Symmetry and Golden Ratio in the Analysis of Regular Pentagon Amelia Carolina Sparavigna, Mauro Maria Baldi
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The Age of Addiction David T. Courtwright Belknap (2019) Opioids
The Age of Addiction David T. Courtwright Belknap (2019) Opioids, processed foods, social-media apps: we navigate an addictive environment rife with products that target neural pathways involved in emotion and appetite. In this incisive medical history, David Courtwright traces the evolution of “limbic capitalism” from prehistory. Meshing psychology, culture, socio-economics and urbanization, it’s a story deeply entangled in slavery, corruption and profiteering. Although reform has proved complex, Courtwright posits a solution: an alliance of progressives and traditionalists aimed at combating excess through policy, taxation and public education. Cosmological Koans Anthony Aguirre W. W. Norton (2019) Cosmologist Anthony Aguirre explores the nature of the physical Universe through an intriguing medium — the koan, that paradoxical riddle of Zen Buddhist teaching. Aguirre uses the approach playfully, to explore the “strange hinterland” between the realities of cosmic structure and our individual perception of them. But whereas his discussions of time, space, motion, forces and the quantum are eloquent, the addition of a second framing device — a fictional journey from Enlightenment Italy to China — often obscures rather than clarifies these chewy cosmological concepts and theories. Vanishing Fish Daniel Pauly Greystone (2019) In 1995, marine biologist Daniel Pauly coined the term ‘shifting baselines’ to describe perceptions of environmental degradation: what is viewed as pristine today would strike our ancestors as damaged. In these trenchant essays, Pauly trains that lens on fisheries, revealing a global ‘aquacalypse’. A “toxic triad” of under-reported catches, overfishing and deflected blame drives the crisis, he argues, complicated by issues such as the fishmeal industry, which absorbs a quarter of the global catch. -
Golden Ratio: a Subtle Regulator in Our Body and Cardiovascular System?
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/306051060 Golden Ratio: A subtle regulator in our body and cardiovascular system? Article in International journal of cardiology · August 2016 DOI: 10.1016/j.ijcard.2016.08.147 CITATIONS READS 8 266 3 authors, including: Selcuk Ozturk Ertan Yetkin Ankara University Istinye University, LIV Hospital 56 PUBLICATIONS 121 CITATIONS 227 PUBLICATIONS 3,259 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: microbiology View project golden ratio View project All content following this page was uploaded by Ertan Yetkin on 23 August 2019. The user has requested enhancement of the downloaded file. International Journal of Cardiology 223 (2016) 143–145 Contents lists available at ScienceDirect International Journal of Cardiology journal homepage: www.elsevier.com/locate/ijcard Review Golden ratio: A subtle regulator in our body and cardiovascular system? Selcuk Ozturk a, Kenan Yalta b, Ertan Yetkin c,⁎ a Abant Izzet Baysal University, Faculty of Medicine, Department of Cardiology, Bolu, Turkey b Trakya University, Faculty of Medicine, Department of Cardiology, Edirne, Turkey c Yenisehir Hospital, Division of Cardiology, Mersin, Turkey article info abstract Article history: Golden ratio, which is an irrational number and also named as the Greek letter Phi (φ), is defined as the ratio be- Received 13 July 2016 tween two lines of unequal length, where the ratio of the lengths of the shorter to the longer is the same as the Accepted 7 August 2016 ratio between the lengths of the longer and the sum of the lengths. -
Music: Broken Symmetry, Geometry, and Complexity Gary W
