Bridges : mathematical connections in art, music, and science : conference proceedings 1998

A major reason for developing the Bridges Conference and this collection of papers is our desire to come together from a diverse set of apparently separated disciplines, to share and recognize abstract similarities, common patterns, and underlying characteristics. The most fundamental patterns in the universe — from the double helix structures of DNA molecules and the spiral arms of galaxies, to the hexagonal packing structures of honeycombs and graphite carbon atoms — appear in the chain of evolution, in the shapes of cultures and civilizations, and in the extreme complexities encountered at high magnifications in high-speed computers. These patterns are the subject of a staggering variety of fields that we study to shape and enrich our expressions. Each field is growing more diverse. Rather than keeping together, they tend to spread, interact and join with other areas. This branching threatens to make one lose the big picture for details that don’t carry the desired essence of the whole. But, the process of growing together may be a natural, inevitable one, which is being accelerated by the works of such people as those included in this collection. We hope the annual Bridges Conference and these proceedings have the potential to connect many more areas of intellectual pursuit, within a variety of disciplines in science, art, and humanities. While emphasizing and encouraging the interconnection of materials in the search for a common ground, we did not ignore those readers who are eager to be challenged with deeper mathematics and led into a higher mastery of the concepts. We are grateful that this publication has occurred on the centennial of the birth of M. C. Escher, whose contributions have impacted a great number of mathematicians, artists, and scientists. We owe gratitude to our friends here and around the world who encouraged us to call for the Conference and enthusiastically participated by submitting stimulating papers. The work of many hands, our humble offering is like one green leaf from a poor dervish. But perhaps our dervish is a prophet in the wilderness. We give what we have, when it blooms. Preface Art, Math, and Computers: New Ways of Creating Pleasing Shapes C. H. Séquin Architecture and Mathematics: Art, Music and Science K. Williams Finding an Integral Equation of Design and Mathematics 21 B. Collins New Directions for Evolving Expressions 29 G. R. Greenfield Automatic Interval Naming Using Relative Pitch 37 D. Gerhard Continuum, Broken Symmetry, and More 49 C. O. Perry Origami Tessellations 55 H. Verrill The Poetics of Mathematics in Music 69 P. Escot Bridges of Mathematics, Art, and Physics 73 D. D. Peden Chaos Theory and the Fall of the Aztec Empire 83 J. Carrera Bolafios The Mathematics of Steve Reich’s Clapping Music 87 J. K. Haack The Circle: A Paradigm for Paradox 93 R. Sarhangi and B. D. Martin Let the Mirrors Do the Thinking 113 G. Clark and S. Zellweger Pythagorean and Platonic Bridges between Geometry and Algebra 121 S. Eberhart Subjective Fidelity and Problems of Measurement in Audio Reproduction 129 J. V.C. Nye Hyperseeing, Hypersculptures and Space Curves 139 N. Friedman Problems with Holbein’s Ambassadors and the Anamorphosis of the Skull 157 J. Sharp Conveying Large Numbers to General Audiences 167 R. McCluney Mathematics and Poetry: Discrepancies within Similarities 175 S. Marcus Spontaneous Patterns in Disk Packings 181 B. D. Lubachevsky, R. L. Graham, and F. H. Stillinger Icosahedral Constructions 195 G. W. Hart Mathematics in Three Dimensional Design: The Integration of Mathematical Thinking into the Design Core 203 D. R. Schol A Symmetry Classification of Columns 209 M. Golubitsky and I. Melbourne Math and Metaphor 225 D. F. Daniel A Visual Presentation of Rank-Ordered Sets 237 M. Alagié Symmetry, Chemistry, and Escher’s Tiles 245 B. D. Martin and R. Sarhangi A Taxonomy of Ancient Geometry Based on the Hidden Pavements of Michelangelo’s Laurentian Library 255 B. Nicholson, J. Kappraff, and S. Hisano Abstracts Symmetry, Causality, Mind 273 M. Leyton Sliceform Sculptures-a Bridge between Art and Mathematics 275 J. Sharp Interpersonal Hypothesis and the GUHA Method 277 J. Doubravova Eco-Mathematic/Geometric Aspects of a Design Proposal for Landscape/Public Art 279 T. M. Stephens Traces of the Geometrical Ordering of Roman Florence 281 C. M. Watts and D. J. Watts The Violin Surface Fitting 282 A. K. Mitra Geometrical Poetry 283 C. Singer Geometry as Libera(l)ting Art in Wren’s St. Paul’s Cathedral 287 S. Padget Relationships of Science, Mathematics and My Constructive Art: A Personal Survey 289 T. M. Stephens What is the Difference between a Banjo and a Harley-Davidson Motorcycle? Answer: You Can Tune the Harley 291 D. Fitzgerald