DOCSLIB.ORG

Views of Polyhedra Were Created by 1.46: Alexander Graham Bell and the Octet Truss

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

UNIVERSITY OF CINCINNATI Date:___________________February 16, 2005 I, _________________________________________________________,Nicola Laila D’souza hereby submit this work as part of the requirements for the degree of: Master of Architecture in: the School of Architecture & Interior Design at the College of Design, Architecture, Art & Planning It is entitled: Natural Forms Through Geometry and Structure: Design of the Parachute Pavilion This work and its defense approved by: Chair: _______________________________Barry Stedman _______________________________Robert Burnham _______________________________Thomas Bible _______________________________ _______________________________ Natural Forms Through Geometry And Structure : Design of the Parachute Pavilion. A thesis submitted to the Division Of Research and Advanced Studies of the University of Cincinnati In partial fulfillment of the requirements for the degree of Master of Architecture In the School of Architecture and Interior Design of the College of Design, Architecture, Art and Planning 2005 by Nicola Laila D’souza Bachelor of Architecture, University of Mumbai, 1999 Thesis Committee: Barry Stedman Robert Burnham Thomas Bible ABSTRACT “Mathematics is only a means for expressing the laws that govern phenomena” stated Albert Einstein. Similarly, geometry is only a means of defining the laws that govern natural structures, both organic and inorganic. Hence it is a powerful and ubiquitous explanator of form in the natural world. Nature is abundant with the most profound examples of delicate strength. All natural forms have an intrinsic geometrical order with an inherent flexibility for variable outcomes. This thesis is inspired by the geometrical order that is responsible for beauty, variety, strength, stability, functionality and economy in nature. It hypothesizes that the geometry that has shaped the natural world, can be used at an intrinsic level to create form in the built environment. The result of this exploration is the Golden Tetrahedral System. It is a triangulated and dimensionally stable geometric system for light weight space frame construction. It is derived from the geometry of the Golden Section. Six tetrahedral blocks, composed from four different lengths and triangular faces, form the basic units of the system. These versatile blocks can be combined to result in planar, spherical, cylindrical and other curved surfaces. With the use of this system, it is also possible to create naturally curved surfaces like those found in a flower petal or a leaf. The modularity and use of repetitive elements enables economic manufacture and construction. The system is tested out in the design project, ‘The Parachute Pavilion’. It is used to form a space frame structure that derives its form from a collapsing parachute, frozen in time, just before it meets the ground at the base of the tower in whose shadow it stands. The intricate complexity of the geometry in this building is integrated with simplicity of form and construction technique so as to be easily constructable. 3 4 ACKNOWLEDGEMENTS For their love and support, I would like to thank my family; my parents, Lawrence D’souza and Lynette D’souza and my sisters and brothers-in-law; Sunita D’souza and Paul Crosser, Liezel D’souza and Are Andreson. For their academic guidance, I would like to thank my thesis committee: Barry Stedman, Robert Burnham and Thomas Bible. I would also like to thank Vincent Sanzalone and Masud Taj and all the faculty at The School of Architecture and Interior Design and Rizvi College of Architecture. For the professional learning that I have had, I would like to thank the staff from Kamal S. Malik (Bombay), Zimmer Gunsul & Frasca Partnership (Washington DC) and Perkins Eastman (New York) I would like to thank Steven Albert, Kevin Schreur and Brandon Kelly for their support and help. For all his support, love and patience I would like to thank John Gallagher. I would like to thank all the scientists and mathematicians who have tried to explain the wonders in nature. And of course, the creator of it all. 5 TABLE OF CONTENTS. Abstract 2 Chapter III: The Golden Ratio & Fivefold Acknowledgements 4 Symmetry. Table of Contents 5 Introduction 59 List of Tables 7 History of the Golden Ratio 60 List of Illustrations 8 Mathematical Properties of the Golden Ratio 62 Introduction 17 Observations of the Golden Ratio in Nature 64 Conclusions 66 Chapter I: Plane and Solid Geometry Introduction 21 Chapter IV: Penrose Tilings and Quasicrystals. Concepts of Space and Geometry 22 Introduction 67 Dimensional Aspects of Space 24 Penrose Tilings 68 Planar Geometry 25 Quasicrystals 70 Curves 26 Surfaces from Quasicrystal Geometry Polygons 28 Planar Surfaces 72 Two Dimensional Tessellations 30 Non Planar Surfaces 74 Three Dimensional Geometry Closed Seed Patterns 76 Classification 34 Open Seed Patterns 82 Three Dimensional Tessellations 36 Replacing Planes by a Pattern 84 Polyhedra 38 Joining of the Modules Together 85 Curved Surfaces 43 Use of Quasicrystal Geometry in Art & Dimensional Stability 44 Architecture 87 Dimensionally Stable Configurations in Conclusions 88 Architecture The Octet Truss 46 Chapter V: The Golden Tetrahedral System. Geodesic Domes 48 Introduction 89 Twisted Domes 50 The Golden Tetrahedral System 90 Conclusion. 