Music: Broken Symmetry, Geometry, and Complexity Gary W. Don, Karyn K. Muir, Gordon B. Volk, James S. Walker he relation between mathematics and Melody contains both pitch and rhythm. Is • music has a long and rich history, in- it possible to objectively describe their con- cluding: Pythagorean harmonic theory, nection? fundamentals and overtones, frequency Is it possible to objectively describe the com- • Tand pitch, and mathematical group the- plexity of musical rhythm? ory in musical scores [7, 47, 56, 15]. This article In discussing these and other questions, we shall is part of a special issue on the theme of math- outline the mathematical methods we use and ematics, creativity, and the arts. We shall explore provide some illustrative examples from a wide some of the ways that mathematics can aid in variety of music. creativity and understanding artistic expression The paper is organized as follows. We first sum- in the realm of the musical arts. In particular, we marize the mathematical method of Gabor trans- hope to provide some intriguing new insights on forms (also known as short-time Fourier trans- such questions as: forms, or spectrograms). This summary empha- sizes the use of a discrete Gabor frame to perform Does Louis Armstrong’s voice sound like his • the analysis. The section that follows illustrates trumpet? the value of spectrograms in providing objec- What do Ludwig van Beethoven, Ben- • tive descriptions of musical performance and the ny Goodman, and Jimi Hendrix have in geometric time-frequency structure of recorded common? musical sound. Our examples cover a wide range How does the brain fool us sometimes • of musical genres and interpretation styles, in- when listening to music? And how have cluding: Pavarotti singing an aria by Puccini [17], composers used such illusions? the 1982 Atlanta Symphony Orchestra recording How can mathematics help us create new of Copland’s Appalachian Spring symphony [5], • music? the 1950 Louis Armstrong recording of “La Vie en Rose” [64], the 1970 rock music introduction to Gary W. -
Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture
REPORTS 21. Materials and methods are available as supporting 27. N. Panagia et al., Astrophys. J. 459, L17 (1996). Supporting Online Material material on Science Online. 28. The authors would like to thank L. Nelson for providing www.sciencemag.org/cgi/content/full/315/5815/1103/DC1 22. A. Heger, N. Langer, Astron. Astrophys. 334, 210 (1998). access to the Bishop/Sherbrooke Beowulf cluster (Elix3) Materials and Methods 23. A. P. Crotts, S. R. Heathcote, Nature 350, 683 (1991). which was used to perform the interacting winds SOM Text 24. J. Xu, A. Crotts, W. Kunkel, Astrophys. J. 451, 806 (1995). calculations. The binary merger calculations were Tables S1 and S2 25. B. Sugerman, A. Crotts, W. Kunkel, S. Heathcote, performed on the UK Astrophysical Fluids Facility. References S. Lawrence, Astrophys. J. 627, 888 (2005). T.M. acknowledges support from the Research Training Movies S1 and S2 26. N. Soker, Astrophys. J., in press; preprint available online Network “Gamma-Ray Bursts: An Enigma and a Tool” 16 October 2006; accepted 15 January 2007 (http://xxx.lanl.gov/abs/astro-ph/0610655) during part of this work. 10.1126/science.1136351 be drawn using the direct strapwork method Decagonal and Quasi-Crystalline (Fig. 1, A to D). However, an alternative geometric construction can generate the same pattern (Fig. 1E, right). At the intersections Tilings in Medieval Islamic Architecture between all pairs of line segments not within a 10/3 star, bisecting the larger 108° angle yields 1 2 Peter J. Lu * and Paul J. Steinhardt line segments (dotted red in the figure) that, when extended until they intersect, form three distinct The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in polygons: the decagon decorated with a 10/3 star medieval Islamic architecture were conceived by their designers as a network of zigzagging lines, line pattern, an elongated hexagon decorated where the lines were drafted directly with a straightedge and a compass. -
The Golden Ratio and the Diagonal of the Square
