose Elements became one of the most widely read books in the world. The geometry taught in high school today is essentially a part of the Elements, with a few unimportant changes. According to the best editions, Euclid's basic assumptions consist of five ācommon notionsā concerning magnitudes, and the following five Postulates: I. A straight line may be drawn from any one point to any other point. II. A finite straight line may be produced to any length in a straight line. III. A circle may be described with any centre at any distance from that centre. IV. All right angles are equal. V. If a straight line meet two other straight lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles .
Certain projections of the real polytopes {3, 3, 4}, {3, 4, 3}, {3, 3, 5} suggest highly symmetric coordinates for the self-reciprocal complex polygons 3{3}3, 4{3}4, 3{4}3, 5{3}5 and 3{5}3. ā¦
Certain projections of the real polytopes {3, 3, 4}, {3, 4, 3}, {3, 3, 5} suggest highly symmetric coordinates for the self-reciprocal complex polygons 3{3}3, 4{3}4, 3{4}3, 5{3}5 and 3{5}3. Although there are a number of interesting complications, this suggestion is essentially correct and leads to elegant coordinates for all the sporadic complex polygons. Among the by-products of producing these coordinates we count most significant our new insights about 2{6}3 and our simple proof that the 600 vertices of the real polytope {5, 3, 3} are quite unrelated to the 600 vertices of either 5{6}2 or 5{4}3.
Abstract For positive integers p and q with ( p ā 2)( q ā 2) > 4 there is, in the hyperbolic plane, a group [ p, q ] generated ā¦
Abstract For positive integers p and q with ( p ā 2)( q ā 2) > 4 there is, in the hyperbolic plane, a group [ p, q ] generated by reflections in the three sides of a triangle ABC with angles Ļ/ p , Ļ/ q , Ļ/2. Hyperbolic trigonometry shows that the side AC has length Ļ, where cosh Ļ = c/s , c = cos Ļ/ q , s = sin Ļ/ p . For a conformal drawing inside the unit circle with centre A , we may take the sides AB and AC to run straight along radiiwhile BC appears as an arc of a circle orthogonal to the unit circle. The circle containing this arc is found to have radius 1/ sinh Ļ = s/z , where z = , while its centre is at distance 1/ tanh Ļ = c/z from A . In the hyperbolic triangle ABC , the altitude from AB to the right-angled vertex C is Ī¶, where sinh Ī¶ = z.
A regular q-gon, denoted by {ql or 2{q}2, has q vertices (or corners) and q edges (or sides). Each vertex belongs to two edges, and each edge joins two vertices. ā¦
A regular q-gon, denoted by {ql or 2{q}2, has q vertices (or corners) and q edges (or sides). Each vertex belongs to two edges, and each edge joins two vertices. The vertices on an edge may be identified with the points on the real line with coordinates 1 and -1, the two square roots of 1. Accordingly we call this a 2-edge (or, sometimes, an ordinary edge) and generalize it [ 1 ] to a p-edgewhose pvertices may be identified with the p points on the complex line whose coordinates are the pth roots of 1, namely
Regular complex polytopes in unitary space have been studied since the 1950s. The advent of computer graphics has enabled us to represent these on the real plane in a meaningful ā¦
Regular complex polytopes in unitary space have been studied since the 1950s. The advent of computer graphics has enabled us to represent these on the real plane in a meaningful way with very little effort. Here we are chiefly concerned with a particular sequence of complex polytopes in two, three and four dimensions and with some associated real polytopes.
(1989). Sphere Packings, Lattices and Groups. By J. H. Conway and N. J. A. Sloane. The American Mathematical Monthly: Vol. 96, No. 6, pp. 538-544.
(1989). Sphere Packings, Lattices and Groups. By J. H. Conway and N. J. A. Sloane. The American Mathematical Monthly: Vol. 96, No. 6, pp. 538-544.
(1988). A Challenging Definite Integral. The American Mathematical Monthly: Vol. 95, No. 4, pp. 330-330.
(1988). A Challenging Definite Integral. The American Mathematical Monthly: Vol. 95, No. 4, pp. 330-330.
