From Networking to Tensegrity Organization

Total Page:16

File Type:pdf, Size:1020Kb

From Networking to Tensegrity Organization 207. VII. BEYOND NETWORKING TO TENSEORITY ALTERNATION - Transcending duality through tensional integrity Part 1: A lesson in organization from building design 209 Part 2: From systems-versus-networks to tensegrity organization 220 - Groupware configurations of challenge and harmony; an alternative approach to "alternative organization" 229 - Implementing principles by balancing configurations of functions; a tensegrity organization approach 239 -A new global organizational order 245 - Networking alternation (Part 1) 247 208. f/e//Clli,lfic 01' ROII/UIl ('op\' of f!Ie \rd/llic Oill/)!Id/m (i/liI'cI-,l/onc). It i,l COl'act! /)y the iigrcnon. 11 kind orthid IIcf, olld \t'l{\ ,\ll,.,IIOllllfct! h\' {he t\\'o ~()ldell cog/e,\ o(/.ell,\' thllf.flcw.fi·011l the fll'O ellds (~I' fhe world to !IIeet of it\ (,(,lItrc. lit f)('//Ihi, VII. BEYOND NETWORKING TO TENSEGRITY ALTERNATION - Transcending duality through tensional integrity Part 1: A lesson in organization from building design 209 Part 2: From systems-versus-networks to tensegrity organization 220 - Groupware configurations of challenge and harmony; an alternative approach to "alternative organization" 229 - Implementing principles by balancing configurations of functions; a tensegrity organization approach 239 -A new global organizational order 245 - Networking alternation (Part 1) 247 IU tl • lIelleJli,\/ic 01' f<O/lI"1I COII\ 011/1/' :II'I'!loic 1II1I11!tlllo,\ (//II\'e!-s/o//e),lt i,\ ('ol'cred /Jy t!le 1/:~r(,11011, (/ kind Ilflhi('k //I't, 01/11 \\'1/\ \/11'1I/0/111I1 'd />\'Ihe 1\1'0 goldI'll cl/gII'\ 0(/.1'11.1' Ihl/t.fle\l'.I;'ol1lthe 1\1'0 ellds (~(r!le \I'orld 10 /lied 111 if\ Cl'lIlrc, ul f)1'!/lhi. 209. Alesson in organization from building design - Transcending dua'lity through tensional integrity: part 1 A geodesic dome, Photo: Courtesy ot Kaiser Aluminium (Temcor) Introduction « bureaucratic» view which emphasi­ The fundamental problem in bUilding zes the importance of «rational, ef­ is to support a load, whether in the It is characteristic of society today that fective» structures, and (d) relation­ case of a bridge or simply to establish any area of thought, belief or action ships between an organizational system a covered space. As in diagram 1 thiS is viewed in a dualistic manner, either . and any corresponding (or complemen­ can be done with beams on columns. positively or negatively. Much of the tary) network. The question is whether But in order to increase the floor space dynamics of society is determined by this approach offers us clues to a new and reduce the materials reqUired for interaction, competition or conflict kind of «psycho-social architecture». a given load, arches of increasing so­ between those holding such polarized This is explored in Part 2 of the article phistication have been constructed perspectives on any issue. The struc­ (see pages 258-265). from Roman times. tures currently available reflect this situation by either (a) focusing attention « The arch is essentially a device for on one pole of the duality, ignoring the The Lesson from dispensing with a center post, by split­ other, or treating it with varying degrees Building Design ting the thrusts a center post would of hostility, or (b) integrating the dual­ It would be desirable to precede this support and by deflecting them to the ity into the structure and permtting a section by a development of the fami­ sides. The downward pull of gravity change over time in response to a ma­ liar argument that there is an interplay on the keystone is converted into paired jority (such as at elections) between between the principles governing the outward thrusts, which the face angles 'acting in terms of one pole or of the physical architecture and designed of successive stages transform into other, or (c) integrating the duality into structures current in a society and those doWnward thrusts once more, but the structure such that one division fa­ governing the psycho-social organiza­ downward thrusts now borne by the vours the approach 'indicated by one tions of that society. Unfortunately side columns. Thus the columns pole of the duality and another favours space limitations preclude doing so in actually support the weight of the key­ that of the second pole. Since the dual­ any adequate manner and it is only stone and its neighbours, without having istic approach is extremely divisive, the possible to develop the following argu­ to be located directly under the stones question is whether more adequate ap­ ment in such a way as to stress that whose weight they bear. So a central proaches are not available as a basis correspondence. space is cleared beneath the arch. for new kinds of structures which could A bUilding, like any other structure It is clear that everything is held in