DOCSLIB.ORG

Exhausting All Exact Solutions of BPS Domain Wall Networks in Arbitrary Dimensions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Exhausting All Exact Solutions of BPS Domain Wall Networks in Arbitrary Dimensions PHYSICAL REVIEW D 101, 105020 (2020) Exhausting all exact solutions of BPS domain wall networks in arbitrary dimensions † ‡ Minoru Eto ,1,2,* Masaki Kawaguchi ,1, Muneto Nitta,2,3, and Ryotaro Sasaki1 1Department of Physics, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata, Yamagata 990-8560, Japan 2Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan 3Department of Physics, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan (Received 15 April 2020; accepted 15 May 2020; published 28 May 2020) We obtain full moduli parameters for generic nonplanar Bogomol’nyi-Prasad-Sommerfield networks of domain walls in an extended Abelian-Higgs model with N complex scalar fields and exhaust all exact solutions in the corresponding CPN−1 model. We develop a convenient description by grid diagrams which are polytopes determined by mass parameters of the model. To illustrate the validity of our method, we work out nonplanar domain wall networks for lower N in 3 þ 1 dimensions. In general, the networks can have compact vacuum bubbles, which are finite vacuum regions surrounded by domain walls, when the polytopes of the grid diagrams have inner vertices, and the size of bubbles can be controlled by moduli parameters. We also construct domain wall networks with bubbles in the shapes of the Platonic, Archimedean, Catalan, and Kepler-Poinsot solids. DOI: 10.1103/PhysRevD.101.105020 I. INTRODUCTION equations called BPS equations. In such cases, one can often embed the theories to supersymmetric (SUSY) It sometimes happens that systems have multiple discrete vacua or ground states, which is inevitable when a discrete theories by appropriately adding fermion superpartners, symmetry is spontaneously broken. In such a case, there in which BPS solitons preserve some fractions of SUSY. appear domain walls (or kinks) in general [1–3] which are Their topological charges are central (or tensorial) charges inevitably created during second order phase transitions of corresponding SUSY algebras. The BPS domain walls 3 1 [4–7]. They are the simplest topological solitons appearing in þ dimensions were studied extensively in field N 1 – N 2 in various condensed matter systems such as magnets [8], theories with both ¼ SUSY [20 35] and ¼ graphenes [9], carbon nanotubes, superconductors [10], SUSY [36–56]; see Refs. [57–60] as a review. They 1 atomic Bose-Einstein condensates [11], and helium super- preserve a half of SUSY and thereby are called 2 BPS fluids [6,7,12], as well as high density nuclear matter states, accompanied by SUSY central (tensorial) charges m [13,14], quark matter [15], and early Universe [4,16]. Zm ðm ¼ 1, 2; labeling spatial coordinates x perpen- In cosmology, cosmological domain wall networks are dicular to the domain wall) as domain wall topological suggested as a candidate of dark matter and/or dark charges [22,24,61]. In general, if several domain walls meet energy [17]. along a line, it forms a planar domain wall junction. In As the cases of other topological solitons, domain walls SUSY models, the planar domain wall junctions preserve a can become Bogomol’nyi-Prasad-Sommerfield (BPS) 1 quarter SUSY [62–64], therefore are called 4 BPS states, states [18,19], attaining the minimum energy for a fixed accompanied by a junction topological charge Y in addition boundary condition and satisfy first order differential 1 to Zm (m ¼ 1, 2). The 4 BPS domain wall junctions have been studied in theories with N ¼ 1 SUSY [32,65–73] and *[email protected] † N ¼ 2 SUSY [74–83]. In the N ¼ 2 SUSY gauge models, [email protected] ‡ not only planar domain wall junctions, but also planar [email protected] 1 domain wall networks as 4 BPS states were constructed Published by the American Physical Society under the terms of [75,76]. The low-energy effective action for normalizable the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to modes within the networks was obtained [78] and applied the author(s) and the published article’s title, journal citation, to study of low-energy dynamics [79]. Non-BPS planer and DOI. Funded by SCOAP3. domain wall networks were also studied in Refs. [84,85]. 