Icons of Mathematics an EXPLORATION of TWENTY KEY IMAGES Claudi Alsina and Roger B
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Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur Si Muove: Biomimetic Embedding of N-Tuple Helices in Spherical Polyhedra - /
Alternative view of segmented documents via Kairos 23 October 2017 | Draft Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur si muove: Biomimetic embedding of N-tuple helices in spherical polyhedra - / - Introduction Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension? Dynamic bonding patterns in n-tuple helices engendering n-fold rotating symbols Embedding the triple helix in a spherical octahedron Embedding the quadruple helix in a spherical cube Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum Embedding six-fold, eight-fold and ten-fold helices in appropriately encircled polyhedra Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra Neglected recognition of logical patterns -- especially of opposition Dynamic relationship between polyhedra engendered by circles -- variously implying forms of unity Symbol rotation as dynamic essential to engaging with value-inversion References Introduction The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (1564-1642). His much-cited phrase, " And yet it moves" (E pur si muove or Eppur si muove) was allegedly pronounced in 1633 when he was forced to recant his claims that the Earth moves around the immovable Sun rather than the converse -- known as the Galileo affair. Such a shift in perspective might usefully inspire the recognition that the stasis attributed so widely to logos and other much-valued cultural and heraldic symbols obscures the manner in which they imply a fundamental cognitive dynamic. Cultural symbols fundamental to the identity of a group might then be understood as variously moving and transforming in ways which currently elude comprehension. -
Compact Design of Modified Pentagon-Shaped Monopole Antenna for UWB Applications
International Journal of Electrical and Electronic Engineering & Telecommunications Vol. 7, No. 2, April 2018 Compact Design of Modified Pentagon-shaped Monopole Antenna for UWB Applications Sanyog Rawat1, Ushaben Keshwala2, and Kanad Ray3 1 Manipal University Jaipur, India. 2 Amity University Uttar Pradesh, India. 3 Amity University Rajasthan, India. Email: [email protected]; [email protected]; [email protected] Abstract—In this work detailed analysis of a modified, introduction, and in Section II the initial antenna pentagon-shaped planar antenna is presented for ultra-wide geometry is presented and discussed. In the preceding bandwidth (UWB) applications. The proposed antenna was section, the modified pentagon-shaped antenna is designed on an FR-4 substrate and has a compact size of 12 elaborated and the results are discussed. × 22 × 1.6 mm3. It achieves an impedance bandwidth of 8.73 GHz (3.8–12.53 GHz) in the UWB range. The design has a II. ANTENNA GEOMETRY uniform gain and stable radiation pattern in the operating bandwidth. A. Initial Pentagram Star-shaped Antenna Index Terms—Golden angle, pentagram, pentagon, star- The antenna presented in this paper was designed on a shaped antenna FR-4 substrate, which was a partial ground on one side and a conducting patch on the opposite side. The antenna design is initially started with a pentagram star shape and I. INTRODUCTION then the shape was modified into a pentagon shape. The Nowadays, wireless communications and their high- pentagram is also called a pentangle, i.e., 5-pointed star speed data rate are becoming increasingly popular. The [13]. The geometry of the microstrip line fed star-shaped research in the field of microstrip antennas has also been monopole antenna is shown in Fig. -
Indirect Proof
95698_ch02_067-120.qxp 9/5/13 4:53 PM Page 76 76 CHAPTER 2 ■ PARALLEL LINES In Exercises 28 to 30, write a formal proof of each theorem. 34. Given: Triangle MNQ with M obtuse ∠MNQ 28. If two parallel lines are cut by a transversal, then the pairs Construct: NE Ќ MQ, with E of alternate exterior angles are congruent. on MQ. N Q 29. If two parallel lines are cut by a transversal, then the pairs 35. Given: Triangle MNQ with of exterior angles on the same side of the transversal are Exercises 34, 35 obtuse ∠MNQ supplementary. Construct: MR Ќ NQ 30. If a transversal is perpendicular to one of two parallel lines, (HINT: Extend NQ.) then it is also perpendicular to the other line. 36. Given: A line m and a point T not on m 31. Suppose that two lines are cut by a transversal in such a way that the pairs of corresponding angles are not congruent. T Can those two lines be parallel? 32. Given: Line ᐍ and point P P m not on ᐍ —! Suppose that you do the following: Construct: PQ Ќ ᐍ i) Construct a perpendicular line r from T to line m. ii) Construct a line s perpendicular to line r at point T. 33. Given: Triangle ABC with B three acute angles What is the relationship between lines s and m? Construct: BD Ќ AC, with D on AC. A C 2.2 Indirect Proof KEY CONCEPTS Conditional Contrapositive Indirect Proof Converse Law of Negative Inverse Inference Let P S Q represent the conditional statement “If P, then Q.” The following statements are related to this conditional statement (also called an implication). -