by-pass the conflict situation built into (including psycho-social structures) has place by weight, so that the continuities the dualistic approach without denying forces acting on it, or stresses acting of stress are chiefly compressive. Two its reality. within it, which are trying to deform or more intersecting arches will define The purpose of Part 1 of this article it or cause it to move. These forces a dome-shaped space, again clear of is to draw attention to the manner in pass through the structure, pushing on supporters because the work of support which this problem has been examined some elements (hence « compressive » has been transferred to peripheral co­ in architecture and how dualities are force) and pulling on others (hence lumns ... balanced and transcended within what « tensile» force). Until the last century, Though the visible continuities are com­ are known as« tensegrity »structures(1). most of the bUilding materials availa­ pressive, there is in fact an invisible Such an approach suggests a number of ble were effective in resisting compres­ tension network which analysis cannot interesting possibilities by which to sive forces (e.g. brick, stone), but few ignore. Each component of a stone dome handle: (a) sets of dualities, such as durable materials were capable of with­ is held in place by the earth's gravita­ opposing values or viewpoints, (b) standing even moderate tensile stress tional field, pulling tensionally « down­ groups pursuing 'opposing objectives, (with the limited exception of wood). ward» through the structure. If the (c) relationships between structures ba­ Consequently buildings and bridges dome were inverted, the force that sed on the «organic, wholistic» view were designed so that large tensile stres­ pulls it together would pull it apart. If that «all men are brothers» and the ses did not occur in them. it could be placed in orbit, it would (1) There are few sources of intormation on tensegrity, originally inves­ milian, 1975 (Tensegrity, pp. 372-432). tigated by R. Buckminster Fuller. This article reviews the insights in Anthony Pugh. An Introduction to Tensegrity. Los Angeles. University the three principle sources currently available, although their authors of Calitornia Press, f976 (See also: Polyhedra; a visual approach. are not responsible for any interpretations here of the possible psycho­ Same author and publisher). social implications of these principles: R. Buckminster Fuller. Syn­ Hugh Kenner. Geodesic Malh and How 10 Use 11. Los Angeles, Uni­ ergetics~ explorations in the geometry of thinking. New York, Mac- versity ot California Press. 1976 (ParI 1 : Tensegrity. pp. 3-44). 210. « L'affaire de tous » de la degradation de ce Comite. Mais la bande, de I'interieur, en reduisant la beaucoup plus que pour les ONG ecar­ I'intention des Etats importe ici pius consultation a des taches politiquement tees qUi y trouveraient un avantage pu­ que le sort du Comite. inoffensives ou de complaissance ou blicitaire et un accroissement d'atten­ encore au simple r61e de diffuseur de tion? Car le fait associatif non-gou­ Un observateur attentif de la consulta­ recommandations des Etats. vernemental existe en soi et non par la grace du statut consultatlf. On doit tion nous disait dernierement que si la On en lira les textes ci-apres. Charte des Nations-Unies etait Et re­ savoir que les ONG, associations inde­ faire, i! !J'est pas sOr que I'art. 71 y Les reponses de la France et de I'Au­ pendantes, qui se sont vouees Et la pro­ trouverait encore place. Faute d'une re­ triche concernant I'application par les tection des droits humains n'attendent vision de la Charte qUi prevoit, en son ONG des principes regissant leur statut, pas I'autorisatlon des Etats pour se article 109, que toute modification exi­ objectent Et cette tendance contraire manifester Et travers les frontiElreS et gera la ratification de tous les membres aux buts et aux principes de la Charte. que le sixieme continent non territorial permanents du Conseil de securite, le cher a Johan Galtung n'a besoin ni Conseil economique et social sera Faut-il dire que le dommage serait pour d'agrements ni de passeports. tente, dans sa majorite, de proceder par I'avenir de I'organisation internationale Robert FENAUX Le statut consultatif et les droits de I'homme Les reponses de la France et de I'Autriche a I'enquete de I'Ecosoc sur I'application par les ONG des principes regissant leur statuto AUTRICHE FRANCE L' Autriche fait grand cas du r61e des organisations non gou­ Le Gouvernement francais attache .Ia plus grande importance vernementales dans le domaine des droits de I'homme. Le sta­ au r61e que le statut consultatif confare aux organisations non tut consultatif qui peut leur etre accorde en vertu de la resolu­ gouvernementales internationales accreditees aupres du Con­ tion 1296 (XLIV) du Conseil economique et social est donc en­ seil economique et social.