52 Mathematical Formulae 91 The Geometry of the System 101 Chapter II: Aperiodic Geometry Basic Combinations of the Tetrahedra 102 Introduction 53 Surfaces with Uniform Curvature 103 Projections from Higher Dimensional Space. 54 Surfaces with Non-Uniform Curvature 105 The Infinite Family of Aperiodic Tessellations. 56 Conforming the System to a Predetermined 106 Conclusion: The Fivefold Case 58 Form. Conclusions 107 6 Bibliography 109 Appendix I: Space Frame Systems Introduction 113 Concepts of SpaceFrames 114 Advantages of SpaceFrames 115 Disadvantages of SpaceFrames 116 Types of SpaceFrames 117 Connector Plates Hub Type 118 Spherical Hub Type 119 Extruded Hub Type 120 The N55 Spaceframe (Hub-less Type) 121 The Aurodyne Hyperframe System 123 Bambutec Technology 124 Conclusion. 125 Appendix II: The Competition Brief Introduction 126 Coney Island History. 127 Site History. 129 Site Description 130 Program Requirements 132 Site Plan 133 Site Images 134 7 LIST OF TABLES Chapter I: Plane and Solid Geometry Table 1.1: Operatons for the Creation of Three Dimensional Objects. Table 1.2: The relation between the values of p and q and the geometry of the tessellation{p,q} Chapter II: Aperiodic Geometry Table 2.1 : A table showing the infinite family of rhombi projected from dimension ‘n’. Chapter IV: Penrose Tilings and Quasicrystals. Table 4.1 The 17 Different Symmetries Possible Around a Node Chapter V: The Golden Tetrahedral System. Table 5.1: Comparision of the lengths Obtained with Different Starting Assumptions Table 5.2 : Ratios of the Different Lengths. 8 LIST OF ILLUSTRATIONS Introduction as Addison Wesley 0.1 Flowers exhibiting Geometry. Royalty Free images from Burton Networks 1.3: Unfolding of Spatial Dimensions. Retrieved on May 12, 2005 from http:// Keith Critchlow. Order in Space, pg 7 www.burtonnetworks.com/downloads/ photos/flowers/ 1.4.a: Open and Closed Curves. Eric W. Weisstein. “Closed Curve.” Retrieved Chapter I: Plane and Solid Geometry May 12,2005 From MathWorld--A Wolfram 1.1.a: Circle Limit III Web Resource. http:// M. C. Escher, pg 97 mathworld.wolfram.com/ClosedCurve.html Photograph from the Catalogue compiled by the authors, F.H. Bool, J.L. Locher & F. Wierda. 1.4.b: Simple and Non Simple Curves. Eric W. Weisstein. “Simple Curve.” Retrieved 1.1.b: Circle Limit IV From MathWorld--A Wolfram Web Resource. M. C. Escher, pg 98 http://mathworld.wolfram.com/ Photograph from the Catalogue compiled by SimpleCurve.html the authors, F.H. Bool, J.L. Locher & F. Wierda. 1.4.c: Jordan and Non Jordan Curves. 1.2.a: Flat Euclidean Space Eric W. Weisstein. “Jordan Curve.” Retrieved http://astro.uwaterloo.ca/~mjhudson/teaching/ May 12, 2005 from MathWorld--A Wolfram sci238/notes/node17.html Web Resource. http://mathworld.wolfram.com/ Copyright©2004 Pearson Education, Publishing JordanCurve.html as Addison Wesley 1.5: Circle. 1.2.b: Spherical Space Eric W. Weisstein. “Circle.” Retrieved May http://astro.uwaterloo.ca/~mjhudson/teaching/ 12, 2005 from MathWorld--A Wolfram Web sci238/notes/node17.html Resource. http:// mathworld.wolfram.com/ Copyright©2004 Pearson Education, Publishing Circle.html as Addison Wesley 1.6: Conics. 1.2.c: Hyperbolic Space Eric W. Weisstein. “Conic Section.” Retrieved http://astro.uwaterloo.ca/~mjhudson/teaching/ May 12, 2005 from MathWorld--A Wolfram sci238/notes/node17.html Web Resource. http://mathworld.wolfram.com/ Copyright©2004 Pearson Education, Publishing ConicSection.html 9 1.7: Ellipse. 1.13: Skew Polygons. Eric W. Weisstein. “Ellipse.” Retrieved May Robert Williams. Natural Structure, pg 34 12, 2005 from MathWorld--A Wolfram Web Resource. http:// mathworld.wolfram.com/ 1.14: Regular Tessellations. Ellipse.html & 1.15: Archimedian Tessellations. Eric W. Weisstein. “Tessellation.” Retrieved 1.8: Parabola. May 12, 2005 from MathWorld--A Wolfram Eric W. Weisstein. “Parabola.” Retrieved May Web Resource. http://mathworld.wolfram.com/ 12, 2005 from MathWorld--A Wolfram Web Tessellation.html Resource. http://mathworld.wolfram.com/ Parabola.html 1.16: Duals of Regular Tessellations. & 1.17: Duals of Archimedian Tessellations. 1.9: Hyperbola. Eric W. Weisstein. “Dual Tessellation.” Retrieved Eric W. Weisstein. “Hyperbola.” Retrieved May 12, 2005 from
Recommended publications
  • PERFORMED IDENTITIES: HEAVY METAL MUSICIANS BETWEEN 1984 and 1991 Bradley C. Klypchak a Dissertation Submitted to the Graduate