On the Standard Lengths of Angle Bisectors and the Angle Bisector Theorem
Global Journal of Advanced Research on Classical and Modern Geometries ISSN: 2284-5569, pp.15-27 ON THE STANDARD LENGTHS OF ANGLE BISECTORS AND THE ANGLE BISECTOR THEOREM G.W INDIKA SHAMEERA AMARASINGHE ABSTRACT. In this paper the author unveils several alternative proofs for the standard lengths of Angle Bisectors and Angle Bisector Theorem in any triangle, along with some new useful derivatives of them. 2010 Mathematical Subject Classification: 97G40 Keywords and phrases: Angle Bisector theorem, Parallel lines, Pythagoras Theorem, Similar triangles. 1. INTRODUCTION In this paper the author introduces alternative proofs for the standard length of An- gle Bisectors and the Angle Bisector Theorem in classical Euclidean Plane Geometry, on a concise elementary format while promoting the significance of them by acquainting some prominent generalized side length ratios within any two distinct triangles existed with some certain correlations of their corresponding angles, as new lemmas. Within this paper 8 new alternative proofs are exposed by the author on the angle bisection, 3 new proofs each for the lengths of the Angle Bisectors by various perspectives with also 5 new proofs for the Angle Bisector Theorem. 1.1. The Standard Length of the Angle Bisector Date: 1 February 2012 . 15 G.W Indika Shameera Amarasinghe The length of the angle bisector of a standard triangle such as AD in figure 1.1 is AD2 = AB · AC − BD · DC, or AD2 = bc 1 − (a2/(b + c)2) according to the standard notation of a triangle as it was initially proved by an extension of the angle bisector up to the circumcircle of the triangle. -
Polygrams and Polygons
Poligrams and Polygons THE TRIANGLE The Triangle is the only Lineal Figure into which all surfaces can be reduced, for every Polygon can be divided into Triangles by drawing lines from its angles to its centre. Thus the Triangle is the first and simplest of all Lineal Figures. We refer to the Triad operating in all things, to the 3 Supernal Sephiroth, and to Binah the 3rd Sephirah. Among the Planets it is especially referred to Saturn; and among the Elements to Fire. As the colour of Saturn is black and the Triangle that of Fire, the Black Triangle will represent Saturn, and the Red Fire. The 3 Angles also symbolize the 3 Alchemical Principles of Nature, Mercury, Sulphur, and Salt. As there are 3600 in every great circle, the number of degrees cut off between its angles when inscribed within a Circle will be 120°, the number forming the astrological Trine inscribing the Trine within a circle, that is, reflected from every second point. THE SQUARE The Square is an important lineal figure which naturally represents stability and equilibrium. It includes the idea of surface and superficial measurement. It refers to the Quaternary in all things and to the Tetrad of the Letter of the Holy Name Tetragrammaton operating through the four Elements of Fire, Water, Air, and Earth. It is allotted to Chesed, the 4th Sephirah, and among the Planets it is referred to Jupiter. As representing the 4 Elements it represents their ultimation with the material form. The 4 angles also include the ideas of the 2 extremities of the Horizon, and the 2 extremities of the Median, which latter are usually called the Zenith and the Nadir: also the 4 Cardinal Points. -
The Arbelos in Wasan Geometry: Chiba's Problem