Recommended publications
  • 7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok
    7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes.
    [Show full text]
  • Snubs, Alternated Facetings, & Stott-Coxeter-Dynkin Diagrams
    Symmetry: Culture and Science Vol. 21, No.4, 329-344, 2010 SNUBS, ALTERNATED FACETINGS, & STOTT-COXETER-DYNKIN DIAGRAMS Dr. Richard Klitzing Non-professional mathematician, (b. Laupheim, BW., Germany, 1966). Address: Kantstrasse 23, 89522 Heidenheim, Germany. E-mail: [email protected] . Fields of interest: Polytopes, geometry, mathematical quasi-crystallography. Publications: (2002) Axial Symmetrical Edge-Facetings of Uniform Polyhedra. In: Symmetry: Cult. & Sci., vol. 13, nos. 3- 4, pp. 241-258. (2001) Convex Segmentochora. In: Symmetry: Cult. & Sci., vol. 11, nos. 1-4, pp. 139-181. (1996) Reskalierungssymmetrien quasiperiodischer Strukturen. [Symmetries of Rescaling for Quasiperiodic Structures, Ph.D. Dissertation, in German], Eberhard-Karls-Universität Tübingen, or Hamburg, Verlag Dr. Kovač, ISBN 978-3-86064-428- 7. – (Former ones: mentioned therein.) Abstract: The snub cube and the snub dodecahedron are well-known polyhedra since the days of Kepler. Since then, the term “snub” was applied to further cases, both in 3D and beyond, yielding an exceptional species of polytopes: those do not bow to Wythoff’s kaleidoscopical construction like most other Archimedean polytopes, some appear in enantiomorphic pairs with no own mirror symmetry, etc. However, they remained stepchildren, since they permit no real one-step access. Actually, snub polytopes are meant to be derived secondarily from Wythoffians, either from omnitruncated or from truncated polytopes. – This secondary process herein is analysed carefully and extended widely: nothing bars a progress from vertex alternation to an alternation of an arbitrary class of sub-dimensional elements, for instance edges, faces, etc. of any type. Furthermore, this extension can be coded in Stott-Coxeter-Dynkin diagrams as well.
    [Show full text]
  • Convex Polytopes and Tilings with Few Flag Orbits
    Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.
    [Show full text]
  • Symmetric Networks with Geometric Constraints As Models of Visual Illusions
    S S symmetry Article Symmetric Networks with Geometric Constraints as Models of Visual Illusions Ian Stewart 1,*,† and Martin Golubitsky 2,† 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK 2 Department of Mathematics, Ohio State University, Columbus, OH 43210, USA; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Received: 17 May 2019; Accepted: 13 June 2019; Published: 16 June 2019 Abstract: Multistable illusions occur when the visual system interprets the same image in two different ways. We model illusions using dynamic systems based on Wilson networks, which detect combinations of levels of attributes of the image. In most examples presented here, the network has symmetry, which is vital to the analysis of the dynamics. We assume that the visual system has previously learned that certain combinations are geometrically consistent or inconsistent, and model this knowledge by adding suitable excitatory and inhibitory connections between attribute levels. We first discuss 4-node networks for the Necker cube and the rabbit/duck illusion. The main results analyze a more elaborate model for the Necker cube, a 16-node Wilson network whose nodes represent alternative orientations of specific segments of the image. Symmetric Hopf bifurcation is used to show that a small list of natural local geometric consistency conditions leads to alternation between two global percepts: cubes in two different orientations. The model also predicts brief transitional states in which the percept involves impossible rectangles analogous to the Penrose triangle. A tristable illusion generalizing the Necker cube is modelled in a similar manner.
