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Finite Geometry Notes
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GROUP THEORY GENERALLY --- BOOKS Artin, E. Geometric algebra Wiley Interscience, 1957 A classic. Material on transvections used in Cullinane's "Binary Coordinate Systems." Biggs, N. L., and White, A. T. Permutation groups and combinatorial structures (London Math. Soc. lecture note series, 23) Cambridge U. Press, 1979 Chapter headings -- Permutation Groups, Finite Geometries, Designs, Groups and Graphs, and Maps. Cameron, Peter J. Permutation groups Cambridge U. Press, 1999 Permutation groups generally, with little on Mathieu groups. But the remark (p. 192) is useful: "Mathieu found his groups in the 1870s.... Mathieu simply wrote down generating permutations; it is not clear how he found them." Carmichael, Robert D. Introduction to the theory of groups of finite order Ginn, 1937 (Dover Books reprint, 1956) Excellent on the group theory needed as background for diamond theory. Dickson, L. E. Linear groups with an exposition of the Galois theory Dover (1958 reprint of 1900 book) Pioneering work on the background of the Mathieu group M24 and the isomorphism of A8 with GL(4,2). Kleidman, P. B., and Liebeck, M. W. The subgroup structure of the finite classical groups Cambridge U. Press, 1988 (London Math. Soc. Lecture Notes Series, 129) Cited in Sphere Packings.... Passman, D. S. Permutation groups Benjamin, 1968 From Biggs and White, 1979: "Most books on group theory contain an introduction to permutation groups, but there are few books wholly devoted to the subject. The two standard works are by Wielandt [1964] and Passman [1968]." Weyl, Hermann The classical groups Princeton U. Press, 1946 The "relativity problem" (coordinatizing a reference frame) appears on page 16. Wielandt, Helmut Finite permutation groups Academic Press, 1964 Cited by Griess and others. GROUP THEORY GENERALLY --- PAPERS Cameron, P. J., and Hall, J. I. Some groups generated by transvection subgroups J. Algebra 140 (1991), 184-209 Cited by Cuypers and Steinbach in "Special linear groups generated by transvections...." http://www.win.tue.nl/~hansc/anja2403.pdf Held, Dieter A characterization of the alternating groups of degree 8 and 9 J. Alg. 31 (1974) 91-116 Cited by Griess in Twelve Sporadic Groups. Praeger, Cheryl E. Finite primitive permutation groups: A survey Springer Lecture Notes in Mathematics 1456 (1989), 63-84 Cited in Sphere Packing.... GROUP THEORY GENERALLY --- PREPRINTS/THESES Doherty, Faun C. C. A history of finite simple groups Citeseer, 1997 General background reading. http://citeseer.nj.nec.com/doherty97history.html Kantor, William M., and Penttila, Tim Reconstructing simple group actions ResearchIndex ( year unknown) Abstract: . Let G be a simple primitive subgroup of Sn , specified in terms of a set of generating permutations. If jGj n 5 , efficient algorithms are presented that find "the most natural permutation representation" of G. For example, if G is a classical group then we find a suitable projective space underlying G. A number of related questions are considered. Our notion of "efficiency" takes into account many existing notions, ranging from practical to theoretical ones. 1991 Mathematics Subject Classification: 20B40, 20D08, 20G40, 51N30. http://citeseer.nj.nec.com/139315.html GROUP THEORY GENERALLY --- WEBSITES Baez, John Some thoughts on the number six Author, 1992 The exceptional outer automorphisms of S6, which play an important role in diamond theory, are described. http://math.ucr.edu/home/baez/six.html Bogart, Kenneth, and Orrison, Michael Problems and lecture notes on the Polya-Burnside enumeration theorem (PDF) Dartmouth sites, 1999 Math 68, Fall 1999 -- Algebraic Combinatorics --- Instructors: Kenneth Bogart, Michael Orrison http://www.math.dartmouth.edu/~m68f99/m68f99QuestionsNov5.pdf Cuypers, Hans, and Steinbach, Anja Special linear groups generated by transvections and embedded projective spaces (PDF) Technische universiteit, Eindhoven, websites, 2000 Abstract: We give a characterization of the "special linear groups"... as linear groups generated by a non-degenerate class S of abstract root groups such that the members of elements of S are transvections. http://www.win.tue.nl/~hansc/anja2403.pdf Goss, Jonathan Symmetry of the cube U. of Exeter sites, 2000 Symmetry group O of the cube. U. of Exeter website. http://newton.ex.ac.uk/people/goss/symmetry/O.html Lyons, Richard Math 555: Theory of Finite Groups (PDF) Rutgers U. websites Lecture notes for a graduate-level course on finite groups. http://www.math.rutgers.edu/~lyons/math555/nnotes.pdf Rusin, Dave Group theory and generalizations Northern Illinois University mathematics websites, 2001 An overview of group theory in a single website. From this site: "Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way." http://www.math.niu.edu/~rusin/known-math/index/20-XX.html XIAO Gang PermGroup University of Nice sites, 1999 Online tool allows you to make computations on finite permutation groups, based on the software GAP4. http://wims.unice.fr/~wims/it_tool~algebra~permgroup.html