Global Journal of Advanced Research on Classical and Modern Geometries ISSN: 2284-5569, Vol.8, (2019), Issue 2, pp.98-104 THE ARBELOS IN WASAN GEOMETRY: CHIBA’S PROBLEM HIROSHI OKUMURA Abstract. We consider a sangaku problem involving an arbelos with two congruent circles, and give several conditions where the two congruent circles appear. Also we show several pairs of congruent circles and Archimedean circles. 2010 Mathematical Subject Classification: 01A27, 51M04, 51N20. Keywords and phrases: arbelos, Archimedean circle, non-Archimedean congruent circles. 1. INTRODUCTION We consider the arbelos appeared in Wasan geometry, and consider an arbelos formed by three semicircles α, β and γ with diameters AO , BO and AB , respectively for a point O on the segment AB . The radii of α and β are denoted by a and b, respectively. We consider the following sangaku .(in 1880 [5] (see Figure 1 (نproblem proposed by Chiba ( ઍ༿ӫ࣏҆ δ ε γ α h β B O H A Figure 1. Problem 1. For a point H on the segment AO, let h be the perpendicular to AB at H. Let δ be the circle touching α and β externally and h, and let ε be the incircle of the curvilinear triangle made by α, γ and h. If δ and ε touch and δ is congruent to the circle of diameter AH, find the radius of ε in terms of a and b. Circles of radius rA = ab /(a + b) are said to be Archimedean. In this paper we generalize the problem and consider two circles touching at a point on a perpendicular to the line AB . -
Symmetry Is a Manifestation of Structural Harmony and Transformations of Geometric Structures, and Lies at the Very Foundation
Bridges Finland Conference Proceedings Some Girihs and Puzzles from the Interlocks of Similar or Complementary Figures Treatise Reza Sarhangi Department of Mathematics Towson University Towson, MD 21252 E-mail: [email protected] Abstract This paper is the second one to appear in the Bridges Proceedings that addresses some problems recorded in the Interlocks of Similar or Complementary Figures treatise. Most problems in the treatise are sketchy and some of them are incomprehensible. Nevertheless, this is the only document remaining from the medieval Persian that demonstrates how a girih can be constructed using compass and straightedge. Moreover, the treatise includes some puzzles in the transformation of a polygon into another one using mathematical formulas or dissection methods. It is believed that the document was written sometime between the 13th and 15th centuries by an anonymous mathematician/craftsman. The main intent of the present paper is to analyze a group of problems in this treatise to respond to questions such as what was in the mind of the treatise’s author, how the diagrams were constructed, is the conclusion offered by the author mathematically provable or is incorrect. All images, except for photographs, have been created by author. 1. Introduction There are a few documents such as treatises and scrolls in Persian mosaic design, that have survived for centuries. The Interlocks of Similar or Complementary Figures treatise [1], is one that is the source for this article. In the Interlocks document, one may find many interesting girihs and also some puzzles that are solved using mathematical formulas or dissection methods. Dissection, in the present literature, refers to cutting a geometric, two-dimensional shape, into pieces that can be rearranged to compose a different shape. -
Archimedes and the Arbelos1 Bobby Hanson October 17, 2007
Archimedes and the Arbelos1 Bobby Hanson October 17, 2007 The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. — G.H. Hardy, A Mathematician’s Apology ACBr 1 − r Figure 1. The Arbelos Problem 1. We will warm up on an easy problem: Show that traveling from A to B along the big semicircle is the same distance as traveling from A to B by way of C along the two smaller semicircles. Proof. The arc from A to C has length πr/2. The arc from C to B has length π(1 − r)/2. The arc from A to B has length π/2. ˜ 1My notes are shamelessly stolen from notes by Tom Rike, of the Berkeley Math Circle available at http://mathcircle.berkeley.edu/BMC6/ps0506/ArbelosBMC.pdf . 1 2 If we draw the line tangent to the two smaller semicircles, it must be perpendicular to AB. (Why?) We will let D be the point where this line intersects the largest of the semicircles; X and Y will indicate the points of intersection with the line segments AD and BD with the two smaller semicircles respectively (see Figure 2). Finally, let P be the point where XY and CD intersect. D X P Y ACBr 1 − r Figure 2 Problem 2. Now show that XY and CD are the same length, and that they bisect each other. -
Fourth Lecture
