DOCSLIB.ORG

A Comparative Study of Platonic Solid Loudspeakers As Directivity Controlled Sound Sources

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Comparative Study of Platonic Solid Loudspeakers As Directivity Controlled Sound Sources Proc. of the 2nd International Symposium on Ambisonics and Spherical Acoustics May 6-7, 2010, Paris, France A COMPARATIVE STUDY OF PLATONIC SOLID LOUDSPEAKERS AS DIRECTIVITY CONTROLLED SOUND SOURCES Alexander Mattioli Pasqual Philippe Herzog Jose´ Roberto de Franc¸a Arruda Faculty of Mechanical Engineering Laboratory of Mechanics and Acoustics State University of Campinas National Center for Scientific Research Rua Mendeleiev 200 31 chemin Joseph-Aiguier 13083-970, Campinas – SP, Brazil 13402 Marseille cedex 20 France [pasqual,arruda]@fem.unicamp.br [email protected] ABSTRACT ARMs are real orthogonal functions (or vectors, if the vibrating body possesses a finite number of degrees of freedom) describ- Compact spherical loudspeaker arrays in the shape of Platonic ing velocity patterns over the surface of a vibrating body. As far solids have been used as directivity controlled sound sources. as compact loudspeaker arrays are concerned, one of the main Such devices are commonly used to synthesize spherical har- advantages of the ARMs over the SHs is that they span a fi- monic (SH) radiation patterns, which might be later combined nite dimension subspace (the subspace dimension is the number to produce different directivities. The SH synthesis presents of drivers in the array) on which any radiation pattern the loud- two main difficulties whose extent depends on the Platonic solid speaker array is able to produce can be projected with no approx- used: large synthesis error at high frequencies due to spatial imation error, whereas SHs span an infinite dimension subspace aliasing and huge membrane velocities at low frequencies due so that truncation error generally arises. On the other hand, un- to low radiation efficiency. This work presents a comparative like the ARMs, the SHs lead to a directivity format that does not theoretical study of the five Platonic loudspeakers. Their ability depend on the reproduction device and therefore the sound mate- to reproduce SHs and the radiation efficiencies of their acoustic rial so encoded is compatible to different loudspeaker arrays [8]. radiation modes are evaluated. It is shown that the dodecahe- In fact, the real SHs are the ARMs of the continuous dron presents the best compromise between complexity of the sphere [9], which is defined here as the sphere able to assume controllable patterns, number of channels and sound power. any surface velocity pattern (infinite number of degrees of free- dom). Moreover, at low frequencies, the ARMs of spherical ar- 1. INTRODUCTION rays give rise to far-field radiation patterns that match the real SHs, although the radiation efficiencies of an ARM and its cor- Compact spherical arrays of independent loudspeakers operating responding SH are different [7]. In other words, at low frequen- at the same frequency range have been used as directivity con- cies, the synthesis error in mimicking a given target directivity in trolled sound sources. The transducers are usually distributed the far-field remains unchanged whether the ARMs or the SHs over the spherical frame according to a Platonic solid geometry are used. However, only the ARMs provide the correct radiation to obtain a highly symmetrical configuration, so that the occur- efficiencies associated to each channel of the loudspeaker array. rence of preferred regions in the 3-D rendition space is reduced. This work presents a comparative theoretical study of the Hexahedral [1], dodecahedral [2, 3] and icosahedral [4] five Platonic solid loudspeakers (with one driver per face) as di- sources with one driver per face have been reported in the liter- rectivity controlled sound sources. Their radiation patterns are ature (an icosahedron with six drivers per face was proposed in derived analytically by approximating each driver as a convex [5]). Such devices are commonly used to synthesize pure spher- spherical cap that oscillates with a constant radial velocity am- ical harmonics (SHs) in the far-field under free-field conditions, plitude over its surface, as described in [6]. The ability of the which might be later combined to produce different directivities. Platonic sources in reproducing pure SH patterns in the least- The synthesis of pure SHs by a compact loudspeaker array squares sense is reviewed. In addition, the radiation efficiencies presents two main difficulties, namely, spatial aliasing in the of the ARMs of each Platonic source are computed in order to high-frequency range and low radiation efficiency in the low- evaluate the low-frequency constraint mentioned before. frequency range. The former degrades the SHs synthesis [1, 2,6] and the latter