CODING THEORY GENERALLY --- BOOKS CODING THEORY GENERALLY --- PAPERS Kaikkonen, M. K. Codes from affine permutation groups Designs, Codes, and Cryptography (to appear, as of 1998) Cited in Sphere Packings.... Landrock, P., and Manz, O. Classical codes as ideals in group algebras Designs, Codes, and Cryptography 2 (1992) 273-285 Cited in Sphere Packing.... CODING THEORY GENERALLY --- PREPRINTS/THESES CODING THEORY GENERALLY --- WEBSITES Heumann, Sharon Coding theory and its application to the theory of sphere packing DUM sites, 1998 General reference on coding theory; a brief discussion of Golay code. http://www.mdstud.chalmers.se/~md7sharo/coding/main/main.html National University of Singapore MA 3218 Coding Theory National University of Singapore websites, 2001 Course notes for introductory course on coding theory, including material on finite fields and golay codes. Links to related material, including obituaries of Shannon and Golay. http://www.math.nus.edu.sg/~ma3218/ "Torquemada" What is coding theory? Dejanews, 1999 Posting describing coding theory. http://angelfire.com/pa4/steve/codingth.html
MATHIEU GROUPS AND GOLAY CODES --- BOOKS Aschbacher, Michael Sporadic groups Cambridge U. Press, 1994 From the preface: "... the first step in a program to provide a uniform, self-contained treatment of the foundational material on the sporadic groups." Cameron, Peter J. The geometry of the Mathieu groups (PDF) (Ch. 9 of Projective and polar spaces) QMW websites, U. of London, 2000 The geometry of the Mathieu groups and the Golay codes. http://www.maths.qmw.ac.uk/~pjc/pps/pps9.pdf Conway, John Horton, and Sloane, Neil J. A. Sphere packings, lattices, and groups Springer, 1998 Contains the classic discussion of Mathieu groups and the Miracle Octad Generator of R. T. Curtis. http://www.amazon.com/exec/obidos/ASIN/0387985859/qid=995052959/sr=2-1/ref=aps_sr_b_1_1/103-4353791 Dixon, John D., and Mortimer, Brian Permutation Groups Springer, 1996 Chapter 6 deals with Mathieu groups and Steiner systems. Greenberg, P. J. Mathieu groups Courant Institute, NYU, New York, 1973 See MR 50#4731. Griess, Robert L. Twelve sporadic groups Springer, 1998 Heavy use of the 4x6 Miracle Octad Generator format of Robert T. Curtis for studying the actions of M24, among other sporadic groups. Luneburg Transitive Erweiterungen endlicher Permutationsgruppen Springer-Verlag Lecture Notes in Mathematics, vol. 84, 1969 Cited by Biggs and White, who describe the one-point-extension method of constructing Mathieu groups. MacWilliams, F. J., and Sloane, N. J. A. The theory of error-correcting codes North-Holland Publishing, 1977 Contains excellent material on the Golay codes and the Curtis MOG (Miracle Octad Generator). Praeger, Cheryl E., and Soicher, Leonard H. Low rank representations and graphs for sporadic groups Cambridge U. Press, 1997 Information on Mathieu and other sporadic groups. Thompson, Thomas M. From error-correcting codes through sphere packings to simple groups Mathematical Association of America, 1983 Good overview of designs, codes, Mathieu groups. MATHIEU GROUPS AND GOLAY CODES --- PAPERS Adem, A., Maginnis, J., and Milgram, R. J. The geometry and cohomology of the Mathieu group M12 J. Alg. 139 (1991) 90-133 Mathieu group M12. Cited in Sphere Packings... Assmus, E. F., Jr., and Mattson, H. F., Jr. Perfect codes and the Mathieu groups Archiv. Math. 17 (1966), 121-135 Cited in Sphere Packings.... Beth, T., Fumy, W., and Reiss, H. P. Der wunderschone Oktaden-Generator Mitt. Math. Sem. Giessen 163 (1984), 169-179 The wonderful octad generator. Conway, John H. Three lectures on exceptional groups Academic Press, 1971 Paper: "Three Lectures on Exceptional Groups," by J. H. Conway, in Finite Simple Groups, ed. by M. B. Powell and G. Higman, Academic Press, 1971. This paper is reprinted in the book Sphere Packings, Lattices, and Groups, by Conway and Sloane. Conway, J. H., and Sloane, N. J. A. Orbit and coset analysis of the Golay and related codes IEEE Transactions on Information Theory, 1990 Cited in Sphere Packings.... Curtis, R. T. A new combinatorial approach to M24 Math. Proc. Camb. Phil. Soc. 79 (1976), 25-42 The defining paper for the Miracle Octad Generator. Curtis, R. T. The maximal subgroups of M24 Math. Proc. Camb. Math. Soc. 81 (1977), 185-192 Cited by Griess in Twelve Sporadic Groups. Curtis, R. T. Eight octads suffice Journal of Combinatorial Theory A36 (1984), 116-123 Miracle Octad Generator octads. Curtis, R. T. The Steiner system S(5,6,12), the Mathieu group M12, and the .... Academic Press, 1984 Paper in book Computational Group Theory. edited by M. D. Atkinson, Academic Press, 1984. Curtis, R. T. Further elementary techniques using the Miracle Octad Generator Proc. Edin. Math. Soc. 32 (1989) 345-353 Cited in Sphere Packings.... Curtis, R. T. Natural constructions of the Mathieu groups Math. Proc. Camb. Phil. Soc. 106 (1989) 423-429 Cited in Sphere Packings.... Curtis, R. T. Geometric interpretations of the 'natural' generators of the Mathieu groups Math. Proc. Camb. Phil. Soc. 107 (1990), 19-26 Cited in Sphere Packings.... Cuypers, Hans The Mathieu groups and their geometries (PDF) Technische Universiteit Eindhoven sites, year unknown Mathieu groups and their geometries. http://www.win.tue.nl/~hansc/mathieu.pdf Elkies, Noam D. Lattices, linear codes, and invariants, Part I (PDF) American Mathematical Society Notices, 47 (2000), 1238-1245 Golay codes and Mathieu groups in a wider mathematical context. http://www.ams.org/notices/200010/fea-elkies-1.pdf Elkies, Noam D. Lattices, linear codes, and invariants, Part II (PDF) American Mathematical Society Notices, 47 (2000), 1382-1391 Golay codes and Mathieu groups in a wider mathematical context. http://www.ams.org/notices/200011/fea-elkies-2.pdf Gibson, I. B., and Blake, I. F. Decoding the binary Golay code with miracle octad generators IEEE Transactions on Information Theory 24 (1978), 261-264 Miracle octad generators. Golay, M. Notes on digital coding Proc. IRE 37 (1949), 657 Golay's seminal 1949 work. Hasan, Z. Intersection of the Steiner systems of M24 Discr. Math. 78 (1989), 267-289 Cited in Sphere Packings.... Held, Dieter The simple groups related to M24 J. Alg. (1969) 253-296 Cited by Griess in Twelve Sporadic Groups. Higgs, R. J., and Humphreys, J. F. Decoding the ternary Golay code IEEE Transactions on Information Theory, 1993 Cited in Sphere Packings.... Hughes, D. R. A combinatorial construction of the small Mathieu designs and groups Ann. Discr. Math. 15 (1982), 259-264 Mathieu groups. Jonsson, W. On the Mathieu groups M22, M23, M24, and the uniqueness of the associated Steiner systems Math. Z. 125 (1972), 193-214 Mathieu groups, Steiner systems. Kondo, S. Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of K9 surfaces Duke Math. J. 92 (1998) 593-598 Cited in Sphere Packings.... Kramer, E. S., Magliveras, S. S., and Mesner, D. M. t-designs from the large Mathieu groups Discr. Math. 36 (1981), 171-189 Mathieu groups and designs. Mason, D. R. On the construction of the Steiner system S(5,8,24) J. Alg. 47 (1977), 77-79 Steiner system S(5, 8, 24). Mason, Geoffrey M24 and certain automorphic forms A.M.S., Contemporary Mathematics 45 (1985) 223-244 Cited by Griess in Twelve Sporadic Groups. Mathieu, Emile Memoire sur la nombre de valeurs que peut acquirer une fonction.... Liouville's J. 5 (1860) 9-42 First of 3 defining papers on Mathieu groups, according to Griess in Twelve Sporadic Groups. Mathieu, Emile Memoire sur l'etude des functions des plusieurs quantites..... J. Math. Pures. Appl. (Liouville) (2) 6 (1861), 241-323 The first of the two papers of Mathieu cited by Dixon and Mortimer as defining the Mathieu groups. Mathieu, Emile Sur la fonction cinq fois transitive de 24 quantites.... J. Math. Pures. Appl. (Liouville) (2) 18 (1873), 25-46 Second of the two papers cited by Dixon and Mortimer as defining the Mathieu groups. Murray, John Constructing the isomorphism of A8 with GL(4,2) (PDF) Queen's University, Belfast, websites, 1999 This isomorphism plays an essential role in the structure of the Mathieu group M24. http://journals.eecs.qub.ac.uk/RIA/ProcAI/1999/PA99I2/PDF/99201AI.PDF Paige, N. A note on the Mathieu groups Canadian J. Math. 9 (1957), 15-18 Cited by Aschbacher in Sporadic Groups as recognizing the 12-dimensional invariant submodule VC of the 24-dimensional binary permutation module. Pless, Vera Decoding the Golay codes IEEE Transactions on Information Theory 32 (1986), 561-567 Golay decoding. Stanton, R. G. The Mathieu groups Canadian Journal of Mathematics 3 (1951), 164-174 Mathieu groups. Todd, John On representations of the Mathieu groups as collineation groups J. London Math. Soc. 34 (1959), 406-16 Pioneering work on octads, sextets, etc. Todd, John A representation of the Mathieu group M24 as a collineation group Annali di Math. Pura e Applicata 71 (1966) 199-238 Cited by Griess in Twelve Sporadic Groups. Tonchev, V. D. (see also Pless, Vera) A characterization of designs related to the Witt system S(5,8,24) Math. Z. 191 (1986), 225-230 S(5,8,24). Witt, E. Die 5-fach transitiven Gruppen von Mathieu Abh. Math. Sem. Univ. Hamburg 12 (1938), 256-264 Seminal paper. Witt, E. Uber Steinersche Systemen Abh. Math. Sem. Univ. Hamburg, 12 (1938), 265-275 Seminal paper. Wolfmann, J. A new construction of binary Golay code (24,12,8) using a group algebra over a finite field Discrete Mathematics 3 (1980) Cited on web page with URL given here. http://www.univ-tln.fr/~wolfmann/publi.html Wolfmann, J. A permutation decoding of the Golay code I.E.E.E. International Symposium on Information Theory, Les Arcs, France, 1982 Cited on website whose URL is given. http://www.univ-tln.fr/~wolfmann/publi.html MATHIEU GROUPS AND GOLAY CODES --- PREPRINTS/THESES Chapman, Robin Constructions of the Golay codes -- A survey (PDF) U. of Exeter sites, year unknown Contains material on the Curtis Miracle Octad Generator. http://www.maths.ex.ac.uk/~rjc/etc/golay11.pdf Conder, Marston, and McKay, John Markings of the Golay code ResearchIndex (year unknown) Abstract: The simple Mathieu group M 24 is the automorphism group of a Steiner system S(5; 8; 24), acting naturally on its 24 points. A marking of M 24 (or the associated Golay code) is defined to be a linear ordering x 1 ! x 2 ! : : : ! x 24 of these points, and with any such marking one may associate a 5-tuple (c 0 ; c 1 ; c 2 ; c 3 ; c 4 ), where c k denotes the number of blocks of the given Steiner system containing exactly k pairs of the form fx 2iGamma1 ; x 2i g. In this paper we give a solution to the marking problem for M 24, namely the determination of the spectrum of these 5-tuples, showing that precisely 90 different 5-tuples occur. From the introduction: The Steiner system S(5; 8; 24) is a block design made up of 24 points and 759 blocks, each of size 8, with the property that every 5 points lie in exactly one block. This design is naturally associated with the Golay code, and its automorphism group is the simple Mathieu group M 24. http://citeseer.nj.nec.com/290522.html