Chirality & Polyhedra
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Chirality in the Plane
Chirality in the plane Christian G. B¨ohmer1 and Yongjo Lee2 and Patrizio Neff3 October 15, 2019 Abstract It is well-known that many three-dimensional chiral material models become non-chiral when reduced to two dimensions. Chiral properties of the two-dimensional model can then be restored by adding appropriate two-dimensional chiral terms. In this paper we show how to construct a three-dimensional chiral energy function which can achieve two-dimensional chirality induced already by a chiral three-dimensional model. The key ingredient to this approach is the consideration of a nonlinear chiral energy containing only rotational parts. After formulating an appropriate energy functional, we study the equations of motion and find explicit soliton solutions displaying two-dimensional chiral properties. Keywords: chiral materials, planar models, Cosserat continuum, isotropy, hemitropy, centro- symmetry AMS 2010 subject classification: 74J35, 74A35, 74J30, 74A30 1 Introduction 1.1 Background A group of geometric symmetries which keeps at least one point fixed is called a point group. A point group in d-dimensional Euclidean space is a subgroup of the orthogonal group O(d). Naturally this leads to the distinction of rotations and improper rotations. Centrosymmetry corresponds to a point group which contains an inversion centre as one of its symmetry elements. Chiral symmetry is one example of non-centrosymmetry which is characterised by the fact that a geometric figure cannot be mapped into its mirror image by an element of the Euclidean group, proper rotations SO(d) and translations. This non-superimposability (or chirality) to its mirror image is best illustrated by the left and right hands: there is no way to map the left hand onto the right by simply rotating the left hand in the plane. -
A New Calculable Orbital-Model of the Atomic Nucleus Based on a Platonic-Solid Framework
A new calculable Orbital-Model of the Atomic Nucleus based on a Platonic-Solid framework and based on constant Phi by Dipl. Ing. (FH) Harry Harry K. K.Hahn Hahn / Germany ------------------------------ 12. December 2020 Abstract : Crystallography indicates that the structure of the atomic nucleus must follow a crystal -like order. Quasicrystals and Atomic Clusters with a precise Icosahedral - and Dodecahedral structure indi cate that the five Platonic Solids are the theoretical framework behind the design of the atomic nucleus. With my study I advance the hypothesis that the reference for the shell -structure of the atomic nucleus are the Platonic Solids. In my new model of the atomic nucleus I consider the central space diagonals of the Platonic Solids as the long axes of Proton - or Neutron Orbitals, which are similar to electron orbitals. Ten such Proton- or Neutron Orbitals form a complete dodecahedral orbital-stru cture (shell), which is the shell -type with the maximum number of protons or neutrons. An atomic nucleus therefore mainly consists of dodecahedral shaped shells. But stable Icosahedral- and Hexagonal-(cubic) shells also appear in certain elements. Consta nt PhI which directly appears in the geometry of the Dodecahedron and Icosahedron seems to be the fundamental constant that defines the structure of the atomic nucleus and the structure of the wave systems (orbitals) which form the atomic nucelus. Albert Einstein wrote in a letter that the true constants of an Universal Theory must be mathematical constants like Pi (π) or e. My mathematical discovery described in chapter 5 shows that all irrational square roots of the natural numbers and even constant Pi (π) can be expressed with algebraic terms that only contain constant Phi (ϕ) and 1 Therefore it is logical to assume that constant Phi, which also defines the structure of the Platonic Solids must be the fundamental constant that defines the structure of the atomic nucleus. -
The Roundest Polyhedra with Symmetry Constraints