implies huge membrane velocities in order to pro- duce meaningful sound power levels [7]. The extent of these 2. THEORY problems is greatly affected by the position of each driver on the array and by the number of transducers, i.e., the choice of the This section gives an overview of the main theory related to the Platonic solid is of major importance to the array performance. radiation control by a compact spherical loudspeaker array. Instead of using SHs as preprogrammed basic directivities, the acoustic radiation modes (ARMs) of the loudspeaker array 2.1. Acoustic radiation modes (ARMs) can be used as a basis for directivity representation [7]. The This subsection presents a short introduction to the ARMs. For This work was sponsored by CAPES (Brazil) and CNRS (France). further details, refer to [7, 9]. Proc. of the 2nd International Symposium on Ambisonics and Spherical Acoustics May 6-7, 2010, Paris, France For a vibrating body with L degrees of freedom, the ARMs where mn = n2 + n +1+ m is used for linear indexing of the (1,2) form a set of L real orthogonal vectors that span the subspace SHs, k is the wave number, a is the sphere radius and hn ( ) of the achievable velocity patterns over the body surface. The are the spherical Hankel functions of the first and second kind.· ARMs can be obtained by the eigenvalue decomposition of an Equation (2) shows that σmn does not depend on m and is operator related to the radiation efficiency of the vibrating body. only a function of the non-dimensional parameter ka for SHs of The radiation efficiency, σ, of a vibrating structure with L a given order n. The radiation efficiencies for the first 49 ARMs degrees of freedom can be written as (SHs up to order n = 6) are presented in Fig. 1. H u Cu 0 σ(u)= (1) 10 uHVu where u is a column vector of velocity amplitude coefficients, −5 C is an L L real symmetric matrix which couples the power 10 radiated by× the elements of u, V is an L L real positive diag- × onal matrix which leads to the spatial mean-square velocity on mn −10 10 the structure surface and the superscript H indicates the complex σ mn=1 (n=0) mn=2−4 (n=1) conjugate transpose. Throughout this work, unless otherwise mn=5−9 (n=2) −15 specified, lower case bold letters indicate vectors, while upper 10 mn=10−16 (n=3) case bold letters indicate matrices. mn=17−25 (n=4) mn=26−36 (n=5) Notice that the radiation efficiency is in the form of the gen- mn=37−49 (n=6) −20 eralized Rayleigh quotient. Thus, the solution of the generalized 10 −2 −1 0 1 eigenvalue problem Cψ Vψ leads to a set of real orthog- 10 10 10 10 = λ L ka onal eigenvectors ψ1, ψ2,..., ψL corresponding to real eigen- values, ordered as λ1 λ2 . λL. These eigenvectors ≥ ≥ ≥ Figure 1: Radiation efficiencies of the first 49 acoustic radiation are the ARMs and the eigenvalues are their radiation efficiency modes of the continuous sphere (spherical harmonics). coefficients, i.e., σl σ(ψ )= λl. ≡ l It can be shown that these modes (ARMs) radiate sound en- Figure 1 illustrates the grouping characteristic of the acous- ergy independently, i.e., the total radiated sound power is given tic radiation modes discussed in [9]. The number of modes by a linear combination of the sound power produced by each within each group is 2n+1, i.e.,1(n = 0),3(n = 1),5(n = 2), mode. Because such a modal decomposition is closely related 7(n = 3), 9(n = 4), etc. It is shown that σmn increases with to the radiation efficiency of the vibrating body, it permits rank- ka and decreases with n. Moreover, at low ka values, the radia- ing the expansion terms (ARMs) with respect to their radiation tion efficiency is strongly affected by n, so that simple directiv- efficiencies, so that a reduced number of active channels can be ity patterns (lower order SHs) radiate much more efficiently than obtained by not driving inefficient modes. In addition, the ARMs complex ones. This result pinpoints the main difficulty concern- are only a function of frequency, the radiating structure geome- ing sound radiation reproduction in the low-frequency range: the try and the constraints the body is subjected to; they do not de- sphere surface must present a huge velocity amplitude in order pend on the source of excitation and on the mass and stiffness of to produce complex directivity patterns with meaningful sound the structure. Furthermore, the ARMs of some radiators (e.g., a power levels. continuous sphere) are proven to be frequency independent. For It is worth noting that σmn can be increased over a given these reasons, the ARMs are a useful representation of vibration frequency range by increasing a. However, for compact spheri- patterns when one is mainly interested in the sound field radiated cal loudspeaker arrays, it is known that a larger sphere limits the by a vibrating structure. radiation control in the high-frequency range due to spatial alias- ing artifacts.