    PERFORMED IDENTITIES: HEAVY METAL MUSICIANS BETWEEN 1984 and 1991 Bradley C. Klypchak a Dissertation Submitted to the Graduate

    PERFORMED IDENTITIES: HEAVY METAL MUSICIANS BETWEEN 1984 AND 1991 Bradley C. Klypchak A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2007 Committee: Dr. Jeffrey A. Brown, Advisor Dr. John Makay Graduate Faculty Representative Dr. Ron E. Shields Dr. Don McQuarie © 2007 Bradley C. Klypchak All Rights Reserved iii ABSTRACT Dr. Jeffrey A. Brown, Advisor Between 1984 and 1991, heavy metal became one of the most publicly popular and commercially successful rock music subgenres. The focus of this dissertation is to explore the following research questions: How did the subculture of heavy metal music between 1984 and 1991 evolve and what meanings can be derived from this ongoing process? How did the contextual circumstances surrounding heavy metal music during this period impact the performative choices exhibited by artists, and from a position of retrospection, what lasting significance does this particular era of heavy metal merit today? A textual analysis of metal- related materials fostered the development of themes relating to the selective choices made and performances enacted by metal artists. These themes were then considered in terms of gender, sexuality, race, and age constructions as well as the ongoing negotiations of the metal artist within multiple performative realms. Occurring at the juncture of art and commerce, heavy metal music is a purposeful construction. Metal musicians made performative choices for serving particular aims, be it fame, wealth, or art. These same individuals worked within a greater system of influence. Metal bands were the contracted employees of record labels whose own corporate aims needed to be recognized.
  • The Conway Space-Filling