Abstract Buckminster Fuller has coined the name tetrahelix for a column of regular tetrahedra, each sharing two faces with neighbours, one 'below' and one 'above' [A. H. Boerdijk, Philips Research ā¦
Abstract Buckminster Fuller has coined the name tetrahelix for a column of regular tetrahedra, each sharing two faces with neighbours, one 'below' and one 'above' [A. H. Boerdijk, Philips Research Reports 7 (1952), p. 309]. Such a column could well be employed in architecture, because it is both strong and attractive. The ( n ā 1)-dimensional analogue is based on a skew polygon such that every n consecutive vertices belong to a regular simplex. The generalized twist which shifts this polygon one step along itself is found to have the characteristic equation (Ī» - 1) 2 {( n - 1)Ī» n -2 + 2( n - 2)Ī» n -3 + 3( n - 3)Ī» n -4 + . . . + ( n - 2)2Ī» + ( n - 1)} = 0, which can be derived from tan n Īø = n tan Īø by setting Ī» = exp (2Īø i ).
The regular polytopes in two and three dimensions (polygons and polyhedra) and the āArchimedean solids ā have been known since ancient times. To these, KEPLER and POINSOT added the regular ā¦
The regular polytopes in two and three dimensions (polygons and polyhedra) and the āArchimedean solids ā have been known since ancient times. To these, KEPLER and POINSOT added the regular star-polyhedra. About the middle of last century, L. SCHLAFLI* discovered the (convex) regular polytopes in more dimensions. As he was ignorant of two of the four KEPLER-POINSOT polyhedra, his enumeration of the analogous star-polytopes in four dimensions remained to be completed by E. HESS. Recently, D. M. Y. SOMMERVILLE interpreted the (convex) regular polytopes as partitions of elliptic space, and considered the analogous partitions of hyperbolic space. Some particular processes, for constructing ā uniform ā polytopes analogous to the Archimedean solids, were discovered by Mrs. BOOLE STOTT and discussed in great detail (with the help of co-ordinates) by Prof. SCHOUTE. Further, E. L. ELTE completely enumerated all the uniform polytopes having a certain ā degree of regularity,ā these including seven new ones (in six, seven and eight dimensions).
Among the many beautiful and nontrivial theorems in geometry found here are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morley's remarkable ā¦
Among the many beautiful and nontrivial theorems in geometry found here are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morley's remarkable theorem on angle trisectors. The transformational point of view is emphasized: reflections, rotations, translations, similarities, inversions, and affine and projective transformations. Many fascinating properties of circles, triangles, quadrilaterals, and conics are developed.
A reflection in Euclidean n-dimensional space is a particular type of congruent transformation which is of period two and leaves a prime (i.e., hyperplane) invariant. Groups generated by a number ā¦
A reflection in Euclidean n-dimensional space is a particular type of congruent transformation which is of period two and leaves a prime (i.e., hyperplane) invariant. Groups generated by a number of these reflections have been extensively studied [5, pp. 187-212]. They are of interest since, with very few exceptions, the symmetry groups of uniform polytopes are of this type. Coxeter has also shown [4] that it is possible, by Wythoff's construction, to derive a number of uniform polytopes from any group generated by reflections. His discussion of this construction is elegantly illustrated by the use of a graphical notation [4, p. 328; 5, p. 84] whereby the properties of the polytopes can be read off from a simple graph of nodes, branches, and rings.
A book which applies some notions of algebra to geometry is a useful counterbalance in the present trend to generalization and abstraction. It should give a basis for the geometrical ā¦
A book which applies some notions of algebra to geometry is a useful counterbalance in the present trend to generalization and abstraction. It should give a basis for the geometrical aspects and help to extend understanding of the connections between some classical branches of geometry.
January 3.6.Definitions for Gm'n,p (m odd) in terms of two generators.112 3.7.Cases in which m, n, p are all odd.3.8.The criterion for finiteness; Theorems E and F; Figs. 3, 4. ā¦
January 3.6.Definitions for Gm'n,p (m odd) in terms of two generators.112 3.7.Cases in which m, n, p are all odd.3.8.The criterion for finiteness; Theorems E and F; Figs. 3, 4. 117 3.9.G'-"'" and the Fibonacci numbers; Theorem G; Fig. 5. Chapter IV.Graphical representation 4.1.Dyck's general group picture.124 4.2.The group picture for Gm-n-p; Figs. 6, 7. 125 4.3.The regular map {m, n}p; Fig. 8. 4.4.The semi-regular map {m/n}p; Fig. 9.1939] * Schreier and van der Waerden [l].Their Theorem 1 shows that fy,(p) is the group of isomorphisms of fyiip) (p prime).We shall make frequent use of this theorem, t Hamilton [lj.