Bridges Finland Conference Proceedings The Golden Ratio and the Diagonal of the Square Gabriele Gelatti Genoa, Italy [email protected] www.mosaicidiciottoli.it Abstract An elegant geometric 4-step construction of the Golden Ratio from the diagonals of the square has inspired the pattern for an artwork applying a general property of nested rotated squares to the Golden Ratio. A 4-step Construction of the Golden Ratio from the Diagonals of the Square For convenience, we work with the reciprocal of the Golden Ratio that we define as: φ = √(5/4) – (1/2). Let ABCD be a unit square, O being the intersection of its diagonals. We obtain O' by symmetry, reflecting O on the line segment CD. Let C' be the point on BD such that |C'D| = |CD|. We now consider the circle centred at O' and having radius |C'O'|. Let C" denote the intersection of this circle with the line segment AD. We claim that C" cuts AD in the Golden Ratio. B C' C' O O' O' A C'' C'' E Figure 1: Construction of φ from the diagonals of the square and demonstration. Demonstration In Figure 1 since |CD| = 1, we have |C'D| = 1 and |O'D| = √(1/2). By the Pythagorean Theorem: |C'O'| = √(3/2) = |C''O'|, and |O'E| = 1/2 = |ED|, so that |DC''| = √(5/4) – (1/2) = φ. Golden Ratio Pattern from the Diagonals of Nested Squares The construction of the Golden Ratio from the diagonal of the square has inspired the research of a pattern of squares where the Golden Ratio is generated only by the diagonals. -
Examples of the Golden Ratio You Can Find in Nature
Examples Of The Golden Ratio You Can Find In Nature It is often said that math contains the answers to most of universe’s questions. Math manifests itself everywhere. One such example is the Golden Ratio. This famous Fibonacci sequence has fascinated mathematicians, scientist and artists for many hundreds of years. The Golden Ratio manifests itself in many places across the universe, including right here on Earth, it is part of Earth’s nature and it is part of us. In mathematics, the Fibonacci sequence is the ordering of numbers in the following integer sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… and so on forever. Each number is the sum of the two numbers that precede it. The Fibonacci sequence is seen all around us. Let’s explore how your body and various items, like seashells and flowers, demonstrate the sequence in the real world. As you probably know by now, the Fibonacci sequence shows up in the most unexpected places. Here are some of them: 1. Flower petals number of petals in a flower is often one of the following numbers: 3, 5, 8, 13, 21, 34 or 55. For example, the lily has three petals, buttercups have five of them, the chicory has 21 of them, the daisy has often 34 or 55 petals, etc. 2. Faces Faces, both human and nonhuman, abound with examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. -
The Golden Relationships: an Exploration of Fibonacci Numbers and Phi
The Golden Relationships: An Exploration of Fibonacci Numbers and Phi Anthony Rayvon Watson Faculty Advisor: Paul Manos Duke University Biology Department April, 2017 This project was submitted in partial fulfillment of the requirements for the degree of Master of Arts in the Graduate Liberal Studies Program in the Graduate School of Duke University. Copyright by Anthony Rayvon Watson 2017 Abstract The Greek letter Ø (Phi), represents one of the most mysterious numbers (1.618…) known to humankind. Historical reverence for Ø led to the monikers “The Golden Number” or “The Devine Proportion”. This simple, yet enigmatic number, is inseparably linked to the recursive mathematical sequence that produces Fibonacci numbers. Fibonacci numbers have fascinated and perplexed scholars, scientists, and the general public since they were first identified by Leonardo Fibonacci in his seminal work Liber Abacci in 1202. These transcendent numbers which are inextricably bound to the Golden Number, seemingly touch every aspect of plant, animal, and human existence. The most puzzling aspect of these numbers resides in their universal nature and our inability to explain their pervasiveness. An understanding of these numbers is often clouded by those who seemingly find Fibonacci or Golden Number associations in everything that exists. Indeed, undeniable relationships do exist; however, some represent aspirant thinking from the observer’s perspective. My work explores a number of cases where these relationships appear to exist and offers scholarly sources that either support or refute the claims. By analyzing research relating to biology, art, architecture, and other contrasting subject areas, I paint a broad picture illustrating the extensive nature of these numbers. -