Curtis, R. On the Mathieu group M24 and related topics Cambridge University (thesis), 1972 Thesis by Curtis. Tanner, R. Michael The tetrahedral Golay code (PDF) UCSC sites, 2000 Golay code reference... Two PDF files are given, one for text and one for figures, on this parent HTML site. http://www.cse.ucsc.edu/~tanner/pubs.html MATHIEU GROUPS AND GOLAY CODES --- WEBSITES Cullinane, S. H. Generating the octad generator Author, 1985 The Miracle Octad Generator of R. T. Curtis is pictured, and a simple way of generating it is described. http://m759.freeservers.com/cgi-bin/i/DTgentheog.jpg Cuypers, Hans The Mathieu groups and designs (PDF) Technical University of Eindhoven websites A 13-page PDF document. http://www.win.tue.nl/~hansc/eidmamathieu.pdf Forney, David Codes on graphs (PDF) MIT sites, 2001 Includes material on Tanner graphs and Golay codes. http://truth.mit.edu/~eyeh/6.451/handouts/L5G.pdf Havlicek, Hans, and Lenz, Hanfried Another simple proof for the existence of the small Witt design (PDF) Technische Universitat, Wien, websites, year unknown Abstract: We give a short proof for the existence of the small Witt design which is based on the projective plane of order three with one point deleted. http://www.geometrie.tuwien.ac.at/havlicek/anothersimple.pdf Pfeiffer, Gotz The subgroups of M24, or how to compute the table of marks of a finite group National University of Ireland, Galway (websites), 1996 Abstract: Let G be a finite group. The table of marks of G arises from a characterization of the permutation representations of G by certain numbers of fixed points. It provides a compact description of the subgroup lattice of G and enables explicit calculations in the Burnside ring of G. In this article we introduce a method for constructing the table of marks of G from tables of marks of proper subgroups of G. An implementation of this method is available in the GAP language. These computer programs are used to construct the table of marks of the sporadic simple Mathieu group. The final section describes how to derive information about the structure of G from its table of marks via the investigation of certain Möbius functions and the idempotents of the Burnside ring of G. The appendix contains tables with detailed information.... http://schmidt.nuigalway.ie/~goetz/pub/marks/marks.html Luers, Ann Dodecahedral faces of M12 U. S. Naval Academy sites, 1997 A 13-page essay which, with its bibliography, is an excellent introduction to the Mathieu groups. The Miracle Octad Generator of R. T. Curtis is briefly discussed. Embedded in the MOG is the 4x4 case of the diamond theorem. http://web.usna.navy.mil/~wdj/m_12.htm Winstead, Chris An introduction to Golay codes Center for Asynchronous Circuit and System Design, University of Utah, 2000 Final section discusses the Miracle Octad Generator. http://www.async.elen.utah.edu/publications/golay-intro.pdfFINITE GEOMETRY --- BOOKS Cameron, Peter J. Parallelisms of complete designs Cambridge U. Press, 1976 Good background on constructions of Mathieu groups. Coxeter, H. S. M. Projective geometry U. of Toronto Press, 1974 Page 93 contains material related to symmetries of finite projective planes -- cited in Cullinane's "Duality and Symmetry" website. http://m759.freeservers.com/duality.html Dieudonne, Jean La geometrie des groupes classiques Springer, 1971 Cited by Griess in Twelve Sporadic Groups. Hirschfeld, J. W. P. Projective geometries over finite fields Oxford U. Press, 1979 Extensive bibliography. From the publisher: "This book is an account of the combinatorics of projective spaces over a finite field, with special emphasis on one and two dimensions. With its successor volumes, Finite projective spaces over three dimensions (1985), which is devoted to three dimensions, and General Galois geometries (1991), on a general dimension, it provides the only comprehensive treatise on this area of mathematics. The area is interesting in itself, but is important for its applications to coding theory and statistics, and its use of group theory, algebraic geometry, and number theory. This new edition is a complete reworking, containing extensive revisions, particularly in the chapters on generalities, the geometry of arcs in ovals, the geometry of arcs of higher degree, and blocking sets. Part I gives a survey of finite fields and an outline of the fundamental properties of projective spaces and their automorphisms; it includes the properties of algebraic varieties and curves used throughout the book and in the companion volumes. Part II covers, in an arbitrary dimension, the properties of subspaces, of partitions into both subspaces and subgeometries, and of quadrics and Hermitian varieties, as well as polarities. Part III is a detailed account of the line and plane; with little reference to the generalities from Parts I and II, the author revisits fundamental properties of the plane and then describes the structure of arcs and their relation to curves. This part includes chapters on blocking sets and on small planes (those with orders up to thirteen). With a comprehensive bibliography containing over 3,000 items, this volume will prove invaluable to researchers in finite geometry, coding theory and combinatorics. This is the only comprehensive work on finite projective spaces available. Bibliography of more than 3000 items. 