(1967). On a Theorem in Geometry. The American Mathematical Monthly: Vol. 74, No. 6, pp. 627-640.
(1967). On a Theorem in Geometry. The American Mathematical Monthly: Vol. 74, No. 6, pp. 627-640.
In complex affine n-space with a unitary metric, a reflection is a congruent transformation leaving invariant all the points of a hyperplane. Thus the characteristic roots of a unitary reflection ā¦
In complex affine n-space with a unitary metric, a reflection is a congruent transformation leaving invariant all the points of a hyperplane. Thus the characteristic roots of a unitary reflection of period p consist of a primitive p th root of unity and n ā 1 unities. A group generated by n reflections is conveniently represented by a graph having a node for each generator and a branch for each pair of non-commutative generators.
An Archimedean solid (in three dimensions) may be defined as a polyhedron whose faces are regular polygons of two or more kinds and whose vertices are all surrounded in the ā¦
An Archimedean solid (in three dimensions) may be defined as a polyhedron whose faces are regular polygons of two or more kinds and whose vertices are all surrounded in the same way. For example, the āgreat rhombicosidodecahedronā is bounded by squares, hexagons and decagons, one of each occurring at each vertex. Thus any Archimedean solid is determined by the faces which meet at one vertex, and therefore by the shape and size of the āvertex figure,ā which may be defined as follows. Suppose, for simplicity, that the length of each edge of the solid is unity. The further extremities of all the edges which meet at a particular vertex lie on a sphere of unit radius, and also on the circumscribing sphere of the solid, and therefore on a circle. These points form a polygon, called the āvertex figure,ā whose sides correspond to the faces at a vertex and are of length 2 cos Ļ/ n for an n -gonal face. Thus the vertex figure of the great rhombicosi-dodecahedron is a scalene triangle of sides .
We begin with some definitions. A cubical graph is a simplicial 1-complex in which each 0-simplex is incident with just three 1-simplexes.
We begin with some definitions. A cubical graph is a simplicial 1-complex in which each 0-simplex is incident with just three 1-simplexes.
In connection with his work on singularities of surfaces, Du Val asked me to enumerate certain subgroups in the symmetry groups of the āpure Archimedeanā polytopes n 21 ( n ā¦
In connection with his work on singularities of surfaces, Du Val asked me to enumerate certain subgroups in the symmetry groups of the āpure Archimedeanā polytopes n 21 ( n < 5), namely those subgroups which are generated by reflections. For the sake of completeness, I have enumerated such subgroups of all the discrete groups generated by reflections (including the symmetry groups of the regular polytopes). The work involved being somewhat intricate, several slips would have been overlooked but for the information that Du Val was able to supply from the (apparently remote) theory of surfaces.
A beginner who thinks that mathematics is either algebra or geometry may not be so very far wrong: other branches of mathematics belong in one camp or the other, with ā¦
A beginner who thinks that mathematics is either algebra or geometry may not be so very far wrong: other branches of mathematics belong in one camp or the other, with little ambiguity; and mathematicians seem born to be either algebraists or geometers, rather as men are said to be inherently either beer drinkers or wine drinkers. There are fashions in mathematics as elsewhere, and of late the algebraists may have had too much of the upper handāin some academic quarters, 'geometer' can sound almost like a nasty word.
The object of the present paper is the geometrical study of the groups of rotation and reflexion of the regular polytopes in higher space, and the extension to these configurations ā¦
The object of the present paper is the geometrical study of the groups of rotation and reflexion of the regular polytopes in higher space, and the extension to these configurations of known results in the cases of the ordinary regular polyhedra. It will appear from the work that the groups can be defined abstractly in terms of a certain number of operations, the relations connecting which have a particularly simple form. For a polytope in n dimensions this number is n ā 1, if we consider the group composed simply of the rotations of the polytope, while if we consider the extended group, which includes the possible reflective symmetries of the polytope, n operations suffice. It appears, further, that with one exception all the groups so obtained possess the property that their operations are expressible in terms of two , so that the entire group can be generated by two operations. The relations connecting these, however, are in general complicated, and the symmetrical forms involving more operations are more convenient to use.