2470-0010=2020=101(10)=105020(22) 105020-1 Published by the American Physical Society ETO, KAWAGUCHI, NITTA, and SASAKI PHYS. REV. D 101, 105020 (2020) Recently, the present authors proposed a model, a D þ 1- II. THE MODEL 1 dimensional Uð Þ gauge theory [86] admitting novel We study a Uð1Þ gauge theory with N charged complex analytic solutions of the BPS single nonplanar domain scalar fields HA (A ¼ 1; 2; …;N) and N0 real scalar fields wall junctions. This model cannot be made supersymmetric 0 ΣA (A0 ¼ 1; 2; …;N0)inD þ 1-dimensional spacetime. but still admits stable BPS states so that we can use the The Lagrangian is given by well-known Bogomol’nyi completion technique to derive BPS equations. The model consists of N charged complex 1 1 XN0 0 1 μν A0 μ A0 scalar fields and N neutral scalar fields coupled to the Uð Þ L ¼ − FμνF þ ∂μΣ ∂ Σ 4e2 2e2 gauge field. In Ref. [86], we restricted ourself to the special A0¼1 numbers N − 1 ¼ N0 ¼ D and imposed the invariance μ † − S 1 þ DμHðD HÞ V ð2:1Þ under the symmetric group Dþ1 of the rank D þ , which are the symmetry groups of the regular D simplex. 1 XN0 In this paper, we investigate generic nonplanar networks V ¼ Y2 þ ðΣA0 H − HMA0 ÞðΣA0 H − HMA0 Þ†; 2e2 of BPS domain walls in D þ 1 dimensions. We consider the A0¼1 ≥ 1 generic case of N D þ imposing no discrete symmetry ð2:2Þ and exhaust all exact solutions with full moduli of generic BPS nonplanar networks of domain walls in the infinite where H is an N component row vector made of HA, Uð1Þ gauge coupling limit in which the model reduces to N−1 the CP model. These are the first exact solutions of H ¼ðH1;H2; …;HNÞ; ð2:3Þ nonplanar domain wall networks in D dimensions (D ≥ 3). We first derive the BPS equations for generic Abelian Y is a scalar quantity defined by gauge theories in D þ 1 dimensions. Then, we partially solve them by the moduli matrix formalism [58] and find all Y ¼ e2ðv2 − HH†Þ; ð2:4Þ moduli parameters of the generic domain wall network solutions. We then demonstrate several concrete nonplanar and MA0 (A0 ¼ 1; …;N0) are N by N real diagonal mass −1 networks in the CPN model in D ¼ 3 for N ¼ 4,5,6.In matrices defined by the case of N ¼ 4, the solution has only one junction at A0 which four vacua meet. Network structures appear for M ¼ diagðmA0;1;mA0;2; …;mA0;NÞ: ð2:5Þ N>4. In the case of N ¼ 5, we show two different types of networks exist in general. The first type has a vacuum The spacetime index μ runs from 0 to D, and Fμν is an Uð1Þ bubble (a compact vacuum domain) surrounded by semi- gauge field strength. The coupling constants in the infinite vacuum domains. Instead, the second type does not Lagrangian in Eq. (2.1) are taken to be the so-called have any bubbles but all the vacuum domains are semi- Bogomol’nyi limit. For later use, let us define mA by an infinitely extended. In the N ¼ 6 case, there are three N0 vector whose components are the Ath diagonal elements different types according to the number of the vacuum of MA0 s, namely, bubbles, two, one, or zero. Finally, we find a connection to the well-known polyhedra known from ancient times. mA ¼ðm1;A;m2;A; …;mN0;AÞ: ð2:6Þ Indeed, we find the vacuum bubbles which are congruent with five Platonic solids. In addition, the Archimedean and In the following, we will mostly consider generic masses: the Catalan solids appear as the vacuum bubbles. We also construct the Kepler-Poinsot star solids as domain wall mA ≠ mB; if A ≠ B: ð2:7Þ networks. This paper is organized as follows. In Sec. II,we Since the scalar potential V is positive semidefinite, a introduce our model. In Sec. III, we derive the BPS classical vacuum of the theory is determined by V ¼ 0: equations and clarify the moduli space of