Load more

Recommended publications
  • Steinitz's Theorem Project Report §1 Introduction §2 Basic Definitions
    Steinitz's Theorem Project Report Jon Hillery May 17, 2019 §1 Introduction Looking at the vertices and edges of polyhedra gives a family of graphs that we might expect has nice properties. It turns out that there is actually a very nice characterization of these graphs! We can use this characterization to find useful representations of certain graphs. §2 Basic Definitions We define a space to be convex if the line segment connecting any two points in the space remains entirely inside the space. This works for two-dimensional sets: and three-dimensional sets: Given a polyhedron, we define its 1-skeleton to be the graph formed from the vertices and edges of the polyhedron. For example, the 1-skeleton of a tetrahedron is K4: 1 Jon Hillery (May 17, 2019) Steinitz's Theorem Project Report Here are some further examples of the 1-skeleton of an icosahedron and a dodecahedron: §3 Properties of 1-Skeletons What properties do we know the 1-skeleton of a convex polyhedron must have? First, it must be planar. To see this, imagine moving your eye towards one of the faces until you are close enough that all of the other faces appear \inside" the face you are looking through, as shown here: This is always possible because the polyedron is convex, meaning intuitively it doesn't have any parts that \jut out". The graph formed from viewing in this way will have no intersections because the polyhedron is convex, so the straight-line rays our eyes see are not allowed to leave via an edge on the boundary of the polyhedron and then go back inside.
    [Show full text]
  • Geometry and Arithmetic of Crystallographic Sphere Packings
    Geometry and arithmetic of crystallographic sphere packings Alex Kontorovicha,b,1 and Kei Nakamuraa aDepartment of Mathematics, Rutgers University, New Brunswick, NJ 08854; and bSchool of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved November 21, 2018 (received for review December 12, 2017) We introduce the notion of a “crystallographic sphere packing,” argument leading to Theorem 3 comes from constructing circle defined to be one whose limit set is that of a geometrically packings “modeled on” combinatorial types of convex polyhedra, finite hyperbolic reflection group in one higher dimension. We as follows. exhibit an infinite family of conformally inequivalent crystallo- graphic packings with all radii being reciprocals of integers. We (~): Polyhedral Packings then prove a result in the opposite direction: the “superintegral” Let Π be the combinatorial type of a convex polyhedron. Equiv- ones exist only in finitely many “commensurability classes,” all in, alently, Π is a 3-connectedz planar graph. A version of the at most, 20 dimensions. Koebe–Andreev–Thurston Theorem§ says that there exists a 3 geometrization of Π (that is, a realization of its vertices in R with sphere packings j crystallographic j arithmetic j polyhedra j straight lines as edges and faces contained in Euclidean planes) Coxeter diagrams having a midsphere (meaning, a sphere tangent to all edges). This midsphere is then also simultaneously a midsphere for the he goal of this program is to understand the basic “nature” of dual polyhedron Πb. Fig. 2A shows the case of a cuboctahedron Tthe classical Apollonian gasket.
    [Show full text]
  • DIMACS REU 2018 Project: Sphere Packings and Number Theory
    DIMACS REU 2018 Project: Sphere Packings and Number Theory Alisa Cui, Devora Chait, Zachary Stier Mentor: Prof. Alex Kontorovich July 13, 2018 Apollonian Circle Packing This is an Apollonian circle packing: I Draw two more circles, each of which is tangent to the original three Apollonian Circle Packing Here's how we construct it: I Start with three mutually tangent circles I Draw two more circles, each of which is tangent to the original three Apollonian Circle Packing Here's how we construct it: I Start with three mutually tangent circles Apollonian Circle Packing Here's how we construct it: I Start with three mutually tangent circles I Draw two more circles, each of which is tangent to the original three Apollonian Circle Packing I Start with three mutually tangent circles I Draw two more circles, each of which is tangent to the original three Apollonian Circle Packing I Start with three mutually tangent circles I Draw two more circles, each of which is tangent to the original three I Continue drawing tangent circles, densely filling space These two images actually represent the same circle packing! We can go from one realization to the other using circle inversions. Apollonian Circle Packing Apollonian Circle Packing These two images actually represent the same circle packing! We can go from one realization to the other using circle inversions. Circle Inversions Circle inversion sends points at a distance of rd from the center of the mirror circle to a distance of r=d from the center of the mirror circle. Circle Inversions Circle inversion sends points at a distance of rd from the center of the mirror circle to a distance of r=d from the center of the mirror circle.