    The Conway Space-Filling

    Symmetry: Art and Science Buenos Aires Congress, 2007 THE CONWAY SPACE-FILLING MICHAEL LONGUET-HIGGINS Name: Michael S. Longuet-Higgins, Mathematician and Oceanographer, (b. Lenham, England, 1925). Address: Institute for Nonlinear Science, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92037-0402, U.S.A. Email: [email protected] Fields of interest: Fluid dynamics, ocean waves and currents, geophysics, underwater sound, projective geometry, polyhedra, mathematical toys. Awards: Rayleigh Prize for Mathematics, Cambridge University, 1950; Hon.D. Tech., Technical University of Denmark, 1979; Hon. LL.D., University of Glasgow, Scotland, 1979; Fellow of the American Geophysical Union, 1981; Sverdrup Gold Medal of the American Meteorological Society, 1983; International Coastal Engineering Award of the American Society of Civil Engineers, 1984; Oceanography Award of the Society for Underwater Technology, 1990; Honorary Fellow of the Acoustical Society of America, 2002. Publications: “Uniform polyhedra” (with H.S.M. Coxeter and J.C.P. Miller (1954). Phil. Trans. R. Soc. Lond. A 246, 401-450. “Some Mathematical Toys” (film), (1963). British Association Meeting, Aberdeen, Scotland; “Clifford’s chain and its analogues, in relation to the higher polytopes,” (1972). Proc. R. Soc. Lond. A 330, 443-466. “Inversive properties of the plane n-line, and a symmetric figure of 2x5 points on a quadric,” (1976). J. Lond. Math. Soc. 12, 206-212; Part II (with C.F. Parry) (1979) J. Lond. Math. Soc. 19, 541-560; “Nested triacontahedral shells, or How to grow a quasi-crystal,” (2003) Math. Intelligencer 25, 25-43. Abstract: A remarkable new space-filling, with an unusual symmetry, was recently discovered by John H.
  • On the Archimedean Or Semiregular Polyhedra

    On the Archimedean Or Semiregular Polyhedra

    ON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA Mark B. Villarino Depto. de Matem´atica, Universidad de Costa Rica, 2060 San Jos´e, Costa Rica May 11, 2005 Abstract We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler’s polyhedral formula. Contents 1 Introduction 2 1.1 RegularPolyhedra .............................. 2 1.2 Archimedean/semiregular polyhedra . ..... 2 2 Proof techniques 3 2.1 Euclid’s proof for regular polyhedra . ..... 3 2.2 Euler’s polyhedral formula for regular polyhedra . ......... 4 2.3 ProofsofArchimedes’theorem. .. 4 3 Three lemmas 5 3.1 Lemma1.................................... 5 3.2 Lemma2.................................... 6 3.3 Lemma3.................................... 7 4 Topological Proof of Archimedes’ theorem 8 arXiv:math/0505488v1 [math.GT] 24 May 2005 4.1 Case1: fivefacesmeetatavertex: r=5. .. 8 4.1.1 At least one face is a triangle: p1 =3................ 8 4.1.2 All faces have at least four sides: p1 > 4 .............. 9 4.2 Case2: fourfacesmeetatavertex: r=4 . .. 10 4.2.1 At least one face is a triangle: p1 =3................ 10 4.2.2 All faces have at least four sides: p1 > 4 .............. 11 4.3 Case3: threefacesmeetatavertes: r=3 . ... 11 4.3.1 At least one face is a triangle: p1 =3................ 11 4.3.2 All faces have at least four sides and one exactly four sides: p1 =4 6 p2 6 p3. 12 4.3.3 All faces have at least five sides and one exactly five sides: p1 =5 6 p2 6 p3 13 1 5 Summary of our results 13 6 Final remarks 14 1 Introduction 1.1 Regular Polyhedra A polyhedron may be intuitively conceived as a “solid figure” bounded by plane faces and straight line edges so arranged that every edge joins exactly two (no more, no less) vertices and is a common side of two faces.
  • Square Rectangle Triangle Diamond (Rhombus) Oval Cylinder Octagon Pentagon Cone Cube Hexagon Pyramid Sphere Star Circle