Article The Roundest Polyhedra with Symmetry Constraints András Lengyel *,†, Zsolt Gáspár † and Tibor Tarnai † Department of Structural Mechanics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary; [email protected] (Z.G.); [email protected] (T.T.) * Correspondence: [email protected]; Tel.: +36-1-463-4044 † These authors contributed equally to this work. Academic Editor: Egon Schulte Received: 5 December 2016; Accepted: 8 March 2017; Published: 15 March 2017 Abstract: Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the sphere, i.e., the roundest polyhedra. A new numerical optimization method developed previously by the authors has been applied to optimize polyhedra to best approximate a sphere if tetrahedral, octahedral, or icosahedral symmetry constraints are applied. In addition to evidence provided for various cases of face numbers, potentially optimal polyhedra are also shown for n up to 132. Keywords: polyhedra; isoperimetric problem; point group symmetry 1. Introduction The so-called isoperimetric problem in mathematics is concerned with the determination of the shape of spatial (or planar) objects which have the largest possible volume (or area) enclosed with given surface area (or circumference). The isoperimetric problem for polyhedra can be reflected in a question as follows: What polyhedron maximizes the volume if the surface area and the number of faces n are given? The problem can be quantified by the so-called Steinitz number [1] S = A3/V2, a dimensionless quantity in terms of the surface area A and volume V of the polyhedron, such that solutions of the isoperimetric problem minimize S. -
Of Great Rhombicuboctahedron/Archimedean Solid
Mathematical Analysis of Great Rhombicuboctahedron/Archimedean Solid Mr Harish Chandra Rajpoot M.M.M. University of Technology, Gorakhpur-273010 (UP), India March, 2015 Introduction: A great rhombicuboctahedron is an Archimedean solid which has 12 congruent square faces, 8 congruent regular hexagonal faces & 6 congruent regular octagonal faces each having equal edge length. It has 72 edges & 48 vertices lying on a spherical surface with a certain radius. It is created/generated by expanding a truncated cube having 8 equilateral triangular faces & 6 regular octagonal faces. Thus by the expansion, each of 12 originally truncated edges changes into a square face, each of 8 triangular faces of the original solid changes into a regular hexagonal face & 6 regular octagonal faces of original solid remain unchanged i.e. octagonal faces are shifted radially. Thus a solid with 12 squares, 8 hexagonal & 6 octagonal faces, is obtained which is called great rhombicuboctahedron which is an Archimedean solid. (See figure 1), thus we have Figure 1: A great rhombicuboctahedron having 12 ( ) ( ) congruent square faces, 8 congruent regular ( ) ( ) hexagonal faces & 6 congruent regular octagonal faces each of equal edge length 풂 ( ) ( ) We would apply HCR’s Theory of Polygon to derive a mathematical relationship between radius of the spherical surface passing through all 48 vertices & the edge length of a great rhombicuboctahedron for calculating its important parameters such as normal distance of each face, surface area, volume etc. Derivation of outer (circumscribed) radius ( ) of great rhombicuboctahedron: Let be the radius of the spherical surface passing through all 48 vertices of a great rhombicuboctahedron with edge length & the centre O. -
A Technology to Synthesize 360-Degree Video Based on Regular Dodecahedron in Virtual Environment Systems*
A Technology to Synthesize 360-Degree Video Based on Regular Dodecahedron in Virtual Environment Systems* Petr Timokhin 1[0000-0002-0718-1436], Mikhail Mikhaylyuk2[0000-0002-7793-080X], and Klim Panteley3[0000-0001-9281-2396] Federal State Institution «Scientific Research Institute for System Analysis of the Russian Academy of Sciences», Moscow, Russia 1 [email protected], 2 [email protected], 3 [email protected] Abstract. The paper proposes a new technology of creating panoramic video with a 360-degree view based on virtual environment projection on regular do- decahedron. The key idea consists in constructing of inner dodecahedron surface (observed by the viewer) composed of virtual environment snapshots obtained by twelve identical virtual cameras. A method to calculate such cameras’ projec- tion and orientation parameters based on “golden rectangles” geometry as well as a method to calculate snapshots position around the observer ensuring synthe- sis of continuous 360-panorama are developed. The technology and the methods were implemented in software complex and tested on the task of virtual