the BPS HH† ¼ v2, and ΣA0 H − HMA0 ¼ 0. In the generic case of solutions. In Sec. IV, we first consider the infinite Uð1Þ Eq. (2.7), there are N discrete vacua. The Ath vacua which gauge coupling limit in which the model reduces to the we will denote by hAi is given by HB ¼ vδB, and −1 A CPN B0 massive nonlinear sigma model. We then exhaust Σ ¼ mB0;A. Simply, the vacua can be identified to the all exact solutions with full moduli parameters for the BPS discrete points determined by the mass vectors fmAg in the equations. We further give several examples of nonplanar Σ space, networks in D ¼ 3. In Sec. V, we study relations between 3 the domain wall networks in D ¼ and the classic solids, hAi∶Σ ¼ mA: ð2:8Þ like the Platonic, Archimedean, Catalan, and Kepler- Poinsot solids. Finally, we summarize our results and give The Lagrangian (2.1) is motivated by supersymmetry. In 0 2 N 2 a discussion in Sec. VI. fact, when NF ¼ , it is a bosonic part of an ¼ 105020-2 EXHAUSTING ALL EXACT SOLUTIONS OF BPS DOMAIN … PHYS. REV.
Load more

Recommended publications
  • Mazes for Superintelligent Ginzburg Projection
    Izidor Hafner Mazes for Superintelligent Ginzburg projection Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 2 Introduction Let as take an example. We are given a uniform polyhedron. 10 4 small rhombidodecahedron 910, 4, ÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ = 9 3 In Mathematica the polyhedron is given by a list of faces and with a list of koordinates of vertices [Roman E. Maeder, The Mathematica Programmer II, Academic Press1996]. The list of faces consists of a list of lists, where a face is represented by a list of vertices, which is given by a matrix. Let us show the first five faces: 81, 2, 6, 15, 26, 36, 31, 19, 10, 3< 81, 3, 9, 4< 81, 4, 11, 22, 29, 32, 30, 19, 13, 5< 81, 5, 8, 2< 82, 8, 17, 26, 38, 39, 37, 28, 16, 7< The nest two figures represent faces and vertices. The polyhedron is projected onto supescribed sphere and the sphere is projected by a cartographic projection. Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 3 1 4 11 21 17 7 2 8 6 5 20 19 30 16 15 3 14 10 12 18 26 34 39 37 29 23 9 24 13 28 27 41 36 38 22 25 31 33 40 42 35 32 5 8 17 24 13 3 1 2 6 15 26 36 31 19 10 9 4 7 14 27 38 47 43 30 20 12 25 49 42 18 11 16 39 54 32 21 23 2837 50 57 53 44 29 22 3433 35 4048 56 60 59 52 41 45 51 58 55 46 The problem is to find the path from the black dot to gray dot, where thick lines represent walls of a maze.
    [Show full text]
  • Arxiv:Math/9906062V1 [Math.MG] 10 Jun 1999 Udo Udmna Eerh(Rn 96-01-00166)
    Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices ∗ Michel DEZA CNRS and Ecole Normale Sup´erieure, Paris, France Mikhail SHTOGRIN Steklov Mathematical Institute, 117966 Moscow GSP-1, Russia Abstract We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter’s regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m ≥ 7, star-honeycombs m m {m, 2 } are embeddable while { 2 ,m} are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing all nine regular polyhedra on the sphere. In section 4, non-embeddability of all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved. In the last section 5, all cases of embedding for dimension d> 2 are identified. Besides hyper-simplices and hyper-octahedra, they are exactly those with bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3-, 4-, 5-space (only two, {4, 3, 5} and {4, 3, 3, 5}, of those 11 have compact both, facets and vertex figures). 1 Introduction arXiv:math/9906062v1 [math.MG] 10 Jun 1999 We say that given tiling (or honeycomb) T has a l1-graph and embeds up to scale λ into m-cube Hm (or, if the graph is infinite, into cubic lattice Zm ), if there exists a mapping f of the vertex-set of the skeleton graph of T into the vertex-set of Hm (or Zm) such that λdT (vi, vj)= ||f(vi), f(vj)||l1 = X |fk(vi) − fk(vj)| for all vertices vi, vj, 1≤k≤m ∗This work was supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach) and Russian fund of fundamental research (grant 96-01-00166).