    [Show full text]
  • Mathematical Constants and Sequences
    Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) ..
    [Show full text]
  • Stacked 4-Polytopes with Ball Packable Graphs
    STACKED 4-POLYTOPES WITH BALL PACKABLE GRAPHS HAO CHEN Abstract. After investigating the ball-packability of some small graphs, we give a full characterisation, in terms of forbidden induced subgraphs, for the stacked 4-polytopes whose 1-skeletons can be realised by the tangency relations of a ball packing. 1. Introduction A ball packing is a collection of balls with disjoint interiors. A graph is said to be ball packable if it can be realized by the tangency relations of a ball packing. Formal definitions will be given later. The combinatorics of disk packings (2-dimensional ball packings) is well un- derstood thanks to the Koebe{Andreev{Thurston's disk packing theorem, which says that every planar graph is disk packable. However, we know little about the combinatorics of ball packings in higher dimensions. In this paper we study the relation between Apollonian ball packings and stacked polytopes: An Apollonian ball packing is formed by repeatedly filling new balls into holes in a ball packing. A stacked polytope is formed, starting from a simplex, by repeatedly gluing new simplices onto facets. Detailed and formal introductions can be found respectively in Section 2.3 and in Section 2.4. There is a 1-to-1 correspondence between 2-dimensional Apollonian ball packings and 3-dimensional stacked polytopes: a graph can be realised by the tangency relations of an Apollonian disk packing if and only if it is the 1-skeleton of a stacked 3-polytope. As we will see, this relation does not hold in higher dimensions: On one hand, the 1-skeleton of a stacked polytope may not be realizable by the tangency relations of any Apollonian ball packing.
    [Show full text]
  • Platonic and Archimedean Solids
    PLATONIC AND ARCHIMEDEAN SOLIDS Author: Daud Sutton Number of Pages: 64 pages Published Date: 25 Oct 2005 Publisher: Wooden Books Publication Country: Powys, United Kingdom Language: English ISBN: 9781904263395 DOWNLOAD: PLATONIC AND ARCHIMEDEAN SOLIDS Streaming and Download help. Report this track or account. For example, a vertex configuration of 4,6,8 means that a square , hexagon , and octagon meet at a vertex with the order taken to be clockwise around the vertex. Some definitions of semiregular polyhedron include one more figure, the elongated square gyrobicupola or "pseudo-rhombicuboctahedron". The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular. The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra , these are the face-uniform solids with regular vertices. The snub cube and snub dodecahedron are known as chiral , as they come in a left-handed form Latin: levomorph or laevomorph and right-handed form Latin: dextromorph. When something comes in multiple forms which are each other's three-dimensional mirror image , these forms may be called enantiomorphs. This nomenclature is also used for the forms of certain chemical compounds. The different Archimedean and Platonic solids can be related to each other using a handful of general constructions. Get All Access Pass. Ancient Fonts Collection. The usual way of enumerating the semiregular polyhedra is to eliminate solutions of conditions 1 and 2 using several classes of arguments and then prove that the solutions left are, in fact, semiregular Kepler , pp. The following table gives all possible regular and semiregular polyhedra and tessellations.