576 pp.; 20 illus.; 0-19-850295-8" http://www.oup-usa.org/isbn/0198502958.html Hirschfeld, J. W. P. Finite projective spaces of three dimensions Oxford U. Press, 1985 Excellent background for understanding the geometry of the 4x4 array (i.e., PG(3,2)). Taylor, D. E. The geometry of the classical groups Heldermann (Berlin), 1992 Cited by King in Finite Geometries and their Automorphisms (preprint). http://www.amazon.com/exec/obidos/ASIN/3885380099/qid%3D999967703/002-9796887-2109616 Yale, Paul B. Geometry and symmetry Holden-Day, 1968 (later reprinted by Dover) Best background for reading Diamond Theory site. From preface: "This book is an introduction to the geometry of euclidean, affine, and projective spaces with special emphasis on the important groups of symmetries of these spaces." Finite geometry is treated alongside characteristic-zero geometry. There is an introduction to the theory of groups and fields. The Polya-Burnside enumeration theorem is discussed. FINITE GEOMETRY --- PAPERS Assmus, E. F., Jr., and Sardi, J. E. N. Generalized Steiner systems of type 3-(v,{4,6},1) Cambridge U. Press, 1981 In Finite Geometries and Designs, edited by P. J. Cameron, J. W. P. Hirschfeld, and D. R. Hughes, Cambridge U. Press, 1981. Beutelspacher, Albrecht 21-6=15: A connection between two distinguished geometries Am. Math. Monthly 93 (1986) 29-41 An intriguing discussion of group actions on small finite geometries. Conwell, G. M. The 3-space PG(3,2) and its group Ann. of Math. 11 (1910), 60-76 (esp. p. 72) Conwell describes a beautiful correspondence between the 35 lines of PG(3,2) and the 35 partitions of an 8-set into two 4-sets. This correspondence also appears in the Miracle Octad Generator of R. T. Curtis, used to study the Mathieu group. Hall, Marshall Cyclic projective planes Duke Math. J. 14 (1947), pp. 1079 - 1090. Cited in Cullinane's "Duality and Symmetry" website. http://m759.freeservers.com/duality.html Singer, J. A theorem in finite projective geometry and some applications to number theory Trans. American Math. Soc. 43 (1938), 377-385. The origin of Singer cyles. Click on URL given to see citations. http://citeseer.nj.nec.com/context/715156/0 FINITE GEOMETRY --- PREPRINTS/THESES King, O. H. Finite geometries and their automorphisms -- Classical groups ResearchIndex.org, 1999 Abstract: We start with linear algebra (on vector spaces) and use it to obtain results in geometry (on projective spaces). The main references for Classical Groups are Dieudonne ([6], [7]), Taylor ([15]) and Dickson ([5]), though I have used Taylor more than others. http://citeseer.nj.nec.com/king99finite.html FINITE GEOMETRY --- WEBSITES Cameron, Peter J. Finite geometry after Aschbacher's theorem: PGL(n,q) from a Kleinian viewpoint QMW websites, U. of London, year unknown Abstract: Studying the geometry of a group G leads us to questions about its maximal subgroups and primitive permutation representations (the G-invariant relations and similar structures, the base size, recognition problems, and so on). Taking the point of view that finite projective geometry is the geometry of the groups PGL(n; q), Aschbacher's theorem gives us eight natural families of geometric objects, with greater or smaller degrees of familiarity. This paper presents some speculations on how the subject could develop from this point of view. http://citeseer.nj.nec.com/245338.html Cameron, Peter J. Projective and polar spaces U. of London websites, 2000 Cameron's description: "Here is the second edition of Projective and Polar Spaces, by Peter J. Cameron. The first edition was published as QMW Maths Notes 13 in 1991.... Separate chapters are available as PDF files: Preface and table of contents 1. Projective spaces 2. Projective planes 3. Coordinatisation of projective spaces 4. Various topics 5. Buekenhout geometries 6. Polar spaces 7. Axioms for polar spaces 8. The Klein quadric and triality 9. The geometry of the Mathieu groups 10. Exterior powers and Clifford algebras References and index A related set of notes on classical groups is also available." http://www.maths.qmul.ac.uk/~pjc/pps/ Cherowitzo, William Edward Singer cycles (An introduction to the applications of geometry in cryptography) U. of Denver sites, 1999 Of interest for its definition and theorem on Singer cycles. http://www-math.cudenver.edu/~wcherowi/courses/m6221/Geometryproject.htm Cherowitzo, William Edward Bibliography for projective geometry U. of Colorado at Denver websites, 2001 Bibliography for the U. of Colorado at Denver course Math 6221, Projective Geometry. Text: Beutelspacher & Rosenbaum, Projective Geometry: From Foundations to Applications , Cambridge University Press, 1998 http://www-math.cudenver.edu/~wcherowi/courses/m6221/m6221f.html Cullinane, S. H. Modeling the 21-point plane with outer automorphisms of S6 Author, 2000 A new way to construct models of PG(2,4) using subsets of a 6-set. http://m759.freeservers.com/modeling.html Cullinane, S. H. Binary coordinate systems Author, 2001 In this 1984 article, coordinate systems for small binary spaces are described. Simply described group actions are shown to generate linear and affine groups on these spaces. http://m759.freeservers.com/coord.html Cullinane, S. H. Duality and symmetry: A matrix symmetrization problem Author, 2001 Can certain (0,1) matrices be symmetrized? http://m759.freeservers.com/duality.html Haberberger, Evi The point-hyperplane design 2-(15, 7, 3) U. of