    [Show full text]
  • Cons=Ucticn (Process)
    DOCUMENT RESUME ED 038 271 SE 007 847 AUTHOR Wenninger, Magnus J. TITLE Polyhedron Models for the Classroom. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB DATE 68 NOTE 47p. AVAILABLE FROM National Council of Teachers of Mathematics,1201 16th St., N.V., Washington, D.C. 20036 ED RS PRICE EDRS Pr:ce NF -$0.25 HC Not Available from EDRS. DESCRIPTORS *Cons=ucticn (Process), *Geometric Concepts, *Geometry, *Instructional MateriAls,Mathematical Enrichment, Mathematical Models, Mathematics Materials IDENTIFIERS National Council of Teachers of Mathematics ABSTRACT This booklet explains the historical backgroundand construction techniques for various sets of uniformpolyhedra. The author indicates that the practical sianificanceof the constructions arises in illustrations for the ideas of symmetry,reflection, rotation, translation, group theory and topology.Details for constructing hollow paper models are provided for thefive Platonic solids, miscellaneous irregular polyhedra and somecompounds arising from the stellation process. (RS) PR WON WITHMICROFICHE AND PUBLISHER'SPRICES. MICROFICHEREPRODUCTION f ONLY. '..0.`rag let-7j... ow/A U.S. MOM Of NUM. INCIII01 a WWII WIC Of MAW us num us us ammo taco as mums NON at Ot Widel/A11011 01116111111 IT.P01115 OF VOW 01 OPENS SIAS SO 101 IIKISAMIT IMRE Offlaat WC Of MANN POMO OS POW. OD PROCESS WITH MICROFICHE AND PUBLISHER'S PRICES. reit WeROFICHE REPRODUCTION Pvim ONLY. (%1 00 O O POLYHEDRON MODELS for the Classroom MAGNUS J. WENNINGER St. Augustine's College Nassau, Bahama. rn ErNATIONAL COUNCIL. OF Ka TEACHERS OF MATHEMATICS 1201 Sixteenth Street, N.W., Washington, D. C. 20036 ivetmIssromrrIPRODUCE TmscortnIGMED Al"..Mt IAL BY MICROFICHE ONLY HAS IEEE rano By Mat __Comic _ TeachMar 10 ERIC MID ORGANIZATIONS OPERATING UNSER AGREEMENTS WHIM U.
    [Show full text]
  • Isotoxal Star-Shaped Polygonal Voids and Rigid Inclusions in Nonuniform Antiplane Shear fields
    Published in International Journal of Solids and Structures 85-86 (2016), 67-75 doi: http://dx.doi.org/10.1016/j.ijsolstr.2016.01.027 Isotoxal star-shaped polygonal voids and rigid inclusions in nonuniform antiplane shear fields. Part I: Formulation and full-field solution F. Dal Corso, S. Shahzad and D. Bigoni DICAM, University of Trento, via Mesiano 77, I-38123 Trento, Italy Abstract An infinite class of nonuniform antiplane shear fields is considered for a linear elastic isotropic space and (non-intersecting) isotoxal star-shaped polygonal voids and rigid inclusions per- turbing these fields are solved. Through the use of the complex potential technique together with the generalized binomial and the multinomial theorems, full-field closed-form solutions are obtained in the conformal plane. The particular (and important) cases of star-shaped cracks and rigid-line inclusions (stiffeners) are also derived. Except for special cases (ad- dressed in Part II), the obtained solutions show singularities at the inclusion corners and at the crack and stiffener ends, where the stress blows-up to infinity, and is therefore detri- mental to strength. It is for this reason that the closed-form determination of the stress field near a sharp inclusion or void is crucial for the design of ultra-resistant composites. Keywords: v-notch, star-shaped crack, stress singularity, stress annihilation, invisibility, conformal mapping, complex variable method. 1 Introduction The investigation of the perturbation induced by an inclusion (a void, or a crack, or a stiff insert) in an ambient stress field loading a linear elastic infinite space is a fundamental problem in solid mechanics, whose importance need not be emphasized.
