Framing Cognitive Space for Higher Order Coherence Toroidal Interweaving from I Ching to Supercomputers and Back? - /
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Equivelar Octahedron of Genus 3 in 3-Space
Equivelar octahedron of genus 3 in 3-space Ruslan Mizhaev ([email protected]) Apr. 2020 ABSTRACT . Building up a toroidal polyhedron of genus 3, consisting of 8 nine-sided faces, is given. From the point of view of topology, a polyhedron can be considered as an embedding of a cubic graph with 24 vertices and 36 edges in a surface of genus 3. This polyhedron is a contender for the maximal genus among octahedrons in 3-space. 1. Introduction This solution can be attributed to the problem of determining the maximal genus of polyhedra with the number of faces - . As is known, at least 7 faces are required for a polyhedron of genus = 1 . For cases ≥ 8 , there are currently few examples. If all faces of the toroidal polyhedron are – gons and all vertices are q-valence ( ≥ 3), such polyhedral are called either locally regular or simply equivelar [1]. The characteristics of polyhedra are abbreviated as , ; [1]. 2. Polyhedron {9, 3; 3} V1. The paper considers building up a polyhedron 9, 3; 3 in 3-space. The faces of a polyhedron are non- convex flat 9-gons without self-intersections. The polyhedron is symmetric when rotated through 1804 around the axis (Fig. 3). One of the features of this polyhedron is that any face has two pairs with which it borders two edges. The polyhedron also has a ratio of angles and faces - = − 1. To describe polyhedra with similar characteristics ( = − 1) we use the Euler formula − + = = 2 − 2, where is the Euler characteristic. Since = 3, the equality 3 = 2 holds true. -
1940-Commencement.Pdf
c~ h' ( c\ '.\.\.\.. ( ~A { I , .f \,.' I f ;' \ . \ J University of Minnesota IJ • COMMENCEMENT CONVOCATION WINTER QUARTER 1940 NORTHROP MEMORIAL AUDITORIUM Thursday, March 21, 1940, Eleven O'Clock I I , ~ \ ' ,i ii, iii, ;, ' PROGRAM PRESIDENT GUY STANTON FORD, Presiding PROCESSIONAL-Finale from the Fourth Symphony Widor ARTHUR B. JENNINGS University Organist HYMN-"America" My country I 'tis of thee, Our fathers' God I to Thee, Sweet land of liberty, Author of Liberty, Of thee I sing; To Thee we sing; Land where our fathers died I Long may our land be bright Land of the Pilgrims' pride, With freedom's holy light; From every mountain side Protect us by Thy might Let freedom ring. Great God, our King I COMMENCEMENT ADDRESS- "Of Human Intercourse" HENRY NOBLE MACCRACKEN, Ph.D., LL.D., L.H.D. President, Vassar College CONFERRING OF DEGREES GUY STANTON FORD, Ph.D., LL.D., Litt.D., L.H.D. President of the University 2 ',' J I SONG-"Hail, Minnesota!" Minnesota, hail to thee I Like the stream that bends to sea, Hail to thee, our College dear I Like the pine that seeks the blue I Thy light shall ever be Minnesota, still for thee, A beacon bright and clear; Thy sons are strong and true. Thy sons and daughters true From thy woods and waters fair, Will proclaim thee near and far; From thy prairies waving far, They will guard thy fame At thy call they throng, And adore thy name; With their shout and song, Thou shalt be their Northern Star. Hailing thee their Northern Star. -
Representing Graphs by Polygons with Edge Contacts in 3D
Representing Graphs by Polygons with Side Contacts in 3D∗ Elena Arseneva1, Linda Kleist2, Boris Klemz3, Maarten Löffler4, André Schulz5, Birgit Vogtenhuber6, and Alexander Wolff7 1 Saint Petersburg State University, Russia [email protected] 2 Technische Universität Braunschweig, Germany [email protected] 3 Freie Universität Berlin, Germany [email protected] 4 Utrecht University, the Netherlands [email protected] 5 FernUniversität in Hagen, Germany [email protected] 6 Technische Universität Graz, Austria [email protected] 7 Universität Würzburg, Germany orcid.org/0000-0001-5872-718X Abstract A graph has a side-contact representation with polygons if its vertices can be represented by interior-disjoint polygons such that two polygons share a common side if and only if the cor- responding vertices are adjacent. In this work we study representations in 3D. We show that every graph has a side-contact representation with polygons in 3D, while this is not the case if we additionally require that the polygons are convex: we show that every supergraph of K5 and every nonplanar 3-tree does not admit a representation with convex polygons. On the other hand, K4,4 admits such a representation, and so does every graph obtained from a complete graph by subdividing each edge once. Finally, we construct an unbounded family of graphs with average vertex degree 12 − o(1) that admit side-contact representations with convex polygons in 3D. Hence, such graphs can be considerably denser than planar graphs. 1 Introduction A graph has a contact representation if its vertices can be represented by interior-disjoint geometric objects1 such that two objects touch exactly if the corresponding vertices are adjacent. -