Books in Brief Hypothesis Might Be That a New Drug Has No Effect
BOOKS & ARTS COMMENT includes not just one correct answer, but all other possibilities. In a medical experiment, the null Books in brief hypothesis might be that a new drug has no effect. But the hypothesis will come pack- The Age of Addiction aged in a statistical model that assumes that David T. Courtwright BELKNAP (2019) there is zero systematic error. This is not Opioids, processed foods, social-media apps: we navigate an necessarily true: errors can arise even in a addictive environment rife with products that target neural pathways randomized, blinded study, for example if involved in emotion and appetite. In this incisive medical history, some participants work out which treatment David Courtwright traces the evolution of “limbic capitalism” from group they have been assigned to. This can prehistory. Meshing psychology, culture, socio-economics and lead to rejection of the null hypothesis even urbanization, it’s a story deeply entangled in slavery, corruption when the new drug has no effect — as can and profiteering. Although reform has proved complex, Courtwright other complexities, such as unmodelled posits a solution: an alliance of progressives and traditionalists aimed measurement error. at combating excess through policy, taxation and public education. To say that P = 0.05 should lead to acceptance of the alternative hypothesis is tempting — a few million scientists do it Cosmological Koans every year. But it is wrong, and has led to Anthony Aguirre W. W. NORTON (2019) replication crises in many areas of the social, Cosmologist Anthony Aguirre explores the nature of the physical behavioural and biological sciences. Universe through an intriguing medium — the koan, that paradoxical Statistics — to paraphrase Homer riddle of Zen Buddhist teaching. -
The Glorious Golden Ratio
Published 2012 by Prometheus Books The Glorious Golden Ratio. Copyright © 2012 Alfred S. Posamentier and Ingmar Lehmann. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, digital, electronic, mechanical, photocopying, recording, or otherwise, or conveyed via the Internet or a website without prior written permission of the publisher, except in the case of brief quotations embodied in critical articles and reviews. Inquiries should be addressed to Prometheus Books 59 John Glenn Drive Amherst, New York 14228–2119 VOICE: 716–691–0133 FAX: 716–691–0137 WWW.PROMETHEUSBOOKS.COM 16 15 14 13 12 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Posamentier, Alfred S. The glorious golden ratio / by Alfred S. Posamentier and Ingmar Lehmann. p. cm. ISBN 978–1–61614–423–4 (cloth : alk. paper) ISBN 978–1–61614–424–1 (ebook) 1. Golden Section. 2. Lehmann, Ingmar II. Title. QA466.P67 2011 516.2'04--dc22 2011004920 Contents Acknowledgments Introduction Chapter 1: Defining and Constructing the Golden Ratio Chapter 2: The Golden Ratio in History Chapter 3: The Numerical Value of the Golden Ratio and Its Properties Chapter 4: Golden Geometric Figures Chapter 5: Unexpected Appearances of the Golden Ratio Chapter 6: The Golden Ratio in the Plant Kingdom Chapter 7: The Golden Ratio and Fractals Concluding Thoughts Appendix: Proofs and Justifications of Selected Relationships Notes Index Acknowledgments he authors wish to extend sincere thanks for proofreading and useful suggestions to Dr. Michael Engber, professor emeritus of mathematics at the City College of the City University of New York; Dr. -
Fractal-Bits