    [Show full text]
  • Parity Check Schedules for Hyperbolic Surface Codes
    Parity Check Schedules for Hyperbolic Surface Codes Jonathan Conrad September 18, 2017 Bachelor thesis under supervision of Prof. Dr. Barbara M. Terhal The present work was submitted to the Institute for Quantum Information Faculty of Mathematics, Computer Science und Natural Sciences - Faculty 1 First reviewer Second reviewer Prof. Dr. B. M. Terhal Prof. Dr. F. Hassler Acknowledgements Foremost, I would like to express my gratitude to Prof. Dr. Barbara M. Terhal for giving me the opportunity to complete this thesis under her guidance, insightful discussions and her helpful advice. I would like to give my sincere thanks to Dr. Nikolas P. Breuckmann for his patience with all my questions and his good advice. Further, I would also like to thank Dr. Kasper Duivenvoorden and Christophe Vuillot for many interesting and helpful discussions. In general, I would like to thank the IQI for its hospitality and everyone in it for making my stay a very joyful experience. As always, I am grateful for the constant support of my family and friends. Table of Contents 1 Introduction 1 2 Preliminaries 2 3 Stabilizer Codes 5 3.1 Errors . 6 3.2 TheBitFlipCode ............................... 6 3.3 The Toric Code . 11 4 Homological Code Construction 15 4.1 Z2-Homology .................................. 18 5 Hyperbolic Surface Codes 21 5.1 The Gauss-Bonnet Theorem . 21 5.2 The Small Stellated Dodecahedron as a Hyperbolic Surface Code . 26 6 Parity Check Scheduling 32 6.1 Seperate Scheduling . 33 6.2 Interleaved Scheduling . 39 6.3 Coloring the Small Stellated Dodecahedron . 45 7 Fault Tolerant Quantum Computation 47 7.1 Fault Tolerance .
    [Show full text]
  • Bridges Conference Proceedings Guidelines Word
    Bridges 2019 Conference Proceedings Helixation Rinus Roelofs Lansinkweg 28, Hengelo, the Netherlands; [email protected] Abstract In the list of names of the regular and semi-regular polyhedra we find many names that refer to a process. Examples are ‘truncated’ cube and ‘stellated’ dodecahedron. And Luca Pacioli named some of his polyhedral objects ‘elevations’. This kind of name-giving makes it easier to understand these polyhedra. We can imagine the model as a result of a transformation. And nowadays we are able to visualize this process by using the technique of animation. Here I will to introduce ‘helixation’ as a process to get a better understanding of the Poinsot polyhedra. In this paper I will limit myself to uniform polyhedra. Introduction Most of the Archimedean solids can be derived by cutting away parts of a Platonic solid. This operation , with which we can generate most of the semi-regular solids, is called truncation. Figure 1: Truncation of the cube. We can truncate the vertices of a polyhedron until the original faces of the polyhedron become regular again. In Figure 1 this process is shown starting with a cube. In the third object in the row, the vertices are truncated in such a way that the square faces of the cube are transformed to regular octagons. The resulting object is the Archimedean solid, the truncated cube. We can continue the process until we reach the fifth object in the row, which again is an Archimedean solid, the cuboctahedron. The whole process or can be represented as an animation. Also elevation and stellation can be described as processes.
    [Show full text]
  • Are Your Polyhedra the Same As My Polyhedra?