Consonance and Dissonance in Visual Music Bill Alves Harvey Mudd College
Claremont Colleges Scholarship @ Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship 8-1-2012 Consonance and Dissonance in Visual Music Bill Alves Harvey Mudd College Recommended Citation Bill Alves (2012). Consonance and Dissonance in Visual Music. Organised Sound, 17, pp 114-119 doi:10.1017/ S1355771812000039 This Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Organised Sound http://journals.cambridge.org/OSO Additional services for Organised Sound: Email alerts: Click here Subscriptions: Click here Commercial reprints: Click here Terms of use : Click here Consonance and Dissonance in Visual Music Bill Alves Organised Sound / Volume 17 / Issue 02 / August 2012, pp 114 - 119 DOI: 10.1017/S1355771812000039, Published online: 19 July 2012 Link to this article: http://journals.cambridge.org/abstract_S1355771812000039 How to cite this article: Bill Alves (2012). Consonance and Dissonance in Visual Music. Organised Sound, 17, pp 114-119 doi:10.1017/ S1355771812000039 Request Permissions : Click here Downloaded from http://journals.cambridge.org/OSO, IP address: 134.173.130.244 on 24 Jul 2014 Consonance and Dissonance in Visual Music BILL ALVES Harvey Mudd College, The Claremont Colleges, 301 Platt Blvd, Claremont CA 91711 USA E-mail: [email protected] The concepts of consonance and dissonance broadly Plato found the harmony of the world in the Pythag- understood can provide structural models for creators of orean whole numbers and their ratios, abstract ideals visual music. -
The Unexpected Number Theory and Algebra of Musical Tuning Systems Or, Several Ways to Compute the Numbers 5,7,12,19,22,31,41,53, and 72
The Unexpected Number Theory and Algebra of Musical Tuning Systems or, Several Ways to Compute the Numbers 5,7,12,19,22,31,41,53, and 72 Matthew Hawthorn \Music is the pleasure the human soul experiences from counting without being aware that it is counting." -Gottfried Wilhelm von Leibniz (1646-1716) \All musicians are subconsciously mathematicians." -Thelonius Monk (1917-1982) 1 Physics In order to have music, we must have sound. In order to have sound, we must have something vibrating. Wherever there is something virbrating, there is the wave equation, be it in 1, 2, or more dimensions. The solutions to the wave equation for any given object (string, reed, metal bar, drumhead, vocal cords, etc.) with given boundary conditions can be expressed as a superposition of discrete partials, modes of vibration of which there are generally infinitely many, each with a characteristic frequency. The partials and their frequencies can be found as eigenvectors, resp. eigenvalues of the Laplace operator acting on the space of displacement functions on the object. Taken together, these frequen- cies comprise the spectrum of the object, and their relative intensities determine what in musical terms we call timbre. Something very nice occurs when our object is roughly one-dimensional (e.g. a string): the partial frequencies become harmonic. This is where, aptly, the better part of harmony traditionally takes place. For a spectrum to be harmonic means that it is comprised of a fundamental frequency, say f, and all whole number multiples of that frequency: f; 2f; 3f; 4f; : : : It is here also that number theory slips in the back door. -
Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library -
2005-2006 Undergraduate Academic Catalog
GORDON OLLEGE CUndergraduate Academic Catalog 2005–2006 Art Durity GORDON COLLEGE UNDERGRADUATE ACADEMIC CATALOG 2005–2006 The United College of Gordon and Barrington 255 Grapevine Road Wenham, Massachusetts 01984 978.927.2300 Fax 978.867.4659 www.gordon.edu Printed on recycled paper Gordon College is in compliance with both the spirit and the letter of Title IX of the Education Amend- ments of 1972 and with Internal Revenue Service Procedure 75–50. This means that the College does not discriminate on the basis of race, color, sex, age, disability, veteran status or national or ethnic origin in administration of its employment policies, admissions policies, recruitment programs (for students and employees), scholarship and loan programs, athletics and other college-administered