1. Introduction . For integers n and k with 2 ā¤ 2k < n , the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-set and ā¦
1. Introduction . For integers n and k with 2 ā¤ 2k < n , the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-set and edge-set E(G(n, k)) to consist of all edges of the form where i is an integer. All subscripts in this paper are to be read modulo n , where the particular value of n will be clear from the context. Thus G(n, k) is always a trivalent graph of order 2 n , and G (5, 2) is the well known Petersen graph. (The subclass of these graphs with n and k relatively prime was first considered by Coxeter ((2), p. 417ff.).)
(1943). A Geometrical Background for De Sitter's World. The American Mathematical Monthly: Vol. 50, No. 4, pp. 217-228.
(1943). A Geometrical Background for De Sitter's World. The American Mathematical Monthly: Vol. 50, No. 4, pp. 217-228.
Configurations of points in higher space have been known and studied for some time. In what follows we shall classify those configurations of points lying on the hyper-sphere in [4] ā¦
Configurations of points in higher space have been known and studied for some time. In what follows we shall classify those configurations of points lying on the hyper-sphere in [4] which may be said to possess the group property. In this classification we shall seek to enumerate only those configurations which are essentially different from this point of view, in order to bring out any geometrical differences which might throw light on the fundamental problems of the theory of linear groups.
(1968). The Problem of Apollonius. The American Mathematical Monthly: Vol. 75, No. 1, pp. 5-15.
(1968). The Problem of Apollonius. The American Mathematical Monthly: Vol. 75, No. 1, pp. 5-15.
The problem of describing a circle to touch three circles, including the nine special cases when one or more of the radii of the given circles are zero or infinite, ā¦
The problem of describing a circle to touch three circles, including the nine special cases when one or more of the radii of the given circles are zero or infinite, was solved by Apollonius of Perga in a work which was lost, but of which Pappus has given some account in his Mathematical Collections. Towards the end of the l6th century the problem was again taken up and solved by F. Vieta, and since that time it has formed the subject of investigations by many mathematicians, from many different points of view.
14. Cayley tMor.de la g&m.dt pvs fon.213 Sur quelques thoor&mes de la geomotrie de position.(Par Mr. A. Cayley de Cambridge.)JtLn prenant pour donne sur Systeme quelconque de points et de ā¦
14. Cayley tMor.de la g&m.dt pvs fon.213 Sur quelques thoor&mes de la geomotrie de position.(Par Mr. A. Cayley de Cambridge.)JtLn prenant pour donne sur Systeme quelconque de points et de droites, on peut mener par deux points donnes des nouvelles droites, ou trouver des points nouveaux, savoir les points d'intersection de deux des droites donnees; et ainsi de suite.On obtient de cette maniere un nouveau Systeme de points et de'droites, qui peut avoir la proprie'te' que plusieurs des points sont situe's dans une meme droite, ou que plusieurs des droites passent par le meme point; ce qui donne lieu a autant de the'oremes de geometrie de position.On a de'j etudie la the'orie de plusieurs de ces systemes; par exemple de celui de quatre points; de six points, situe's deux a deux sur trois droites qui se rencontrent dans un m&ne point; de six points trois a trois sur deux droites, ou plus ge'ne'ralement, de six points sur une conique (ce dernier cas, celui de Thexagramme mystique de Pascal, n'est pas encore epuise'; nous y reviendrons dans la suite), et meme de quelques systernes dans l'espace.Cependant il existe des systemes plus ge'ne'raux que ceux qui ont ete' examines, et dont les propriete's peuvent etre aper^ues d'une maniere presque intiutive, et qui, a ce que se crois, sont nouveaux.Commensons par le cas le plus simple.Imaginons un nornbre n de points situes d'une maniere quelconque dans l'espace, et que nous designerons par l, 2, 3 ... H. Qu'on fasse passer par toutes les combinaisons de deux points, des droites, et par toutes les combinaisons de trois points des plans; puis eoupons ces droites et ces plans par un plan quelconque, les droites selon des points, et les plans selon des droites.Soha le point qui correspond a la droite mene'e par les deux points Ī±,/Ļ; soit de m&ne Ī²Ī³ le point qui correspond a celle mene'e par les points /?, y, et ainsi de suite.Soit de plus Ī±Ī²Ī³ la droite qui correspond au plan passant par les trois points Ī±, Ī², Ī³ etc. ll est clair que les trois points a , Ī±Ī³, Ī²Ī³ seront situes dans la