    Square Rectangle Triangle Diamond (Rhombus) Oval Cylinder Octagon Pentagon Cone Cube Hexagon Pyramid Sphere Star Circle

    SQUARE RECTANGLE TRIANGLE DIAMOND (RHOMBUS) OVAL CYLINDER OCTAGON PENTAGON CONE CUBE HEXAGON PYRAMID SPHERE STAR CIRCLE Powered by: www.mymathtables.com Page 1 what is Rectangle? • A rectangle is a four-sided flat shape where every angle is a right angle (90°). means "right angle" and show equal sides. what is Triangle? • A triangle is a polygon with three edges and three vertices. what is Octagon? • An octagon (eight angles) is an eight-sided polygon or eight-gon. what is Hexagon? • a hexagon is a six-sided polygon or six-gon. The total of the internal angles of any hexagon is 720°. what is Pentagon? • a plane figure with five straight sides and five angles. what is Square? • a plane figure with four equal straight sides and four right angles. • every angle is a right angle (90°) means "right ang le" show equal sides. what is Rhombus? • is a flat shape with four equal straight sides. A rhombus looks like a diamond. All sides have equal length. Opposite sides are parallel, and opposite angles are equal what is Oval? • Many distinct curves are commonly called ovals or are said to have an "oval shape". • Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. Powered by: www.mymathtables.com Page 2 What is Cube? • Six equal square faces.tweleve edges and eight vertices • the angle between two adjacent faces is ninety. what is Sphere? • no faces,sides,vertices • All points are located at the same distance from the center. what is Cylinder? • two circular faces that are congruent and parallel • faces connected by a curved surface.
  • Applying the Polygon Angle

    Applying the Polygon Angle

    POLYGONS 8.1.1 – 8.1.5 After studying triangles and quadrilaterals, students now extend their study to all polygons. A polygon is a closed, two-dimensional figure made of three or more non- intersecting straight line segments connected end-to-end. Using the fact that the sum of the measures of the angles in a triangle is 180°, students learn a method to determine the sum of the measures of the interior angles of any polygon. Next they explore the sum of the measures of the exterior angles of a polygon. Finally they use the information about the angles of polygons along with their Triangle Toolkits to find the areas of regular polygons. See the Math Notes boxes in Lessons 8.1.1, 8.1.5, and 8.3.1. Example 1 4x + 7 3x + 1 x + 1 The figure at right is a hexagon. What is the sum of the measures of the interior angles of a hexagon? Explain how you know. Then write an equation and solve for x. 2x 3x – 5 5x – 4 One way to find the sum of the interior angles of the 9 hexagon is to divide the figure into triangles. There are 11 several different ways to do this, but keep in mind that we 8 are trying to add the interior angles at the vertices. One 6 12 way to divide the hexagon into triangles is to draw in all of 10 the diagonals from a single vertex, as shown at right. 7 Doing this forms four triangles, each with angle measures 5 4 3 1 summing to 180°.
  • Archimedean Solids

    Archimedean Solids

    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat).
  • HÁLÓZATELMÉLET ÉS MŰVÉSZET a Lineáris Információ Nincs Központban