activities. ******** Gordon College supports the efforts of secondary school officials and governing bodies to have their schools achieve regional accreditation to provide reliable assurance of the quality of the educational preparation of its applicants for admission. ******** Any student who is unable, because of religious beliefs, to attend classes or to participate in any examina- tion, study or work requirement on a particular day shall be excused from such activity and be provided with an opportunity to make it up, provided it shall not create an unreasonable burden upon the school. No fees shall be charged nor any adverse or prejudicial effects result. ******** In compliance with the Higher Education Amendments of 1986, Gordon College operates a drug abuse prevention program encompassing general dissemination of informational literature, awareness seminars and individual counseling. Assistance is available to students, staff and faculty. For more information please contact the Center for Student Development. -
Board of Trustees Ohio State University
RECORD OF PROCEEDINGS OF THE BOARD OF TRUSTEES OF THE OHIO STATE UNIVERSITY COLUMBUS July 1, 1926, to June 30, 1927 THE OHIO STATE UNIVERSITY GEORGE W. RIGHTMIRE President BOARD OF TRUSTEES Date of Original Appointment Term Expires MRS. ALMA W. PATERSON, Columbus .... Mar. 27, 1924 May 13, 1926 HERBERT s. ATKINSON, Columbus ....••. Mar. 17, 1925 May 13, 1927 EGBERT H. MACK, Sandusky ............ Dec. 12, 1922 May 13, 1928 JOHN KAISER, Marietta ........•........ Feb. 25, 1915 May 13, 1929 *JULIUS F. STONE, Columbus ........... Mar. 17, 1925 May 13, 1930 LAWRENCE E. LAYBOURNE, Springfield ... May 14, 1921 May 13, 1931 HARRY A. CATON, Coshocton ............ May 14, 1925 May 13, 1932 CARL E. STEEB Secretary of the Board C. F. KETTERING Treasurer of the Board 0. E. BRADFUTE Assistant Treasurer of the Board • Also served as Trustee May 23, 1909 to March 21, 1917. Proceedings of the Board of Trustees The Ohio State University OFFICE OF THE BOARD OF TRUSTEES THE OHIO STATE UNIVERSITY Wooster, Ohio, July 12, 1926. The Board of Trustees met at Wooster, Ohio, pursuant to ad- journment. Present: L. E. Laybourne, Chairman, Egbert Mack, John Kaiser, Mrs. Alma Paterson, Herbert S. Atkinson, Harry A. Caton. * * * * * * The minutes of the last meeting were approved. * * * * * • Upon motion, the Chairman was directed to appoint a committee of three members to make an inquiry into matters pertaining to the University Hospital. The Chairman appointed Messrs. Mack, Atkin- son, and Kaiser as members of this committee. * • * * * * Upon recommendation of the President, the following resigna- tions were accepted and the balances cancelled in accordance with the general rule : Name Title Date Effective Annual Rate Agricultural E"'tension R. -
Musical Tuning Systems As a Form of Expression
Musical Tuning Systems as a Form of Expression Project Team: Lewis Cook [email protected] Benjamin M’Sadoques [email protected] Project Advisor Professor Wilson Wong Department of Computer Science This report represents the work of WPI undergraduate students submitted to the faculty as evidence of completion of a degree requirement. WPI routinely publishes these reports on its website without editorial or peer review. For more information about the projects program at WPI, please see http://www.wpi.edu/academics/ugradstudies/project- learning.html Abstract Many cultures and time periods throughout history have used a myriad of different practices to tune their instruments, perform, and create music. However, most musicians in the western world will only experience 12-tone equal temperament a represented by the keys on a piano. We want musicians to recognize that choosing a tuning system is a form of musical expression. The goal of this project was to help musicians of any skill-level experience, perform, and create music involving tuning systems. We created software to allow musicians to experiment and implement alternative tuning systems into their own music. ii Table of Contents Abstract ................................................................................................................................... ii Table of Figures .................................................................................................................... vii 1 Introduction ........................................................................................................................ -
On Three Classes of Regular Toroids