    [Show full text]
  • Mating of Polyhedrato Evince an Order Ofthe All-Space-Filling Periodic Honeycombs
    International Journal of u- and e- Service, Science and Technology Vol.9, No. 6 (2016), pp.179-192 http://dx.doi.org/10.14257/ijunesst.2016.9.6.17 Mating of Polyhedrato Evince an Order ofthe All-Space-Filling Periodic Honeycombs Robert C. Meurant Director Emeritus, Institute of Traditional Studies; Executive Director Education & Research, Harrisco-Enco [email protected] Abstract Earlier discovery and presentation of an inherent order of the regular and semi- regular polyhedra that displays three interrelated classes, together with consideration of the honeycombs, has led me to posit the existence of a coherent and integral metapattern that should relate the various all-space-filling periodical honeycombs, which I advance in an earlier paper that should be read in conjunction with thispaper, as they form part of a series.Here, I approach the periodic polyhedral honeycombs byexploring how pairs of polyhedra regularly combine or mate, whether proximally or distally, along the √1, √2 and √3 axes of their reference cubic and tetrahedral lattices. This is first performed for pairs of what I elsewhere term the Great Enablers (GEs), the positive and negative tetrahedra and truncated tetrahedra; secondly, for pairs of GEs and the Primary Polytopes (PPs); and thirdly, for pairs of PPs. This reveals that these three forms of mating, GE:GE, GE:PP and PP:PP, correlate with the three symmetry groups {2,3,3|2,3,3}, {2,3,3|2,3,4} and {2,3,4|2,3,4}, respectively, of the periodical honeycombs.These matings typically occur in naturallyoccurring pairs along each axis, so in general, a PP mates with just two PPs, though in certain cases one of these is the same as the original.
    [Show full text]
  • Local Symmetry Preserving Operations on Polyhedra
    Local Symmetry Preserving Operations on Polyhedra Pieter Goetschalckx Submitted to the Faculty of Sciences of Ghent University in fulfilment of the requirements for the degree of Doctor of Science: Mathematics. Supervisors prof. dr. dr. Kris Coolsaet dr. Nico Van Cleemput Chair prof. dr. Marnix Van Daele Examination Board prof. dr. Tomaž Pisanski prof. dr. Jan De Beule prof. dr. Tom De Medts dr. Carol T. Zamfirescu dr. Jan Goedgebeur © 2020 Pieter Goetschalckx Department of Applied Mathematics, Computer Science and Statistics Faculty of Sciences, Ghent University This work is licensed under a “CC BY 4.0” licence. https://creativecommons.org/licenses/by/4.0/deed.en In memory of John Horton Conway (1937–2020) Contents Acknowledgements 9 Dutch summary 13 Summary 17 List of publications 21 1 A brief history of operations on polyhedra 23 1 Platonic, Archimedean and Catalan solids . 23 2 Conway polyhedron notation . 31 3 The Goldberg-Coxeter construction . 32 3.1 Goldberg ....................... 32 3.2 Buckminster Fuller . 37 3.3 Caspar and Klug ................... 40 3.4 Coxeter ........................ 44 4 Other approaches ....................... 45 References ............................... 46 2 Embedded graphs, tilings and polyhedra 49 1 Combinatorial graphs .................... 49 2 Embedded graphs ....................... 51 3 Symmetry and isomorphisms . 55 4 Tilings .............................. 57 5 Polyhedra ............................ 59 6 Chamber systems ....................... 60 7 Connectivity .......................... 62 References
    [Show full text]
  • On the Relations Between Truncated Cuboctahedron, Truncated Icosidodecahedron and Metrics
    Forum Geometricorum b Volume 17 (2017) 273–285. b b FORUM GEOM ISSN 1534-1178 On The Relations Between Truncated Cuboctahedron, Truncated Icosidodecahedron and Metrics Ozcan¨ Gelis¸gen Abstract. The theory of convex sets is a vibrant and classical field of mod- ern mathematics with rich applications. The more geometric aspects of con- vex sets are developed introducing some notions, but primarily polyhedra. A polyhedra, when it is convex, is an extremely important special solid in Rn. Some examples of convex subsets of Euclidean 3-dimensional space are Pla- tonic Solids, Archimedean Solids and Archimedean Duals or Catalan Solids. There are some relations between metrics and polyhedra. For example, it has been shown that cube, octahedron, deltoidal icositetrahedron are maximum, taxi- cab, Chinese Checker’s unit sphere, respectively. In this study, we give two new metrics to be their spheres an Archimedean solids truncated cuboctahedron and truncated icosidodecahedron. 1. Introduction A polyhedron is a three-dimensional figure made up of polygons. When dis- cussing polyhedra one will use the terms faces, edges and vertices. Each polygonal part of the polyhedron is called a face. A line segment along which two faces come together is called an edge. A point where several edges and faces come together is called a vertex. That is, a polyhedron is a solid in three dimensions with flat faces, straight edges and vertices. A regular polyhedron is a polyhedron with congruent faces and identical ver- tices. There are only five regular convex polyhedra which are called Platonic solids. A convex polyhedron is said to be semiregular if its faces have a similar configu- ration of nonintersecting regular plane convex polygons of two or more different types about each vertex.