Bayreuth sites, 1999 Tetrahedral model of PG(3,2). See also the "Smallest Perfect Universe" site. http://www.mathe2.uni-bayreuth.de/~evi/pointhyper15.html Newbold, Mark Stereoscopic animated hypercube Author, 1996 Any 4x4 array (as in diamond theory) has the adjacency structure of the 16 vertices of a hypercube, or tesseract. This site lets you move a hypercube in 4-space and view it from various angles. ("There is such a thing as a tesseract." - Madeleine L'Engle in A Wrinkle in Time) http://dogfeathers.com/java/hyprcube.html Polster, Burkard The smallest perfect universe U. of Adelaide sites, year unknown The finite geometry PG(3,2) is the "smallest perfect universe," according to Polster. The tetrahedral model of this projective 3-space is described. http://www.maths.adelaide.edu.au/pure/bpolster/perfect.html
COMBINATORIAL DESIGNS --- BOOKS Beth, Thomas; Jungnickel, D.; Lenz, Hanfried Design theory, Volume I (Encyclopedia of mathematics and Its applications , Vol. 69) Cambridge University Press, 2000 Hardcover - 700 pages 2nd edition Vol 001 (April 2000) Cambridge Univ. Pr.; ISBN: 0521444322 http://www.amazon.com/exec/obidos/ASIN/0521444322/qid=999964686/sr=1-1/ref=sc_b_1/002-9796887-21096 Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried Design theory, Volume II (Encyclopedia of mathematics and Its applications, Vol. 78) Cambridge University Press, 2000 Hardcover 2nd edition Vol. 2 (February 2000) Cambridge Univ. Pr.; ISBN: 0521772311 http://www.amazon.com/exec/obidos/ASIN/0521772311/qid=999964244/sr=1-3/ref=sc_b_3/002-9796887-21096 Cameron, P. J., and van Lint, J. H. Designs, graphs, codes and their links Cambridge U. Press (London Math. Soc. Student Texts 22), 1991 Excellent background on orthogonality of Latin squares as it relates to finite geometry, exceptional properties of the number six, the Golay codes, etc. Colbourn, C. J., and Dinitz, J. H., editors CRC handbook of combinatorial designs CRC Press, Boca Raton, FL , 1996 Valuable reference manual. http://www.amazon.com/exec/obidos/ASIN/0849389488/qid%3D999964135/002-9796887-2109616 Denes, J., and Keedwell, A. D. Latin squares and their applications Academic Press, 1974 Scholarly study of Latin squares and their connections with finite geometry, etc. Hall, M. Combinatorial theory Blaisdell, 1967 Biggs and White (1979) say that "the standard works of reference [on designs] are by Dembowski [1968] and Hall [1967]." Hedayat, A. S.; Sloane, N. J. A.; and Stufken, John Orthogonal arrays: Theory and applications Springer-Verlag, 1999 From the preface: "This is the first book on the subject since its introduction more than fifty years ago, and can be used as a graduate text or as a reference work. It features all of the key results, many very useful tables, and a large number of exercises and research problems." http://www.research.att.com/~njas/doc/OA.html Ryser, H. J. Combinatorial mathematics Mathematical Assoc. of America, 1963 Pp. 57-58, 77 cited in Cullinane's "Duality and Symmetry" website. http://m759.freeservers.com/duality.html Tonchev, V. D. Combinatorial configurations Longman (London), 1988 Cited in Sphere Packing.... COMBINATORIAL DESIGNS --- PAPERS Abel, R. J. R;, Brouwer, A. E.; Colbourn, C. J.; and Dinitz, J. H. Mutually orthogonal Latin squares (MOLS) CRC Press, Boca Raton, 1996 In CRC Handbook of Combinatorial Designs (C.J. Colbourn and J.H. Dinitz, eds.), CRC Press, Boca Raton, FL (1996), 111--142. http://citeseer.nj.nec.com/context/135241/0 COMBINATORIAL DESIGNS --- PREPRINTS/THESES Colbourn, Charles J., and Dinitz, Jeffrey H. Mutually orthogonal Latin squares: A brief survey of constructions ResearchIndex, 1999 Much more informative than the orthogonal-garage-doors article in the current MAA Focus newsletter (9/8/01). http://citeseer.nj.nec.com/colbourn99mutually.html COMBINATORIAL DESIGNS --- WEBSITES Cherowitzo, William Edward Lecture notes for Math6406, Combinatorial Structures U. of Colorado at Denver websites, 2001 Notes for course on finite combinatorial structures; existence, construction, and applications. Topics include Latin squares, Room squares, Hadamard matrices, block designs and finite geometries and matroid theory. http://www-math.cudenver.edu/~wcherowi/courses/m6406/m6406f.html Colbourn, Charles J., and Dinitz, Jeffrey H. The CRC handbook of combinatorial designs (1995 website) ResearchIndex, 1995 Excellent compact summary of theorems on finite groups and designs, with detailed tables of primitive groups, etc. http://citeseer.nj.nec.com/colbourn95crc.html Colbourn, Charles J., and Dinitz, Jeffrey H. Bibliography of mutually orthogonal Latin squares ResearchIndex , 1999 This ResearchIndex website, as opposed to the Colbourn-Dinitz preprint proper, provides a very convenient list of citations to the literature on mutually orthogonal Latin squares. http://citeseer.nj.nec.com/colbourn99mutually.html
