Computer graphics and geometric ornamental design

Throughout history, geometric patterns have formed an important part of art and ornamental design. Today we have unprecedented ability to understand ornamental styles of the past, to recreate traditional designs, and to innovate with new interpretations of old styles and with new styles altogether. The power to further the study and practice of ornament stems from three sources. We have new mathematical tools: a modern conception of geometry that enables us to describe with precision what designers of the past could only hint at. We have new algorithmic tools: computers and the abstract mathematical processing they enable allow us to perform calculations that were intractable in previous generations. Finally, we have technological tools: manufacturing devices that can turn a synthetic description provided by a computer into a real-world artifact. Taken together, these three sets of tools provide new opportunities for the application of computers to the analysis and creation of ornament. In this dissertation, I present my research in the area of computer-generated geometric art and ornament. I focus on two projects in particular. First I develop a collection of tools and methods for producing traditional Islamic star patterns. Then I examine the tesselations of M. C. Escher, developing an “Escherization” algorithm that can derive novel Escher-like tesselations of the plane from arbitrary user-supplied shapes. Throughout, I show how modern mathematics, algorithms, and technology can be applied to the study of these ornamental styles.

[1]  Syed Jan Abas,et al.  SYMMETRIES OF ISLAMIC GEOMETRICAL PATTERNS , 1994 .

[2]  A. J. Lee,et al.  ISLAMIC STAR PATTERNS , 1986 .

[3]  Douglas Dunham,et al.  Hyperbolic Islamic Patterns -- A Beginning , 2001 .

[4]  David S. Ebert,et al.  Texturing and Modeling , 1998 .

[5]  G. C. Shephard,et al.  Tilings and Patterns , 1990 .

[6]  Hans Hermes,et al.  Introduction to mathematical logic , 1973, Universitext.

[7]  Philip J. Schneider,et al.  An algorithm for automatically fitting digitized curves , 1990 .

[8]  W. Magnus,et al.  Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations , 1966 .

[9]  S. Golomb.  Polyominoes: Puzzles, Patterns, Problems, and Packings , 1994 .

[10]  Douglas Dunham,et al.  Artistic Patterns in Hyperbolic Geometry , 1999 .

[11]  Ian Stewart.  The problems of mathematics , 1987 .

[12]  Craig S. Kaplan,et al.  Escherization , 2000, SIGGRAPH.

[13]  河東 泰之,et al.  A.Connes:Noncommutative Geometry , 1997 .

[14]  Howard Alexander,et al.  The computer/plotter and the 17 ornamental design types , 1975, SIGGRAPH '75.

[15]  Robert Jensen,et al.  Ornamentalism: The New Decorativeness In Architecture & Design , 1982 .

[16]  Kenneth Falconer,et al.  Unsolved Problems In Geometry , 1991 .

[17]  Roman E. Maeder.  Uniform Polyhedra , 1993 .

[18]  A. V. Shubnikov,et al.  Symmetry in Science and Art , 1974 .

[19]  William H. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[20]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[21]  A. Racinet.  The encyclopedia of ornament , 1988 .

[22]  William A. Barrett,et al.  Intelligent scissors for image composition , 1995, SIGGRAPH.

[23]  H. Coxeter,et al.  The Geometric vein : the Coxeter Festschrift , 1981 .

[24]  Peter R. Cromwell,et al.  Celtic knotwork: Mathematical art , 1993 .

[25]  Ernst Gombrich,et al.  The Sense of Order: A Study in the Psychology of Decorative Art , 1979 .

[26]  Ivars Peterson,et al.  The Mathematical Tourist: Snapshots of Modern Mathematics , 1989 .

[27]  Douglas DeCarlo,et al.  Stylization and abstraction of photographs , 2002, ACM Trans. Graph..

[28]  Michael Leyton,et al.  A generative theory of shape , 2004, Proceedings Shape Modeling Applications, 2004..

[29]  William W. Chow,et al.  Interlocking shapes in art and engineering , 1980 .

[30]  S. Umeyama,et al.  Least-Squares Estimation of Transformation Parameters Between Two Point Patterns , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[31]  Cameron Browne.  Font Decoration by Automatic Mesh Fitting , 1998, EP.

[32]  Chaim Goodman-Strauss,et al.  A Small Aperiodic Set of Planar Tiles , 1999, Eur. J. Comb..

[33]  Branko Grünbaum,et al.  Interlace Patterns in Islamic and Moorish Art , 1993 .

[34]  Robert Dixon.  Two Conformal Mappings , 1992 .

[35]  George fl. Bain,et al.  The methods of construction of Celtic art , 1973 .

[36]  Marvin J. Greenberg.  Euclidean and non-Euclidean geometries : development and history , 1977 .

[37]  E. Hanbury Hankin.  Examples of Methods of Drawing Geometrical Arabesque Patterns , 1925 .

[38]  E. H. Hankin,et al.  The drawing of geometric patterns in Saracenic art , 1925 .

[39]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[40]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[41]  Alan Holden,et al.  Shapes, space, and symmetry , 1971 .

[42]  Roberto Bonola,et al.  Non-Euclidean geometry , 1912 .

[43]  Douglas E. Zongker,et al.  CREATION OF OVERLAY TILINGS THROUGH DUALIZATION OF REGULAR NETWORKS , 1999 .

[44]  Tony DeRose,et al.  Subdivision surfaces in character animation , 1998, SIGGRAPH.

[45]  Keith Critchlow,et al.  Islamic patterns : an analytical and cosmological approach , 1976 .

[46]  Charlie Gunn,et al.  Discrete groups and visualization of three-dimensional manifolds , 1993, SIGGRAPH.