    Are Your Polyhedra the Same as My Polyhedra? Branko Gr¨unbaum 1 Introduction “Polyhedron” means different things to different people. There is very little in common between the meaning of the word in topology and in geometry. But even if we confine attention to geometry of the 3-dimensional Euclidean space – as we shall do from now on – “polyhedron” can mean either a solid (as in “Platonic solids”, convex polyhedron, and other contexts), or a surface (such as the polyhedral models constructed from cardboard using “nets”, which were introduced by Albrecht D¨urer [17] in 1525, or, in a more mod- ern version, by Aleksandrov [1]), or the 1-dimensional complex consisting of points (“vertices”) and line-segments (“edges”) organized in a suitable way into polygons (“faces”) subject to certain restrictions (“skeletal polyhedra”, diagrams of which have been presented first by Luca Pacioli [44] in 1498 and attributed to Leonardo da Vinci). The last alternative is the least usual one – but it is close to what seems to be the most useful approach to the theory of general polyhedra. Indeed, it does not restrict faces to be planar, and it makes possible to retrieve the other characterizations in circumstances in which they reasonably apply: If the faces of a “surface” polyhedron are sim- ple polygons, in most cases the polyhedron is unambiguously determined by the boundary circuits of the faces. And if the polyhedron itself is without selfintersections, then the “solid” can be found from the faces. These reasons, as well as some others, seem to warrant the choice of our approach.
    [Show full text]
  • [ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated Into This Format from Data Given In
    ENTRY POLYHEDRA [ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated into this format from data given in http://netlib.bell-labs.com/netlib tetrahedron The [tetrahedron] is a polyhedron with 4 vertices and 4 faces. The dual polyhedron is called tetrahedron. cube The [cube] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. hexahedron The [hexahedron] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. octahedron The [octahedron] is a polyhedron with 6 vertices and 8 faces. The dual polyhedron is called cube. dodecahedron The [dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called icosahedron. icosahedron The [icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called dodecahedron. small stellated dodecahedron The [small stellated dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called great dodecahedron. great dodecahedron The [great dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called small stellated dodecahedron. great stellated dodecahedron The [great stellated dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called great icosahedron. great icosahedron The [great icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called great stellated dodecahedron. truncated tetrahedron The [truncated tetrahedron] is a polyhedron with 12 vertices and 8 faces. The dual polyhedron is called triakis tetrahedron. cuboctahedron The [cuboctahedron] is a polyhedron with 12 vertices and 14 faces. The dual polyhedron is called rhombic dodecahedron.
    [Show full text]
  • Spectral Realizations of Graphs
    SPECTRAL REALIZATIONS OF GRAPHS B. D. S. \DON" MCCONNELL 1. Introduction 2 The boundary of the regular hexagon in R and the vertex-and-edge skeleton of the regular tetrahe- 3 dron in R , as geometric realizations of the combinatorial 6-cycle and complete graph on 4 vertices, exhibit a significant property: Each automorphism of the graph induces a \rigid" isometry of the figure. We call such a figure harmonious.1 Figure 1. A pair of harmonious graph realizations (assuming the latter in 3D). Harmonious realizations can have considerable value as aids to the intuitive understanding of the graph's structure, but such realizations are generally elusive. This note explains and explores a proposition that provides a straightforward way to generate an entire family of harmonious realizations of any graph: A matrix whose rows form an orthogonal basis of an eigenspace of a graph's adjacency matrix has columns that serve as coordinate vectors of the vertices of an harmonious realization of the graph. This is a (projection of a) spectral realization. The hundreds of diagrams in Section 42 illustrate that spectral realizations of graphs with a high degree of symmetry can have great visual appeal. Or, not: they may exist in arbitrarily-high- dimensional spaces, or they may appear as an uninspiring jumble of points in one-dimensional space. In most cases, they collapse vertices and even edges into single points, and are therefore only very rarely faithful. Nevertheless, spectral realizations can often provide useful starting points for visualization efforts. (A basic Mathematica recipe for computing (projected) spectral realizations appears at the end of Section 3.) Not every harmonious realization of a graph is spectral.