    HÁLÓZATELMÉLET ÉS MŰVÉSZET a Lineáris Információ Nincs Központban

    MOHOLY-NAGY MŰVÉSZETI EGYETEM DOKTORI ISKOLA KALLÓ ANGÉLA HÁLÓZATELMÉLET ÉS MŰVÉSZET A Lineáris Információ Nincs Központban. Jöjjön a LINK! DLA ÉRTEKEZÉS TÉMAVEZETŐ: Dr. TILLMANN JÓZSEF BUDAPEST-KOLOZSVÁR 2009 TARTALOMJEGYZÉK DLA ÉRTEKEZÉS – TÉZISEK A DLA értekezés tézisei magyar nyelven – Bevezető – Tézisek Thesis of DLA dissertation (A DLA értekezés tézisei angol nyelven) – Introduction – Thesis DLA ÉRTEKEZÉS Bevezető 1. Hálózatelmélet – Előzmények 1.1. Kiindulópont 1.2. Hálózatelmélet – három név, három cím, három megközelítés 1.2.1. Barabási 1.2.2. Buchanan 1.2.3. Csermely 2. A háló ki van vetve 2.1. A hálózatelmélet hajnala 2.1.1. Sajátos gráfok 2.2. Lánc, lánc… 2.2.1. Az ismeretségi hálózat és a köztéri művészet 2.2.2. Az ismeretségi hálózat és a multimédia művészet 2.2.3. Az ismeretségi hálózat és a net art, avagy ma van a tegnap holnapja 1 2.3. Kis világ 2.3.1. Hidak 2.3.2. Mitől erős egy gyenge kapcsolat? 2.3.3. Centrum és periféria 2.4. Digitális hálózatok 2.4.1. A hálózatok hálózata 2.4.2. A világháló 2.4.3. A Lineáris Információ Nincs Központban. Jöjjön a LINK! 2.4.4. Kicsi világ @ világháló 2.5. Művészet a hálón 2.5.1. A net mint art 2.5.2. Interaktív művészet a hálón és azon túl 2.5.3. A szavak hálózata mint művészet 2.6. Szemléletváltás 2.6.1. A nexus néhány lehetséges módja 2.6.2. Egy lépésnyire a fraktáloktól 2.6.3. Fraktálok – természet, tudomány, művészet 2.6.4. Térképek 3. Összegzés magyar nyelven 4. Summary of DLA dissertation (Összegzés angol nyelven) Bibliográfia Curriculum Vitae 2 HÁLÓZATELMÉLET ÉS MŰVÉSZET A Lineáris Információ Nincs Központban.
  • 1 Dalí Museum, Saint Petersburg, Florida

    1 Dalí Museum, Saint Petersburg, Florida

    Dalí Museum, Saint Petersburg, Florida Integrated Curriculum Tour Form Education Department, 2015 TITLE: “Salvador Dalí: Elementary School Dalí Museum Collection, Paintings ” SUBJECT AREA: (VISUAL ART, LANGUAGE ARTS, SCIENCE, MATHEMATICS, SOCIAL STUDIES) Visual Art (Next Generation Sunshine State Standards listed at the end of this document) GRADE LEVEL(S): Grades: K-5 DURATION: (NUMBER OF SESSIONS, LENGTH OF SESSION) One session (30 to 45 minutes) Resources: (Books, Links, Films and Information) Books: • The Dalí Museum Collection: Oil Paintings, Objects and Works on Paper. • The Dalí Museum: Museum Guide. • The Dalí Museum: Building + Gardens Guide. • Ades, dawn, Dalí (World of Art), London, Thames and Hudson, 1995. • Dalí’s Optical Illusions, New Heaven and London, Wadsworth Atheneum Museum of Art in association with Yale University Press, 2000. • Dalí, Philadelphia Museum of Art, Rizzoli, 2005. • Anderson, Robert, Salvador Dalí, (Artists in Their Time), New York, Franklin Watts, Inc. Scholastic, (Ages 9-12). • Cook, Theodore Andrea, The Curves of Life, New York, Dover Publications, 1979. • D’Agnese, Joseph, Blockhead, the Life of Fibonacci, New York, henry Holt and Company, 2010. • Dalí, Salvador, The Secret life of Salvador Dalí, New York, Dover publications, 1993. 1 • Diary of a Genius, New York, Creation Publishing Group, 1998. • Fifty Secrets of Magic Craftsmanship, New York, Dover Publications, 1992. • Dalí, Salvador , and Phillipe Halsman, Dalí’s Moustache, New York, Flammarion, 1994. • Elsohn Ross, Michael, Salvador Dalí and the Surrealists: Their Lives and Ideas, 21 Activities, Chicago review Press, 2003 (Ages 9-12) • Ghyka, Matila, The Geometry of Art and Life, New York, Dover Publications, 1977. • Gibson, Ian, The Shameful Life of Salvador Dalí, New York, W.W.
  • A Congruence Problem for Polyhedra

