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Selection of Art for Auction W8 on March 30, 2014
Celebrating the 16th Annual Bridges Conference at the University of Twente and Saxion University of Applied Sciences Enschede, the Netherlands Art Exhibition Catalog 2013 Robert Fathauer and Nathan Selikoff, Editors Tessellations Publishing, Phoenix, Arizona Artistic and Scientific Conference Organizers Dave H. A. Blank George W. Hart Institute for Nanotechnology Stony Brook University of Twente, the Netherlands New York, USA Rinus Roelofs Reza Sarhangi The Foundation Passages Department of Mathematics Hengelo, the Netherlands Towson University, USA Artistic and Scientific Local Organizers Martha J. Haveman Ruben Sinkeldam Qua Art – Qua Science Saxion University of Applied Sciences Enschede, the Netherlands Enschede, the Netherlands Artistic and Scientific Board of Advisors Ina Bode Robert W. Fathauer AKI/ARTEZ Tessellations Company Enschede, the Netherlands Phoenix, Arizona, USA Kristóf Fenyvesi Paul Hildebrandt Jyväskylä University Zometool Inc. Jyväskylä, Finland Longmont, Colorado USA Craig S. Kaplan Rob Marsch Cheriton School of Computer Science The City of Enschede University of Waterloo, Canada The Netherlands Nathan Selikoff Carlo H. Séquin Digital Awakening Studios EECS, Computer Science Division Orlando, Florida, USA UC Berkeley, USA Peter van Roosmalen Henk Vosmer The Centre for Architecture SART Enschede, the Netherlands Enschede, the Netherlands BRIDGES ENSCHEDE 2013 Art Exhibition Catalog v PREFACE The International Bridges Conference, created in 1998 nationally recognized Bachelor’s degrees and Master’s and running annually since, provides a model of how to degrees in a wide range of fields, including engineering, integrate the seemingly diverse disciplines of mathemat- economics, and health care. The other campuses are lo- ics and the arts. Practicing mathematicians, scientists, cated in Deventer and Apeldoorn. artists, teachers, musicians, writers, computer scientists, Enschede also has an academy of arts and design sculptors, dancers, weavers, and model builders come to- combined with a conservatory, named ArtEZ. -
Spirolaterals: Math Is Art
CALIFORNIA STATE SCIENCE FAIR 2008 PROJECT SUMMARY Name(s) Project Number Ben Ackerman J1301 Project Title Thinking in Semicircles: The Area of an Arbelos Abstract Objectives/Goals The purpose of this experiment was to prove that the area of an arbelos is equal to the area of a circle whose diameter is the altitude of a right triangle drawn to the hypotenuse, which is inscribed in a semicircle. The experiment involved creating a mathematical proof using the knowledge of various geometry theorems and postulates, and applying them to the diagram in order to create accurate formulas for both areas. The history of Geometry and the format for a mathematical proof are discussed. Methods/Materials The materials to construct the diagram used for this proof include a pencil, a ruler, and a compass. The area of the arbelos was found by subtracting the areas of the two smaller inner semicircle from the larger one. The diameter of the circle, which was also the altitude drawn to the hypotenuse of a right triangle inscribed in the larger semicircle, was found using the theorem, which states that the altitude drawn to the hypotenuse of a right triangle is the geometric mean of the two segments of the hypotenuse. By dividing the altitude, or the diameter, in two, the radius of the circle was found, thus making the area of the circle possible to calculate. Once the dust settled, the two formulas for the areas were equal. Results The proof held true. Once the areas of the arbelos and the circle were calculated they were equal. -
Knotted Doughnuts and Other Mathematical Entertainments
MARTIN GARDNER KNOTTED DOUGHNUTS KNOTTED DOUGHNUTS AND OTHER MATHEMATICAL ENTERTAINMENTS MARTIN GARDNER KNOTTED DOUGHNUTS AND OTHER MATHEMATICAL ENTERTAINMENTS W. H. Freeman and Company New York Library of Congress Cataloging-in-Publication Data Gardner. Martin, 1914 - Knotted doughnuts and other mathematical entertainments Includes bibliographies and index. 1. Mathematical recreations. I. Title. QA95.GZ7 1986 793.7'4 85-31134 ISBN 0-7 167-1794-8 ISBN 0-7 167-1799-9 (pbk.) Copyright O 1986 by W. H. Freeman and Company No part of this book may be reproduced by any mechanical, photographic, or electronic process, or in the form of a phonographic recording, nor may it be stored in a retrieval system, transmitted, or otherwise copied for public or private use, without written permission from the publisher. Printed in the United States of America To Gerry Pie1 and Dennis Flanagan and all my other good friends at Scientific American during the 25 years that I had the great privilege of writing the magazine's Mathematical Games column contents Preface xi CHAPTER ONE Coincidence 1 CHAPTER TWO The Binary Gray Code 11 CHAPTER THREE Polycubes 2 8 CHAPTER FOUR Bacon's Cipher 45 CHAPTER FIVE Doughnuts: Linked and Knotted 5 5 CONTENTS x CHAPTER SIX The Tour of the Arrows and Other Problems 6 8 CHAPTER SEVEN Napier's Bones 85 Napier's Abacus CHAPTER NINE Sim, Chomp and Racetrack CHAPTER TEN Elevators 123 CHAPTER ELEVEN Crossing Numbers 133 CHAPTER TWELVE Point Sets on the Sphere 145 CONTENTS xi Newcomb's Paradox 155 CHAPTER FOURTEEN Reflections on Newcomb's Paradox CHAPTER FIFTEEN Reverse the Fish and Other Problems 176 Look-See Proofs 192 CHAPTER SEVENTEEN Worm Paths 205 Waring's Problems CHAPTER NINETEEN Cram, Bynum and Quadraphage 232 CONTENTS xii CHAPTER TWENTY The I Ching 244 CHAPTER TWENTY-ONE The La$er Curve 257 Index of Names 273 Because this is the eleventh collection of my Scientijc American columns, there is little to say in a preface that I have not said before. -
Polytiles: Equilateral and Equiangular Polygons (Part 1A)
Polytiles: Equilateral and Equiangular Polygons (part 1a) Thomas L Ruen [email protected] Abstract This introductory paper defines a polytile or p-tile as an equilateral polygon with turn angles as multiples of 360°/p, for even p=4, 6, 8…. Polytile notation is offered as a simple way to describe these polygons, with a geometric interpretation as a cycle of edge-to-edge regular p-gons, with vertices centered on each p-gon. A wide range of examples polygons and tilings are shown in this paper to demonstrate a breadth of value. A next paper, part 1b, will formalize the definition and report an enumeration of convex polytiles. Further papers will elaborate topics introduced in this paper including stars, compounds, extending notations, recursively extended polygons, symmetry, naming of polygons, and partial and fractional polytiles which allow one or two unequal edge lengths. 1. Introduction Equilateral polygons are polygons with equal edge-lengths. Equiangular polygons [2] are polygons with equal vertex angles. Polygons which are both equilateral and equiangular are called regular. This paper primarily looks at equilateral polygons, with section 5 exploring equiangular ones. When we are looking at equilateral polygons, we will count colinear vertices as vertices, while for equiangular polygons, we will ignore these and count as longer edges with integer lengths. The simplest irregular equilateral polygons are isotoxal [12] or edge-transitive, with identical edges within its symmetry, alternating two vertex types. The simplest equiangular polygons are isogonal or vertex-transitive, with identical vertices within its symmetry, alternating two edge types. Figure 1 shows hexagons: regular, isotoxal, and isogonal polygons.