[47]  Dave Witte Morris,et al.  Creating repeating hyperbolic patterns , 1981, SIGGRAPH '81.

[48]  Z. Lučić,et al.  Fundamental domains for planar discontinuous groups and uniform tilings , 1991 .

[49]  P. Locher,et al.  The perceptual value of symmetry , 1989 .

[50]  E. Hanbury Hankin.  Some Difficult Saracenic Designs. III. A Pattern Containing Fifteen-Rayed Stars , 1936 .

[51]  Carlo H. Séquin,et al.  Escher sphere construction kit , 2001, I3D '01.

[52]  Daniel H. Huson,et al.  Visualization of Periodic Tilings , 1997, VisMath.

[53]  José Montesinos,et al.  Classical tessellations and three-manifolds , 1987 .

[54]  A. K. Dewdney,et al.  The Tinkertoy computer and other machinations , 1993 .

[55]  István Hargittai,et al.  Symmetry through the eyes of a chemist , 1986 .

[56]  Judith R. Brown,et al.  Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 2000, New Orleans, LA, USA, July 23-28, 2000 , 2000, SIGGRAPH.

[57]  John Berglund.  Is There a k-Anisohedral Tile for k ≥5? , 1993 .

[58]  Bruno Ernst,et al.  The Magic Mirror of M.C. Escher , 1976 .

[59]  C. Tyler.  1 The Human Expression of Symmetry : Art and Neuroscience , 2004 .

[60]  Olaf Delgado-Friedrichs,et al.  Data structures and algorithms for tilings I , 2003, Theor. Comput. Sci..

[61]  Branko Grünbaum,et al.  Can All Tiles of a Tiling Have Five-Fold Symmetry? , 1982 .

[62]  Branko Grünbaum,et al.  The Emperor’s New clothes: Full regalia, G string, or nothing? , 1984 .

[63]  Aldous Huxley,et al.  Heaven and Hell , 1957 .

[64]  Daniel H. Huson,et al.  The generation and classification of tile-k-transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane , 1993 .

[65]  Stephen Wolfram,et al.  A New Kind of Science , 2003, Artificial Life.

[66]  John C. Baird,et al.  A quantitative approach to the study of visual symmetry , 1977 .

[67]  George Markowsky.  Introduction To Algorithmic Information Theory , 1996, J. Univers. Comput. Sci..

[68]  David Salesin,et al.  Computer-generated floral ornament , 1998, SIGGRAPH.

[69]  G. C. Shephard,et al.  Tilings and patterns. W. H. Freeman and Co. Ltd., Oxford 1987. IX + 700 p., 1395 figs., price £ 54.95, ISBN 0–7167–1193–1 , 1991 .

[70]  Branko Grünbaum.  Levels of Orderliness: Global and Local Symmetry , 2003 .

[71]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[72]  G. E. Martin.  The foundations of geometry and the non-Euclidean plane , 1984 .

[73]  Esther M. Arkin,et al.  An efficiently computable metric for comparing polygonal shapes , 1991, SODA '90.

[74]  M. C. Escher,et al.  Escher on Escher: Exploring the Infinite , 1989 .

[75]  Paul M. B. Vitányi,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 1993, Graduate Texts in Computer Science.

[76]  E. Howe,et al.  Effects of Partial Symmetry, Exposure Time, and Backward Masking on Judged Goodness and Reproduction of Visual Patterns , 1980, The Quarterly journal of experimental psychology.

[77]  John Stillwell,et al.  Symmetry , 2000, Am. Math. Mon..

[78]  H. C. Williams,et al.  Symmetries of culture , 1990, The Mathematical Gazette.

[79]  Stanley B. Lippman,et al.  C++ Primer , 1993 .

[80]  H. Gold.  The magic will , 1971 .

[81]  Chris Stolte,et al.  Rendering effective route maps: improving usability through generalization , 2001, SIGGRAPH.

[82]  Ellery B. Golos.  Foundations of Euclidean and non-Euclidean geometry , 1968 .

[83]  Craig S. Kaplan and George W. Hart,et al.  Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons , 2001 .

[84]  Andrew Lesniewski,et al.  Noncommutative Geometry , 1997 .

[85]  Andrew S. Glassner,et al.  Aperiodic Tiling , 1998, IEEE Computer Graphics and Applications.

[86]  E. Hanbury Hankin.  Some Difficult Saracenic Designs. II. A Pattern Containing Seven-Rayed Stars , 1934 .

[87]  Craig S. Kaplan,et al.  Computer Generated Islamic Star Patterns , 2000 .

[88]  Aggie Vinik.  Mathematics and humor , 1978 .

[89]  G. C. Shephard,et al.  Spherical Tilings with Transitivity Properties , 1981 .

[90]  Jinny Beyer,et al.  Designing Tessellations: The Secrets of Interlocking Patterns , 1999 .

[91]  William W. Chow,et al.  Automatic generation of interlocking shapes , 1979 .

[92]  Silvio Levy,et al.  Automatic Generation of Hyperbolic Tilings , 1993 .

[93]  Thaddeus Beier,et al.  Feature-based image metamorphosis , 1998 .

[94]  Daniel H. Huson,et al.  The classification of 2-isohedral tilings of the plane , 1992 .

[95]  Hans-Peter Seidel,et al.  Exploring Plane Hyperbolic Geometry , 1997, VisMath.

[96]  Victor Ostromoukhov,et al.  Mathematical Tools for Computer-Generated Ornamental Patterns , 1998, EP.

[97]  George Jean Nathan,et al.  The magic mirror , 1960 .

[98]  D. Hofstadter.  Metamagical Themas: Questing for the Essence of Mind and Pattern , 1985 .