fractal-bits Claude Heiland-Allen 2012{2019 Contents 1 buddhabrot/bb.c . 3 2 buddhabrot/bbcolourizelayers.c . 10 3 buddhabrot/bbrender.c . 18 4 buddhabrot/bbrenderlayers.c . 26 5 buddhabrot/bound-2.c . 33 6 buddhabrot/bound-2.gnuplot . 34 7 buddhabrot/bound.c . 35 8 buddhabrot/bs.c . 36 9 buddhabrot/bscolourizelayers.c . 37 10 buddhabrot/bsrenderlayers.c . 45 11 buddhabrot/cusp.c . 50 12 buddhabrot/expectmu.c . 51 13 buddhabrot/histogram.c . 57 14 buddhabrot/Makefile . 58 15 buddhabrot/spectrum.ppm . 59 16 buddhabrot/tip.c . 59 17 burning-ship-series-approximation/BossaNova2.cxx . 60 18 burning-ship-series-approximation/BossaNova.hs . 81 19 burning-ship-series-approximation/.gitignore . 90 20 burning-ship-series-approximation/Makefile . 90 21 .gitignore . 90 22 julia/.gitignore . 91 23 julia-iim/julia-de.c . 91 24 julia-iim/julia-iim.c . 94 25 julia-iim/julia-iim-rainbow.c . 94 26 julia-iim/julia-lsm.c . 98 27 julia-iim/Makefile . 100 28 julia/j-render.c . 100 29 mandelbrot-delta-cl/cl-extra.cc . 110 30 mandelbrot-delta-cl/delta.cl . 111 31 mandelbrot-delta-cl/Makefile . 116 32 mandelbrot-delta-cl/mandelbrot-delta-cl.cc . 117 33 mandelbrot-delta-cl/mandelbrot-delta-cl-exp.cc . 134 34 mandelbrot-delta-cl/README . 139 35 mandelbrot-delta-cl/render-library.sh . 142 36 mandelbrot-delta-cl/sft-library.txt . 142 37 mandelbrot-laurent/DM.gnuplot . 146 38 mandelbrot-laurent/Laurent.hs . 146 39 mandelbrot-laurent/Main.hs . 147 40 mandelbrot-series-approximation/args.h . 148 41 mandelbrot-series-approximation/emscripten.cpp . 150 42 mandelbrot-series-approximation/index.html . -
Math Morphing Proximate and Evolutionary Mechanisms
Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 2009 Volume V: Evolutionary Medicine Math Morphing Proximate and Evolutionary Mechanisms Curriculum Unit 09.05.09 by Kenneth William Spinka Introduction Background Essential Questions Lesson Plans Website Student Resources Glossary Of Terms Bibliography Appendix Introduction An important theoretical development was Nikolaas Tinbergen's distinction made originally in ethology between evolutionary and proximate mechanisms; Randolph M. Nesse and George C. Williams summarize its relevance to medicine: All biological traits need two kinds of explanation: proximate and evolutionary. The proximate explanation for a disease describes what is wrong in the bodily mechanism of individuals affected Curriculum Unit 09.05.09 1 of 27 by it. An evolutionary explanation is completely different. Instead of explaining why people are different, it explains why we are all the same in ways that leave us vulnerable to disease. Why do we all have wisdom teeth, an appendix, and cells that if triggered can rampantly multiply out of control? [1] A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Beno?t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. http://www.kwsi.com/ynhti2009/image01.html A fractal often has the following features: 1. It has a fine structure at arbitrarily small scales. -
A Classroom Investigation Into the Catenary
A classroom investigation into the catenary Ed Staples Erindale College, ACT <[email protected]> The Catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight… The word catenary is derived from the Latin word catena, which means “chain”. Huygens first used the term catenaria in a letter to Leibniz in 1690… Hooke discovered that the catenary is the ideal curve for an arch of uniform density and thick- ness which supports only its own weight. (Wikipedia, catenary) he quest to find the equation of a catenary makes an ideal investigation Tfor upper secondary students. In the modelling exercise that follows, no knowledge of calculus is required to gain a fairly good understanding of the nature of the curve. This investigation1 is best described as a scientific investi- 1. I saw a similar inves- gation—a ‘hands on’ experience that examines some of the techniques used tigation conducted with students by by science to find models of natural phenomena. It was Joachim Jungius John Short at the time working at (1587–1657) who proved that the curve followed by a chain was not in fact a Rosny College in 2004. The chain parabola, (published posthumously in 1669, Wikipedia, catenary). Wolfram hung along the length of the wall MathWorld also provides relevant information and interesting interactive some four or five metres, with no demonstrations (see http://mathworld.wolfram.com/Catenary.html). more than a metre vertically from the The investigation, to be described here, is underpinned by the fact that the chain’s vertex to the catenary’s equation can be thought of as a real valued polynomial consisting horizontal line Australian Senior Mathematics Journal 25 (2) 2011 between the two of an infinite number of even-powered terms described as: anchor points.