    A Congruence Problem for Polyhedra

    A congruence problem for polyhedra Alexander Borisov, Mark Dickinson, Stuart Hastings April 18, 2007 Abstract It is well known that to determine a triangle up to congruence requires 3 measurements: three sides, two sides and the included angle, or one side and two angles. We consider various generalizations of this fact to two and three dimensions. In particular we consider the following question: given a convex polyhedron P , how many measurements are required to determine P up to congruence? We show that in general the answer is that the number of measurements required is equal to the number of edges of the polyhedron. However, for many polyhedra fewer measurements suffice; in the case of the cube we show that nine carefully chosen measurements are enough. We also prove a number of analogous results for planar polygons. In particular we describe a variety of quadrilaterals, including all rhombi and all rectangles, that can be determined up to congruence with only four measurements, and we prove the existence of n-gons requiring only n measurements. Finally, we show that one cannot do better: for any ordered set of n distinct points in the plane one needs at least n measurements to determine this set up to congruence. An appendix by David Allwright shows that the set of twelve face-diagonals of the cube fails to determine the cube up to conjugacy. Allwright gives a classification of all hexahedra with all face- diagonals of equal length. 1 Introduction We discuss a class of problems about the congruence or similarity of three dimensional polyhedra.
  • Binary Icosahedral Group and 600-Cell

    Binary Icosahedral Group and 600-Cell

    Article Binary Icosahedral Group and 600-Cell Jihyun Choi and Jae-Hyouk Lee * Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-3277-3346 Received: 10 July 2018; Accepted: 26 July 2018; Published: 7 August 2018 Abstract: In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, we show that the binary icosahedral group in H is the set of vertices of a 600-cell by applying the Coxeter–Dynkin diagram of H4. Keywords: binary polyhedral group; icosahedron; dodecahedron; 600-cell MSC: 52B10, 52B11, 52B15 1. Introduction The classification of finite subgroups in SLn(C) derives attention from various research areas in mathematics. Especially when n = 2, it is related to McKay correspondence and ADE singularity theory [1]. The list of finite subgroups of SL2(C) consists of cyclic groups (Zn), binary dihedral groups corresponded to the symmetry group of regular 2n-gons, and binary polyhedral groups related to regular polyhedra. These are related to the classification of regular polyhedrons known as Platonic solids. There are five platonic solids (tetrahedron, cubic, octahedron, dodecahedron, icosahedron), but, as a regular polyhedron and its dual polyhedron are associated with the same symmetry groups, there are only three binary polyhedral groups(binary tetrahedral group 2T, binary octahedral group 2O, binary icosahedral group 2I) related to regular polyhedrons.
  • Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions

    Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions

    SQU Journal for Science, 16 (2011) 82-101 © 2011 Sultan Qaboos University Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions Mehmet Koca, Nazife Ozdes Koca* and Muna Al-Shueili Department of Physics, College of Science, Sultan Qaboos University, P.O.Box 36, Al- Khoud, 123 Muscat, Sultanate of Oman, *Email: [email protected]. متعددات السطوح اليدوية المستمدة من أشكال كوكزتر والرباعيات مهمت كوجا، نزيفة كوجا و منى الشعيلي ملخص: هناك متعددات أسطح ٌدوٌان أرخمٌدٌان: المكعب المسطوح والثنعشري اﻷسطح المسطوح مع أشكالها الكتﻻنٌة المزدوجة: رباعٌات السطوح العشرونٌة الخماسٌة مع رباعٌات السطوح السداسٌة الخماسٌة. فً هذا البحث نقوم برسم متعددات السطوح الٌدوٌة مع مزدوجاتها بطرٌقة منظمة. نقوم باستخدام المجموعة الجزئٌة الدورانٌة الحقٌقٌة لمجموعات كوكستر ( WAAAWAWB(1 1 1 ), ( 3 ), ( 3 و WH()3 ﻻشتقاق المدارات الممثلة لﻷشكال. هذه الطرٌقة تقودنا إلى رسم المتعددات اﻷسطح التالٌة: رباعً الوجوه وعشرونً الوجوه و المكعب المسطوح و الثنعشري السطوح المسطوح على التوالً. لقد أثبتنا أنه بإمكان رباعً الوجوه وعشرونً الوجوه التحول إلى صورتها المرآتٌة بالمجموعة الثمانٌة الدورانٌة الحقٌقٌة WBC()/32. لذلك ﻻ ٌمكن تصنٌفها كمتعددات أسطح ٌدوٌة. من المﻻحظ أٌضا أن رؤوس المكعب المسطوح ورؤوس الثنعشري السطوح المسطوح ٌمكن اشتقاقها من المتجهات المجموعة خطٌا من الجذور البسٌطة وذلك باستخدام مجموعة الدوران الحقٌقٌة WBC()/32 و WHC()/32 على التوالً. من الممكن رسم مزدوجاتها بجمع ثﻻثة مدارات من المجموعة قٌد اﻻهتمام. أٌضا نقوم بإنشاء متعددات اﻷسطح الشبه منتظمة بشكل عام بربط متعددات اﻷسطح الٌدوٌة مع صورها المرآتٌة. كنتٌجة لذلك نحصل على مجموعة البٌروهٌدرال كمجموعة جزئٌة من مجموعة الكوكزتر WH()3 ونناقشها فً البحث. الطرٌقة التً نستخدمها تعتمد على الرباعٌات. ABSTRACT: There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron.
  • How Platonic and Archimedean Solids Define Natural Equilibria of Forces for Tensegrity

    How Platonic and Archimedean Solids Define Natural Equilibria of Forces for Tensegrity

    How Platonic and Archimedean Solids Define Natural Equilibria of Forces for Tensegrity Martin Friedrich Eichenauer The Platonic and Archimedean solids are a well-known vehicle to describe Research Assistant certain phenomena of our surrounding world. It can be stated that they Technical University Dresden define natural equilibria of forces, which can be clarified particularly Faculty of Mathematics Institute of Geometry through the packing of spheres. [1][2] To solve the problem of the densest Germany packing, both geometrical and mechanical approach can be exploited. The mechanical approach works on the principle of minimal potential energy Daniel Lordick whereas the geometrical approach searches for the minimal distances of Professor centers of mass. The vertices of the solids are given by the centers of the Technical University Dresden spheres. Faculty of Geometry Institute of Geometry If we expand this idea by a contrary force, which pushes outwards, we Germany obtain the principle of tensegrity. We can show that we can build up regular and half-regular polyhedra by the interaction of physical forces. Every platonic and Archimedean solid can be converted into a tensegrity structure. Following this, a vast variety of shapes defined by multiple solids can also be obtained. Keywords: Platonic Solids, Archimedean Solids, Tensegrity, Force Density Method, Packing of Spheres, Modularization 1. PLATONIC AND ARCHIMEDEAN SOLIDS called “kissing number” problem. The kissing number problem is asking for the maximum possible number of Platonic and Archimedean solids have systematically congruent spheres, which touch another sphere of the been described in the antiquity. They denominate all same size without overlapping. In three dimensions the convex polyhedra with regular faces and uniform vertices kissing number is 12.