observing the Earth from space. The research findings can be applied in virtual environment systems, video simulators, scientific visualization, virtual laboratories, etc. Keywords: Visualization, Virtual Environment, Regular Dodecahedron, Video 360, Panoramic Mapping, GPU. 1 Introduction In modern human activities the importance of the visualization of virtual prototypes of real objects and phenomena is increasingly grown. In particular, it’s especially required for training specialists in space and aviation industries, medicine, nuclear industry and other areas, where the mistake cost is extremely high [1-3]. The effectiveness of work- ing with virtual prototypes is significantly increased if an observer experiences a feeling of being present in virtual environment. -
Chirality and Projective Linear Groups
DISCRETE MATHEMATICS Discrete Mathematics 131 (1994) 221-261 Chirality and projective linear groups Egon Schultea,l, Asia IviC Weissb**,’ “Depurtment qf Mathematics, Northeastern Uniwrsity, Boston, MA 02115, USA bDepartment qf Mathematics and Statistics, York University, North York, Ontario, M3JIP3, Camda Received 24 October 1991; revised 14 September 1992 Abstract In recent years the term ‘chiral’ has been used for geometric and combinatorial figures which are symmetrical by rotation but not by reflection. The correspondence of groups and polytopes is used to construct infinite series of chiral and regular polytopes whose facets or vertex-figures are chiral or regular toroidal maps. In particular, the groups PSL,(Z,) are used to construct chiral polytopes, while PSL,(Z,[I]) and PSL,(Z,[w]) are used to construct regular polytopes. 1. Introduction Abstract polytopes are combinatorial structures that generalize the classical poly- topes. We are particularly interested in those that possess a high degree of symmetry. In this section, we briefly outline some definitions and basic results from the theory of abstract polytopes. For details we refer to [9,22,25,27]. An (abstract) polytope 9 ofrank n, or an n-polytope, is a partially ordered set with a strictly monotone rank function rank( .) with range { - 1, 0, . , n}. The elements of 9 with rankj are called j-faces of 8. The maximal chains (totally ordered subsets) of .P are calledJags. We require that 6P have a smallest (- 1)-face F_ 1, a greatest n-face F,, and that each flag contains exactly n+2 faces. Furthermore, we require that .Y be strongly flag-connected and that .P have the following homogeneity property: when- ever F<G, rank(F)=j- 1 and rank(G)=j+l, then there are exactly two j-faces H with FcH-cG. -
Chiral Icosahedral Hinge Elastegrity's Geometry of Motion
Chiral Icosahedral Hinge Elastegrity’s Geometry of Motion Eleftherios Pavlides, PhD AIA Roger Williams University1, Peter Fauci, Roger Williams University [email protected] 401 662 7521 Introduction Hinge Elastegrity, Definitions and Transformations Presented at G4G12 2a. 2 pairs 2b. 3 orthogonal H H 3.b 2 hypotenuse elastic of moving isosceles triangle faces hinges link each of 2 triangles for each of 8 irregular tetrahedra frame each tetrahedra 3c. isosceles leg elastic of 6 gates 3.a 4 free isosceles legss hinging pairs of right Fig.1 (left) edges per gate (6 gates) triangles forming 6-strut s 12 springs 24-cable Fig.2 Chiral Icosahedral nodal Hinge elastegrity rigid & Fig.3 Chiral Icosahedral tensegrity moving parts Hinge elastegrity rigid & hinges & gate edges The object that gave rise to the math in this paper is “hinge elastegrities”, a class of structures that originally arose from two Bauhaus exercises assigned at the Yale School of Architecture in the 1970’s and investigated in a series of art projects. The key new object obtained in 1982 involved cutting slits into folded pieces of paper and weaving them into 8 irregular tetrahedra, each with 3 isosceles right-triangle faces outlining an equilateral face fig.2b The 8 tetrahedra are suspended with 12 pairs of moving isosceles-right-triangles, congruent to the tetrahedral face right triangles fig.2a giving rise to an icosahedral shape (not necessarily regular) fig.2. Each pair of right triangles is attached to each other with an elastic hinge, along one of its isosceles legs fig.2a that act as springs. -
A Method for Predicting Multivariate Random Loads and a Discrete Approximation of the Multidimensional Design Load Envelope