WALSH FUNCTIONS --- BOOKS Beauchamp, K. G. Walsh functions and their applications Academic Press, 1975 Out of print as of August 2001. Golomb, Solomon W. Shift register sequences (Revised edition) Aegean Park Press, Laguna Hills, CA, 1982 The fifteen "stencils" in Golomb's Fig. VIII-8, page 219, are the same as the fifteen affine hyperplanes that account for patterns' symmetry in diamond theory. This figure occurs in a discussion of Rademacher-Walsh functions. Golubov, Efimov, and Skvortsov Walsh series and transforms Kluwer Academic Publishers, 1991, 367 pp., $169.00. ISBN 0-7923-1100-0 Reviewed in Bulletin of the A. M. S., Volume 26, Number 2, April 1992, Pages 348-359; Reviewed by W. R. Wade, U. of Tennessee. See review for overview of the place of Walsh functions in modern analysis. http://www.ams.org/bull/pre-1996-data/199226-2/wade.pdf Schipp, F.; Wade, W. R.; and Simon, P. Walsh series : an introduction to dyadic harmonic analysis Adam Hilger, Ltd., Bristol and New York, 1990 See also Wade's review article in AMS Bulletin of April 1992. http://www.ams.org/bull/pre-1996-data/199226-2/wade.pdf WALSH FUNCTIONS --- PAPERS Fine, J. On the Walsh functions Trans. Amer. Math. Soc., 65 (1949), 372-414 Cited by Wade in 1992 review article. Fine, J. The generalized Walsh functions Trans. Amer. Math. Soc. 69 (1950), 66-77 Cited by Wade in April 1992 AMS Bulletin review article. Wade, W. R. Review of the book Walsh Series and Transforms, by Golubov, et al. (PDF) Bulletin of the American Mathematical Society, April 1992, 348-359 Informative article that gives overview of the place of Walsh functions in modern harmonic analysis. http://www.ams.org/bull/pre-1996-data/199226-2/wade.pdf WALSH FUNCTIONS --- PREPRINTS/THESES WALSH FUNCTIONS --- WEBSITES Ritter, Terry Walsh-Hadamard transforms: A literature survey Ritter, 1966 Literature on properties of Walsh functions. Some mention of their relation to the combinatorially interesting M-sequences. http://www.ciphersbyritter.com/RES/WALHAD.HTM University of Ferrara authors Walsh functions bibliography - Part I - Theory U. of Ferrara, 1987 Theory of Walsh functions -- a bibliography. http://felix.unife.it/Root/d-Mathematics/d-Analysis/b-Walsh-functions-theory University of Ferrara authors Walsh functions bibliography - Part II - Applications U. of Ferrara, 2000 Applications of Walsh functions -- a bibliography. http://felix.unife.it/Root/d-Mathematics/d-Analysis/b-Walsh-functions-applications
SYMMETRY IN GENERAL --- BOOKS Shubnikov, A. V., Belov, N. V., et al. Colored symmetry Pergamon Press, 1964 Overview of color symmetry. See also Shubnikov and Koptsik, Symmetry in Science and Art. Plenum Press, New York, London. Shubnikov, A. V., and Koptsik, V. A. Symmetry in science and art Plenum Press, 1974 Overview. http://www.emis.de/monographs/jablan/refer.htm#sb64 Weyl, Hermann Symmetry Princeton U. Press, 1952 A classic. SYMMETRY IN GENERAL --- PAPERS SYMMETRY IN GENERAL --- PREPRINTS/THESES Cullinane, S. H. Diamond theory Author, 1976 The original version of the material discussed on the diamond theory website. http://m759.freeservers.com Cullinane, S. H. Symmetry invariance in a diamond ring (A.M.S. abstract 79T-A37) Notices of the Amer. Math. Soc., February 1979, pages A-193, 194 Original version of the "4x4 case" discussed in diamond theory website. http://m759.freeservers.com/ SYMMETRY IN GENERAL --- WEBSITES Aslaksen, Helmer Mathematics in art and architecture National University of Singapore websites, 2000 Very nicely laid out elementary survey, with considerable material on the aesthetics of symmetry. http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml#Symmetry Brading, Katherine, and Castellani, Elena Symmetries in physics: new reflections Sub-faculty of Philosophy, Oxford University websites, 2001 The intersection of the pure mathematics of symmetry with philosophy and physics. Contributed papers, with abstracts and some bibliography. Excellent starting point for further research. http://units.ox.ac.uk/departments/philosophy/students/graduate/physics/contents.html Cederberg, Judith Bibliography of works on symmetry (72 items) St. Olaf College websites (year unknown) Helpful undergraduate-level survey of works on symmetry. Also lists some software and videos, following the 72-item bibliography. http://www.stolaf.edu/people/cederj/Courses.dir/Geo.dir/bib-356/node16.html Cullinane, S. H. Diamond theory Author, 2000 Diamond Theory home page. http://m759.freeservers.com/ Hammond, Todd Symmetry (Annotated bibliography) Truman State University websites (year unknown) Annotated bibliography of works on symmetry. http://math.truman.edu/~thammond/history/Symmetry.html ISIS (International Society for the Interdisciplinary Study of Symmetry) ISIS-Symmetry website ISIS, 2001 P.O. Box 994, Budapest, (18 Nador St.) H-1245 Hungary Phone: 36 (1) 131-8326, Fax: 36 (1) 131-3161, E-mail: h492dar@ella.hu, sym@freemail.hu http://members.tripod.com/vismath/isis0.htm Jablan, Slavik V. Symmetry and ornament www.emis.de, 1995 Symmetry and Ornament website. http://www.emis.de/monographs/jablan/ Jablan, Slavik V. Bibliography on symmetry www.emis.de, 1995 Bibligraphy from Symmetry and Ornament website. http://www.emis.de/monographs/jablan/refer.htm#sb64 Lungu, Alexandru P. The recent generalizations of colored symmetry Author (year unknown) Recent generalizations of colored symmetry. http://members.tripod.com/vismath5/lungu/
AESTHETICS/PSYCHOLOGY OF ART --- BOOKS Gombrich, E. H. The sense of order Cornell U. Press, 1979 Designs of Dominique Douat (1722) are illustrated and discussed on pages 70-72. AESTHETICS/PSYCHOLOGY OF ART --- PAPERS AESTHETICS/PSYCHOLOGY OF ART --- PREPRINTS/THESES AESTHETICS/PSYCHOLOGY OF ART --- WEBSITES Cullinane, S. H. A mathematician's aesthetics Author, 2000 Quotations about the aesthetics of mathematics. http://m759.freeservers.com/PHaesthetics.html Northern New Mexico Community College Design Foundation (Chapter 11) (PDF) NNMCC sites, year unknown Navajo aesthetics and Douat designs. http://design-foundation.nnm.cc.nm.us/dsf/dsf/edu/designhistory/chapter11.pdf