    [Show full text]
  • Paper Models of Polyhedra
    Paper Models of Polyhedra Gijs Korthals Altes Polyhedra are beautiful 3-D geometrical figures that have fascinated philosophers, mathematicians and artists for millennia Copyrights © 1998-2001 Gijs.Korthals Altes All rights reserved . It's permitted to make copies for non-commercial purposes only email: [email protected] Paper Models of Polyhedra Platonic Solids Dodecahedron Cube and Tetrahedron Octahedron Icosahedron Archimedean Solids Cuboctahedron Icosidodecahedron Truncated Tetrahedron Truncated Octahedron Truncated Cube Truncated Icosahedron (soccer ball) Truncated dodecahedron Rhombicuboctahedron Truncated Cuboctahedron Rhombicosidodecahedron Truncated Icosidodecahedron Snub Cube Snub Dodecahedron Kepler-Poinsot Polyhedra Great Stellated Dodecahedron Small Stellated Dodecahedron Great Icosahedron Great Dodecahedron Other Uniform Polyhedra Tetrahemihexahedron Octahemioctahedron Cubohemioctahedron Small Rhombihexahedron Small Rhombidodecahedron S mall Dodecahemiododecahedron Small Ditrigonal Icosidodecahedron Great Dodecahedron Compounds Stella Octangula Compound of Cube and Octahedron Compound of Dodecahedron and Icosahedron Compound of Two Cubes Compound of Three Cubes Compound of Five Cubes Compound of Five Octahedra Compound of Five Tetrahedra Compound of Truncated Icosahedron and Pentakisdodecahedron Other Polyhedra Pentagonal Hexecontahedron Pentagonalconsitetrahedron Pyramid Pentagonal Pyramid Decahedron Rhombic Dodecahedron Great Rhombihexacron Pentagonal Dipyramid Pentakisdodecahedron Small Triakisoctahedron Small Triambic
    [Show full text]
  • 1 Mar 2014 Polyhedra, Complexes, Nets and Symmetry
    Polyhedra, Complexes, Nets and Symmetry Egon Schulte∗ Northeastern University Department of Mathematics Boston, MA 02115, USA March 4, 2014 Abstract Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zig- zag, or helical. A polyhedron or complex is regular if its geometric symmetry group is transitive on the flags (incident vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all infinite, which are not polyhedra. Their edge graphs are nets well-known to crystallographers, and we identify them explicitly. There also are 6 infinite families of chiral apeirohedra, which have two orbits on the flags such that adjacent flags lie in different orbits. 1 Introduction arXiv:1403.0045v1 [math.MG] 1 Mar 2014 Polyhedra and polyhedra-like structures in ordinary euclidean 3-space E3 have been stud- ied since the early days of geometry (Coxeter, 1973). However, with the passage of time, the underlying mathematical concepts and treatments have undergone fundamental changes. Over the past 100 years we can observe a shift from the classical approach viewing a polyhedron as a solid in space, to topological approaches focussing on the underlying maps on surfaces (Coxeter & Moser, 1980), to combinatorial approaches highlighting the basic incidence structure but deliberately suppressing the membranes that customarily span the faces to give a surface.
    [Show full text]
  • Wythoffian Skeletal Polyhedra
    Wythoffian Skeletal Polyhedra by Abigail Williams B.S. in Mathematics, Bates College M.S. in Mathematics, Northeastern 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 Dedication I would like to dedicate this dissertation to my Meme. She has always been my loudest cheerleader and has supported me in all that I have done. Thank you, Meme. ii Abstract of Dissertation Wythoff's construction can be used to generate new polyhedra from the symmetry groups of the regular polyhedra. In this dissertation we examine all polyhedra that can be generated through this construction from the 48 regular polyhedra. We also examine when the construction produces uniform polyhedra and then discuss other methods for finding uniform polyhedra. iii Acknowledgements I would like to start by thanking Professor Schulte for all of the guidance he has provided me over the last few years. He has given me interesting articles to read, provided invaluable commentary on this thesis, had many helpful and insightful discussions with me about my work, and invited me to wonderful conferences. I truly cannot thank him enough for all of his help. I am also very thankful to my committee members for their time and attention. Additionally, I want to thank my family and friends who, for years, have supported me and pretended to care everytime I start talking about math. Finally, I want to thank my husband, Keith.
    [Show full text]