International Forum on Aeroelasticity and Structural Dynamics IFASD 2017 25-28 June 2017 Como, Italy A METHOD FOR PREDICTING MULTIVARIATE RANDOM LOADS AND A DISCRETE APPROXIMATION OF THE MULTIDIMENSIONAL DESIGN LOAD ENVELOPE Carlo Aquilini1 and Daniele Parisse1 1Airbus Defence and Space GmbH Rechliner Str., 85077 Manching, Germany [email protected] Keywords: aeroelasticity, buffeting, combined random loads, multidimensional design load envelope, IFASD-2017 Abstract: Random loads are of greatest concern during the design phase of new aircraft. They can either result from a stationary Gaussian process, such as continuous turbulence, or from a random process, such as buffet. Buffet phenomena are especially relevant for fighters flying in the transonic regime at high angles of attack or when the turbulent, separated air flow induces fluctuating pressures on structures like wing surfaces, horizontal tail planes, vertical tail planes, airbrakes, flaps or landing gear doors. In this paper a method is presented for predicting n-dimensional combined loads in the presence of massively separated flows. This method can be used both early in the design process, when only numerical data are available, and later, when test data measurements have been performed. Moreover, the two-dimensional load envelope for combined load cases and its discretization, which are well documented in the literature, are generalized to n dimensions, n ≥ 3. 1 INTRODUCTION The main objectives of this paper are: 1. To derive a method for predicting multivariate loads (but also displacements, velocities and accelerations in terms of auto- and cross-power spectra, [1, pp. 319-320]) for aircraft, or parts of them, subjected to a random excitation. -
Symmetry and Chirality in Discrete Structures
Symmetry and chirality in discrete structures Marston Conder University of Auckland, New Zealamd UW Math Day, Seattle, March 2013 Where is New Zealand? What is New Zealand famous for? • Great place to live (South Pacific) • Multi-cultural (European, Maori/Polynesian, Asian ...) • Beautiful scenery (beaches, volcanoes, lakes, mountains) • Kiwis (strange little birds), kiwi fruit (strange little fruit) • Dairy produce (milk, butter, cheese, etc.) • Film sites (Lord of the Rings, The Hobbit, Narnia, etc.) • Rugby football • Extreme sports (bungy-jumping, white-water rafting, etc.) What is symmetry? Symmetry can mean many different things, such as balance, uniform proportion, harmony, or congruence Generally, an object has symmetry if it can be transformed in way that leaves it looking the same as it did originally. Symmetry can be reflective: ... or rotational: ... or translational: ... or a combination of these types Examples of these kinds of symmetry abound in nature ... but have also been manufactured by human fascination and enterprise e.g. the Platonic solids (c. 360BC) or earlier ... the `Neolithic Scots' (c. 2000BC) ... as publicised by Atiyah and Sutcliffe (2003) ... but unfortunately a hoax! The claim that the Scots knew about these five regular solids over 1000 years before Plato was based on the above five `Scottish stones' at the Ashmolean Museum in Oxford | but one has 14 faces, and none of them is an icosahedron [See John Baez's website for the full story] Tilings at the Alhambra Palace { on its walls, floors, ceilings, and even some of the furniture { amazingly exhibit all of the 17 \wallpaper symmetries" (in two dimensions) [Rafael P´erezG´omezand Jos´eMara Montesinos, 1980s] Symmetry can induce strength and stability: .. -
Bridges Conference Proceedings Guidelines
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture Exploring the Vertices of a Triacontahedron Robert Weadon Rollings 883 Brimorton Drive, Scarborough, Ontario, M1G 2T8 E-mail: [email protected] Abstract The rhombic triacontahedron can be used as a framework for locating the vertices of the five platonic solids. The five models presented here, which are made of babinga wood and brass rods illustrate this relationship. Introduction My initial interest in the platonic solids began with the work of Cox [1]. My initial explorations of the platonic solids lead to the series of models Polyhedra through the Beauty of Wood [2]. I have continued this exploration by using a rhombic triacontahedron at the core