TILINGS AND GRAPHIC DESIGNS -- BOOKS Douat, Dominique Methode pour faire une infinité de desseins differents avec des carreaux .... U. of Paris, 1722 Full title: Methode pour faire une infinité de desseins differents avec des carreaux mipartis de deux couleurs par une ligne diagonale : ou observations du Pere Dominique Douat Religieux Carmes de la Province de Toulouse sur un memoire inséré dans l'Histoire de l'Académie Royale des Sciences de Paris l'année 1704, présenté par le Reverend Sebastien Truchet religieux du même ordre, Academicien honoraire, imprimé chez Jacques Quillau, Imprimeur Juré de l'Université, Paris 1722. http://www.irisa.fr/faqtypo/truchet/truchet3E.html Grunbaum, Branko, and Shephard, G. C. Tilings and patterns W. H. Freeman and C ompany, 1987 Grunbaum and Shephard on tilings. Wieting, T. W. The mathematical theory of plane chromatic ornaments Dekker, New York, 1982 Plane chromatic ornaments. http://www.emis.de/monographs/jablan/refer.htm#sb64 TILINGS AND GRAPHIC DESIGNS -- PAPERS Esperet, Philippe, and Girou, Denis Coloriage de pavage dit.... Cahiers GUTenberg no. 31 (December 1998) PDF paper on coloring Truchet tilings. http://www.loria.fr/~roegel/TeX/31-girou.pdf Smith, Cyril S., and Boucher, Pauline The Tiling Patterns of Sebastien Truchet.... Leonardo, vol. 20, num.4 (1987) 373--385. "The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy" http://www.irisa.fr/faqtypo/truchet/truchet3E.html van der Laan, Kees Tiling in PostScript and METAFONT -- Escher's Wink (PDF) Dutch language oriented TeX Users Group, 1997 Tiling by computer. http://www.ntg.nl/maps/pdf/19_12.pdf TILINGS AND GRAPHIC DESIGNS -- PREPRINTS/THESES TILINGS AND GRAPHIC DESIGNS -- WEBSITES Andre, Jacques, and Girou, Denis Truchet & types: tiling systems and ornaments Authors, 1998 Some history of Truchet's tiling systems. http://www.irisa.fr/faqtypo/truchet/truchet3E.html Huff, Kenneth A. The Art of Kenneth A. Huff Author, 2001 Truchet patterns as a basis for artworks. http://www.itgoesboing.com/Presentations/SIGGRAPH99Sketch/3.html ISIS Visual mathematics ISIS, 2001 Electronic Journal of ISIS-Symmetry. http://members.tripod.com/vismath/ Jablan, Slavik V. Tiling patterns derived from Truchet tile Author, year unknown Tiling patterns derived from Truchet tile (see, E.Gombrich: The Sense of Order, Cornell University Press, Ithaca, New York, 1979; C.S.Smith: The Tiling Patterns of Sebastian Truchet and the Topology of Stryctural Hierarchy, Leonardo 20, 4 (1987), 373-385). http://members.tripod.com/~modularity/truch.htm Lyon, Mike Tiling Author, 2000 Lyon, an artist, describes his 4x4 patterns made up of two-color diagonally-divided squares. His patterns include many of diamond theory, but also include many others. http://www.kc-shotokan.com/mike/Tiling/tiling.htm
OTHER --- BOOKS Heath, Thomas L., Sir The thirteen books of Euclid's elements, Volume I Dover Publications reprint, 1956 Plato's diamond is on page 352. Trudeau, R. J. The non-Euclidean revolution Birkhauser Boston, 1987 Trudeau's "story theory" of truth is opposed to the traditional "diamond theory" of truth, exemplified by Plato and Euclid. http://www.birkhauser.com/cgi-win/ISBN/0-8176-3311-1 OTHER --- PAPERS Todd, J. A. As it might have been Bull. London Math. Soc. 2 (1969), 1-4 Cited by King in Finite Geometries and their Automorphisms (preprint). http://citeseer.nj.nec.com/cachedpage/329143/37 OTHER --- PREPRINTS/THESES OTHER --- WEBSITES Cameron, Peter J. Quotations related to combinatorics QMW sites, U. of London, 2000 Quotations on aesthetics of combinatorial mathematics. http://www.maths.qmw.ac.uk/~pjc/comb/quotes.txt Cascoly Software company Program for making quilt-pattern designs Cascoly Softwear, year unknown Graphics software. http://cascoly.com/cascmisc.htm Crowe, Donald W. Symmetries of cultures Crowe (website, year unknown) Diamond theory in Fiji. http://members.tripod.com/vismath6/crowe1/ Cullinane, S. H. The diamond theory of truth Author, 2000 Non-technical discussion of philosophical and literary issues related to diamond theory. http://math16.com/ Eschgfäller, Josef, et al. Mathematical BBS (Excellent math portal at U. of Ferrara) Universita degli Studi di Ferrara, 2001 Perhaps the best general portal to mathematics on the Web. http://felix.unife.it/ Labarthe, Jean-Jacques Hidden angular momenta Universite Paris Sud sites, 2001 Applies the finite geometries PG(2,2) and PG(3,2) to physics. http://www.lac.u-psud.fr/recherche/brochure/anglais/th1ang.html NEC Research Institute Research Index NEC Research Institute, 2001 Excellent bibliographic tool. http://citeseer.nj.nec.com/cs Peterson, Ivars Square of the hypotenuse Math. Assoc. of America sites, 2000 Plato's diamond may have been the basis of an earlier, elegant proof of the Pythagorean theorem. See also Aldous Huxley's story "Young Archimedes." http://www.maa.org/mathland/mathtrek_11_27_00.html Plato Meno ISLAS sites, year unknown The Meno dialogue of ca. 380 BC, with illustrations of Socrates's arguments using the diamond figure. A precursor to Gerard Manley Hopkins's "immortal diamond"? http://www.islas.org/drmcm/english/west_to_Dante/Readings/meno.html
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