    [Show full text]
  • Hedgehog-Lattice Spin Texture in Classical Heisenberg
    Hedgehog-lattice spin texture in classical Heisenberg antiferromagnets on the breathing pyrochlore lattice Kazushi Aoyama1 and Hikaru Kawamura2 1Department of Earth and Space Science, Graduate School of Science, Osaka University, Osaka 560-0043, Japan 2Toyota Physical and Chemical Research Institute, Aichi, 480-1118, Japan (Dated: January 1, 2021) The hedgehog lattice, a three-dimensional periodic array of magnetic monopoles and anti- monopoles, is known to be realized in the presence of the Dzyaloshinskii-Moriya (DM) interaction. Here, we demonstrate by means of Monte Carlo simulations that the hedgehog lattice is induced by not the DM interaction but frustration in classical Heisenberg antiferromagnets on the breathing pyrochlore lattice. In the model, the breathing bond-alternation is characterized by the ratio of the nearest-neighbor (NN) antiferromagnetic exchange interaction for large tetrahedra to that for ′ 1 1 1 small ones, J1/J1. A quadruple-q state with the ordering vector of q = (± 2 , ± 2 , ± 2 ), which is realized for a large third NN antiferromagnetic interaction along the bond direction J3, turns out to ′ become the hedgehog-lattice state in the breathing case of J1/J1 < 1, while in the uniform case of ′ J1/J1 = 1, it is a collinear state favored by thermal fluctuations. It is also found that in a magnetic 1 1 1 field, the structure of the ( 2 , 2 , 2 ) hedgehog lattice is changed from cubic to tetragonal, resulting in a nonzero net spin chirality which in a metalilc system, should yield a characteristic topological Hall effect. I. INTRODUCTION of the hedgehogs (monopoles) and anti-hedgehogs (anti- monopoles) each having positive or negative nonzero Topological spin textures such as the magnetic scalar spin chirality.
    [Show full text]
  • Global Insight from Crown Chakra Dynamics in 3D? Strategic Viability Through Interrelating 1,000 Perspectives in Virtual Reality - /
    Alternative view of segmented documents via Kairos 8 June 2020 | Draft Global Insight from Crown Chakra Dynamics in 3D? Strategic viability through interrelating 1,000 perspectives in virtual reality - / - Introduction Simpler crown chakra models Design options for a crown chakra with more complex dynamics Framing crown chakra dynamics in relation to symmetrical polyhedra Rendering crown chakra dynamics through interlocking tori Perspectives, projections and learning processes in 3D modelling Cognitive embodiment in the crown chakra? From helicopter to "psychopter": the role of anti-torque? Crown chakra understood as an axial turbofan -- an "attention breathing" jet engine? Balancing duality: objectivity vs subjectivity? Complementarity of pattern languages -- an all-encompassing meta-pattern? Dependence of viable global governance on pattern management? Meta-pattern, Meta-poesis and Music References Introduction The argument for imaginative exploration in 3D of the so-called "1,000-petalled lotus" -- the Crown Chakra or Sahasrara -- was developed in an earlier exercise (Satellite Constellation and Crown Chakra as Complementary Global Metaphors? Experimental representation of crown chakra in virtual reality, 2020). Various models were presented as animations in 3D, illustrated by video and accessible interactively according to the X3D protocol. Subsequent experiment has resulted in the direct presentation of a series of such 3D models in web pages, as indicated separately (Eliciting Insight from Mandala-style Logos in 3D: interactive engagement with mandalas and yantras in virtual reality, 2020). Distinct from the earlier approach, the models were directly interactive rather than requiring plugins or separate operations. The focus there on logos was inspired by the tendency of many organizations, and notably international institutions, to frame their identity through symbolic, mandala-like, imagery in 2D.