ON THREE CLASSES OF REGULAR TOROIDS SZILASSI Lajos (HU) Abstract. As it is known, in a regular polyhedron every face has the same number of edges and every vertex has the same number of edges, as well. A polyhedron is called topologically regular if further conditions (e.g. on the angle of the faces or the edges) are not imposed. An ordinary polyhedron is called a toroid if it is topologically torus-like (i.e. it can be converted to a torus by continuous deformation), and its faces are simple polygons. A toroid is said to be regular if it is topologically regular. It is easy to see, that the regular toroids can be classified into three classes, according to the number of edges of a vertex and of a face. There are infinitelly many regular toroids in each classes, because the number of the faces and vertices can be arbitrarilly large. Hence, we study mainly those regular toroids, whose number of faces or vertices is minimal, or that ones, which have any other special properties. Among these polyhedra, we take special attention to the so called „Császár-polyhedron”, which has no diagonal, i.e. each pair of vertices are neighbouring, and its dual polyhedron (in topological sense) the so called „Szilassi-polyhedron”, whose each pair of faces are neighbouring. The first one was found by Ákos Császár in 1949, and the latter one was found by the writer of this paper, in 1977. Keywords. regular, torus-like polyhedron As it is known, in regular polyhedra, the same number of edges meet at each vertex, and each face has the same number of edges. -
The Foundations of Scientific Musical Tuning by Jonathan Tennenbaum
Click here for Full Issue of Fidelio Volume 1, Number 1, Winter 1992 The Foundations of Scientific Musical Tuning by Jonathan Tennenbaum want to demonstrate why, from a scientific stand teenth-century physicist and physiologist whose 1863 point, no musical tuning is acceptable which is not book, Die Lehre von den Tonempfindungen als physiolo I based on a pitch value fo r middle C of 256Hz (cycles gische Grundlage fu r die Theorie der Musik (The Theory per second), corresponding to A no higher than 432Hz. of the Sensations of Tone as a Foundation of Music Theory) In view of present scientificknowl edge, all other tunings became the standard reference work on the scientific including A =440 must be rejected as invalid and arbi bases of music, and remains so up to this very day. trary. Unfortunately, every essential assertion in Helmholtz's Those in fa vor of constantly raising the pitch typically book has been proven to be fa lse. argue, "What diffe rence does it make what basic pitch Helmholtz's basic fallacy-still taught in most music we choose, as long as the other notes are properly tuned conservatories and universities today-was to claim that relative to that pitch? After all, musical tones are just the scientificbasis of music is to be fo und in the properties frequencies, they are all essentially alike. So, why choose of vibrating, inert bodies, such as strings, tuning fo rks, one pitch rather than another?" To these people, musical pipes, and membranes. Helmholtz definedmusical tones tones are like paper money, whose value can be inflated merely as periodic vibrations of the air. -
Foundations in Music Psychology
Foundations in Music Psy chol ogy Theory and Research edited by Peter Jason Rentfrow and Daniel J. Levitin The MIT Press Cambridge, Mas sa chu setts London, England © 2019 Mas sa chu setts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. This book was set in Stone Serif by Westchester Publishing Ser vices. Printed and bound in the United States of Amer i ca. Library of Congress Cataloging- in- Publication Data Names: Rentfrow, Peter J. | Levitin, Daniel J. Title: Foundations in music psy chol ogy : theory and research / edited by Peter Jason Rentfrow and Daniel J. Levitin. Description: Cambridge, MA : The MIT Press, 2019. | Includes bibliographical references and index. Identifiers: LCCN 2018018401 | ISBN 9780262039277 (hardcover : alk. paper) Subjects: LCSH: Music— Psychological aspects. | Musical perception. | Musical ability. Classification: LCC ML3830 .F7 2019 | DDC 781.1/1— dc23 LC rec ord available at https:// lccn . loc . gov / 2018018401 10 9 8 7 6 5 4 3 2 1 Contents I Music Perception 1 Pitch: Perception and Neural Coding 3 Andrew J. Oxenham 2 Rhythm 33 Henkjan Honing and Fleur L. Bouwer 3 Perception and Cognition of Musical Timbre 71 Stephen McAdams and Kai Siedenburg 4 Pitch Combinations and Grouping 121 Frank A. Russo 5 Musical Intervals, Scales, and Tunings: Auditory Repre sen ta tions and Neural Codes 149 Peter Cariani II Music Cognition 6 Musical Expectancy 221 Edward W. Large and Ji Chul Kim 7 Musicality across the Lifespan 265 Sandra E.