of my models and illustrating its relationship to all five platonic solids. Rhombic Triacontahedron The rhombic triacontahedron has 30 rhombic faces where the ratio of the diagonals is the golden ratio. The models shown in Figures 1, 2, 4, and 5 illustrate that the vertices of an icosahedron, dodecahedron, hexahedron and tetrahedron all lie on the circumsphere of the rhombic triacontahedron. In addition, Figure 3 shows that the vertices of the octahedron lie on the inscribed sphere of the rhombic triacontahedron. Creating the Models Each of the five rhombic triacontahedrons are made of 1/4 inch thick babinga wood. The edges of the rhombi were cut at 18 degrees on a band saw. After the assembly of the triacontahedron 1/8 inch brass rods were inserted a specific vertices and then fitted with 1/2 inch cocabola wood spheres. Different vertices (or faces in the case of the octahedron) were chosen for each platonic solid. -
Polytopes and Nuclear Structure by Roger Ellman Abstract
Polytopes and Nuclear Structure by Roger Ellman Abstract While the parameters Z and A indicate a general structural pattern for the atomic nuclei, the exact nuclear masses in their fine differences appear to vary somewhat randomly, seem not to exhibit the orderly kind of logical system that nature must exhibit. It is shown that separation energy [the mass of the nucleus before decay less the mass of a decay’s products], not mass defect [the sum of the nuclear protons and neutrons masses less the nuclear mass], is the “touchstone” of nuclear stability. When not part of an atomic nucleus the neutron decays into a proton and an electron indicating that it can be deemed a combination of those two. Considering that, the nucleus can be analyzed as an assembly of A protons and N = A - Z electrons, where N of the protons form neutrons with the N electrons. Resulting analysis discloses a comprehensive orderly structure among the actual nuclear masses of all the nuclear types and isotopes. The analysis examines in detail the conditions for nuclear stability / instability. An interesting secondary component of that analysis and the resulting logical order is the family of geometric forms called polytopes, in particular the regular polyhedrons. Roger Ellman, The-Origin Foundation, Inc. http://www.The-Origin.org 320 Gemma Circle, Santa Rosa, CA 95404, USA [email protected] 1 Polytopes and Nuclear Structure Roger Ellman While the parameters, Z and A, of atomic nuclei indicate a general structural pattern for the nuclei, the exact nuclear masses in their various differences seem not to exhibit the orderly kind of logical system that nature must exhibit. -
On the Study of Chirality of Knots Through Polynomial Invariants
Treball final de grau GRAU DE MATEMÀTIQUES Facultat de Matemàtiques i Informàtica Universitat de Barcelona KNOT THEORY: On the study of chirality of knots through polynomial invariants Autor: Sergi Justiniano Claramunt Director: Dr. Javier José Gutiérrez Marín Realitzat a: Departament de Matemàtiques i Informàtica Barcelona, January 18, 2019 Contents Abstract ii Introduction iii 1 Mathematical bases 1 1.1 Definition of a knot . .1 1.2 Equivalence of knots . .4 1.3 Knot projections and diagrams . .6 1.4 Reidemeister moves . .8 1.5 Invariants . .9 1.6 Symmetries, properties and generation of knots . 11 1.7 Tangles and Conway notation . 12 2 Jones Polynomial 15 2.1 Introduction . 15 2.2 Rules of bracket polynomial . 16 2.3 Writhe and invariance of Jones polynomial . 18 2.4 Main theorems and applications . 22 3 HOMFLY and Kauffman polynomials on chirality detection 25 3.1 HOMFLY polynomial . 25 3.2 Kauffman polynomial . 28 3.3 Testing chirality . 31 4 Conclusions 33 Bibliography 35 i Abstract In this project we introduce the theory of knots and specialize in the compu- tation of the knot polynomials. After presenting the Jones polynomial, its two two-variable generalizations are also introduced: the Kauffman and HOMFLY polynomial. Then we study the ability of these polynomials on detecting chirality, obtaining a knot not detected chiral by the HOMFLY polynomial, but detected chiral by the Kauffman polynomial. Introduction The main idea of this project is to give a clear and short introduction to the theory of knots and in particular the utility of knot polynomials on detecting chirality of knots.