    [Show full text]
  • KNOTS KNOTES Contents 1. Motivation, Basic Definitions And
    KNOTS KNOTES JUSTIN ROBERTS Contents 1. Motivation, basic definitions and questions 3 1.1. Basic definitions 3 1.2. Basic questions 4 1.3. Operations on knots 6 1.4. Alternating knots 7 1.5. Unknotting number 8 1.6. Further examples of knots and links 9 1.7. Methods 11 1.8. A table of the simplest knots and links 12 2. Formal definitions and Reidemeister moves 14 2.1. Knots and equivalence 14 2.2. Projections and diagrams 17 2.3. Reidemeister moves 18 2.4. Is there an algorithm for classifying and tabulating knots? 22 3. Simple invariants 25 3.1. Linking number 25 3.2. Invariants 27 3.3. 3-colourings 29 3.4. p-colourings 36 3.5. Optional: colourings and the Alexander polynomial. (Unfinished section!) 39 3.6. Some additional problems 43 4. The Jones polynomial 46 4.1. The Kauffman bracket 47 4.2. Correcting via the writhe 52 4.3. A state-sum model for the Kauffman bracket 53 4.4. The Jones polynomial and its skein relation 55 4.5. Some properties of the Jones polynomial 60 4.6. A characterisation of the Jones polynomial 61 4.7. How powerful is the Jones polynomial? 64 4.8. Alternating knots and the Jones polynomial 66 4.9. Other knot polynomials 71 5. A glimpse of manifolds 74 5.1. Metric spaces 74 5.2. Topological spaces 76 5.3. Hausdorff and second countable 78 5.4. Reconsidering the definition of n-manifold 79 6. Classification of surfaces 81 6.1. Combinatorial models for surfaces 81 Date: January 27, 2015.
    [Show full text]
  • Symmetry Groups of Regular Polytopes in Three and Four Dimensions
    Delft University of Technology Faculty of Electrical Engineering, Mathematics & Computer Science Delft Institute of Applied Mathematics Symmetry groups of regular polytopes in three and four dimensions. The Platonic Solids, Binary Groups and Regular Polytopes in four-dimensional space. Thesis submitted to the Delft Intstitute of Applied Mathematics as partial fulfillment of the requirements for the degree of BACHELOR OF SCIENCE in APPLIED MATHEMATICS by G.M.C. van Ittersum Supervisors Dr. J.G. Spandaw Dr. P.M. Visser Commitee members Dr. K.P. Hart Delft, The Netherlands August 29, 2020 Copyright © 2020 by G.M.C. van Ittersum. All rights reserved. Abstract A pentagon is an example of a highly symmetric polygon in two-dimensional space. The three-and four-dimensional analogue of these polygons are the regular polyhedra and the regular polytopes. There exist five regular polyhedra in three- dimensional space and these are called the Platonic solids. These five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron and the icosahedron. In four-dimensional space, the regular polytopes are the 5-cell, the 8-cell, also called the tesseract, the 16-cell, the 24-cell, the 120-cell and the 600-cell. The aim of this thesis is to give an introduction to some symmetry groups of the regular polytopes in three and four dimensional space at undergraduate math- ematical level. The main focus of this thesis is to describe the symmetry group of the icosahedron, to introduce the Icosians, which are related to the rotation group of the icosahedron, and to study the action of the symmetries of the 600-cell on the twenty-five 24-cells it circumscribes.
    [Show full text]
  • Instructions for Formatting Papers for the 12Th
    PERIODIC CORPUSCLE STRUCTURES AND THE SPACES IN BETWEEN Eva WOHLLEBEN1, Felix HEDIGER2, Wolfram LIEBERMEISTER3 1Muthesius-Kunsthochschule Kiel, Germany 2Artist – www.kuenstlermensch.de 3Charité - Universitätsmedizin Berlin, Germany ABSTRACT: Corpuscles are flexible geometric elements formed by regular triangles. Corpuscle elements can be joined, thus serving as building blocks for polyhedral chains and complex three- dimensional structures. Here we describe a periodic three-dimensional corpuscle grid whose nega- tive space consists of periodically arranged octahedra. As a recurrent local structure, the grid con- tains the octahedron-centric corpuscle ball, an arrangement of 18 corpuscle elements surrounding an empty, octahedron-shaped space. Similarly, we construct corpuscle balls with empty cubes or icosa- hedra at their centres and the according rotation symmetries. Both corpuscle balls can be extended to form periodic grids, but only if slight deformations of the edge lengths are allowed. The corpuscle grids and balls contain rigid, ring-like substructures, which stabilise them against deformation. Keywords: Corpuscle, flexible polyhedron, tiling 1. INTRODUCTION Corpuscles are flexible structure that extends periodically throughout geometric elements that can serve as building the entire three-dimensional space. Here we blocks for polyhedral clusters, chains, and describe this structure in terms of two archi- periodic three-dimensional grids [1]. Each tectures: one architecture, the “positive” cor- corpuscle consists of several triangles puscle grid itself, is formed by the intercon- connected by flexible edges. Between these nected corpuscles. The second architecture is triangles, there may be open slits, called mouths, which permit to link several cor- formed by the empty “negative” space in puscle elements. Some of the resulting larger between.
    [Show full text]