Fourth Knowledge Lecture of the New Hermetics The Symbol Of Venus and The Tree Of Life It embraces all ten Sephiroth on the Tree. It is a fitting emblem of the Isis of Nature. Since it contains all the Sephiroth its circle should be made larger then that of Mercury shown in a previous diagram. The Geomantic figures and their Zodiacal Attributions Planets, Colors, Lineal Figures Etc. No. Planet Color Shape Odor 3. Saturn Black Triangle Myrrh 4. Jupiter Blue Square Cedar 5. Mars Red Pentagram Pepper 6. Sun Yellow Hexagram Frankincense 7. Venus Green Septagram Benzoin 8. Mercury Orange Octagram Sandalwood 9. Moon Violet Enneagram Camphor Element Color Odor Fire Red Cinnamon Water Blue Cedar Air Yellow Sandalwood Earth Black Myrrh DIAGRAM 66 The Triangle The first linear shape, associated with the sephirah Binah and the planet Saturn. Medieval sorcerers used triangles to bind spirits because they believed the limiting force of the triangle would confine the spirit. The triangle may be used in any effort to restrict, structure or limit anything. The number three also indicates cycles and therefore time, also a limiting factor appropriate to Saturn. The Square The square is related to Chesed, whose number is four. A square implies the form of a castle or walled structure, society, prestige, rulership and other Jupiterean qualities. The square also symbolizes structure beyond walls, it represents a completion or perfection; a "square deal" and three "square meals" a day illustrate the archetypal idea verbally. Also the four elements working in balanced harmony. The Pentagram The pentagram is the Force of Mars, and the sephira Geburah. -
Solutions Manual to Accompany Classical Geometry
Solutions Manual to Accompany Classical Geometry Solutions Manual to Accompany CLASSICAL GEOMETRY Euclidean, Transformational, Inversive, and Projective I. E. Leonard J. E. Lewis A. C. F. Liu G. W. Tokarsky Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Canada WILEY Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representation or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. -
Advanced Euclidean Geometry
Advanced Euclidean Geometry Paul Yiu Summer 2016 Department of Mathematics Florida Atlantic University July 18, 2016 Summer 2016 Contents 1 Some Basic Theorems 101 1.1 The Pythagorean Theorem . ............................ 101 1.2 Constructions of geometric mean . ........................ 104 1.3 The golden ratio . .......................... 106 1.3.1 The regular pentagon . ............................ 106 1.4 Basic construction principles ............................ 108 1.4.1 Perpendicular bisector locus . ....................... 108 1.4.2 Angle bisector locus . ............................ 109 1.4.3 Tangency of circles . ......................... 110 1.4.4 Construction of tangents of a circle . ............... 110 1.5 The intersecting chords theorem ........................... 112 1.6 Ptolemy’s theorem . ................................. 114 2 The laws of sines and cosines 115 2.1 The law of sines . ................................ 115 2.2 The orthocenter ................................... 116 2.3 The law of cosines .................................. 117 2.4 The centroid ..................................... 120 2.5 The angle bisector theorem . ............................ 121 2.5.1 The lengths of the bisectors . ........................ 121 2.6 The circle of Apollonius . ............................ 123 3 The tritangent circles 125 3.1 The incircle ..................................... 125 3.2 Euler’s formula . ................................ 128 3.3 Steiner’s porism ................................... 129 3.4 The excircles .................................... -
Chapter 6 the Arbelos
Chapter 6 The arbelos 6.1 Archimedes’ theorems on the arbelos Theorem 6.1 (Archimedes 1). The two circles touching CP on different sides and AC CB each touching two of the semicircles have equal diameters AB· . P W1 W2 A O1 O C O2 B A O1 O C O2 B Theorem 6.2 (Archimedes 2). The diameter of the circle tangent to all three semi- circles is AC CB AB · · . AC2 + AC CB + CB2 · We shall consider Theorem ?? in ?? later, and for now examine Archimedes’ wonderful proofs of the more remarkable§ Theorems 6.1 and 6.2. By synthetic reasoning, he computed the radii of these circles. 1Book of Lemmas, Proposition 5. 2Book of Lemmas, Proposition 6. 602 The arbelos 6.1.1 Archimedes’ proof of the twin circles theorem In the beginning of the Book of Lemmas, Archimedes has established a basic proposition 3 on parallel diameters of two tangent circles. If two circles are tangent to each other (internally or externally) at a point P , and if AB, XY are two parallel diameters of the circles, then the lines AX and BY intersect at P . D F I W1 E H W2 G A O C B Figure 6.1: Consider the circle tangent to CP at E, and to the semicircle on AC at G, to that on AB at F . If EH is the diameter through E, then AH and BE intersect at F . Also, AE and CH intersect at G. Let I be the intersection of AE with the outer semicircle. Extend AF and BI to intersect at D.