    [Show full text]
  • Arxiv:1712.00147V1 [Math.MG] 1 Dec 2017 the Bend of a Sphere Is the Reciprocal of Its (Signed) Radius
    GEOMETRY AND ARITHMETIC OF CRYSTALLOGRAPHIC SPHERE PACKINGS ALEX KONTOROVICH AND KEI NAKAMURA Abstract. We introduce the notion of a \crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit for the first time an infinite family of conformally-inequivalent such with all radii being reciprocals of integers. We then prove a result in the opposite direction: the \superintegral" ones exist only in finitely many \commensurability classes," all in dimensions below 30. The goal of this program, the details of which will appear elsewhere, is to under- stand the basic \nature" of the classical Apollonian gasket. Why does its integral structure exist? (Of course it follows here from Descartes' Kissing Circles Theorem, but is there a more fundamental, intrinsic explanation?) Are there more like it? (Around a half-dozen similarly integral circle and sphere packings were previously known, each given by an ad hoc description.) If so, how many more? Can they be classified? We develop a basic unified framework for addressing these questions, and find two surprising (and opposing) phenomena: (I) there is indeed a whole infinite zoo of integral sphere packings, and (II) up to \commensurability," there are only finitely-many Apollonian-like objects, over all dimensions! Definition 1. By an Sn−1-packing (or just packing) P of Rcn := Rn [ f1g, we mean an infinite collection of oriented (n − 1)-spheres (or co-dim-1 planes) so that: • the interiors of spheres are disjoint, and • the spheres densely fill up space; that is, we require that any ball in Rcn intersects the interior of some sphere in P.
    [Show full text]
  • Apollonian Ball Packings and Stacked Polytopes
    APOLLONIAN BALL PACKINGS AND STACKED POLYTOPES HAO CHEN Abstract. We investigate in this paper the relation between Apollonian d-ball packings and stacked (d + 1)-polytopes for dimension d ≥ 3. For d = 3, the relation is fully described: we prove that the 1-skeleton of a stacked 4-polytope is the tangency graph of an Apollonian 3-ball packing if and only if no six 4-cliques share a 3-clique. For higher dimension, we have some partial results. 1. Introduction A ball packing is a collection of balls with disjoint interiors. A graph is said to be ball packable if it can be realized by the tangency relations of a ball packing. The combinatorics of disk packings (2-dimensional ball packings) is well understood thanks to the Koebe{Andreev{Thurston's disk packing theorem, which asserts that every planar graph is disk packable. However, few is known about the combinatorics of ball packings in higher dimensions. In this paper we study the relation between Apollonian ball packings and stacked polytopes. An Apollonian ball packing is constructed from a Descartes configuration by repeatedly filling new balls into \holes". A stacked polytope is constructed from a simplex by repeatedly gluing new simplices onto facets. See Section 2.3 and 2.4 respectively for formal descriptions. There is a 1-to-1 correspondence between 2-dimensional Apollonian ball packings and 3-dimensional stacked polytopes. Namely, a graph can be realised by the tangency relations of an Apollonian disk packing if and only if it is the 1-skeleton of a stacked 3-polytope.
    [Show full text]
  • Graph Colorings, Flows and Perfect Matchings Louis Esperet
    Graph colorings, flows and perfect matchings Louis Esperet To cite this version: Louis Esperet. Graph colorings, flows and perfect matchings. Combinatorics [math.CO]. Université Grenoble Alpes, 2017. tel-01850463 HAL Id: tel-01850463 https://tel.archives-ouvertes.fr/tel-01850463 Submitted on 27 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. HABILITATION A` DIRIGER DES RECHERCHES Specialit´ e´ : Informatique et Mathematiques´ Appliquees´ Present´ ee´ par Louis Esperet prepar´ ee´ au sein du Laboratoire G-SCOP (UMR5272) Graph colorings, flows and perfect matchings Soutenue le 19 Octobre 2017, devant le jury compose´ de : Victor Chepoi Professeur, Aix-Marseille Universite,´ Examinateur Toma´sˇ Kaiser Professeur, University of West Bohemia, Rapporteur Myriam Preissmann Directrice de recherche CNRS, Grenoble, Examinatrice Bruce Reed Directeur de recherche CNRS, Sophia-Antipolis, Rapporteur Gautier Stauffer Professeur, Grenoble-INP, Examinateur Stephan´ Thomasse´ Professeur, ENS de Lyon, Examinateur Cun-Quan Zhang Professeur, West Virginia University,
    [Show full text]
  • Apollonian Ball Packings and Stacked Polytopes
    Discrete Comput Geom DOI 10.1007/s00454-016-9777-3 Apollonian Ball Packings and Stacked Polytopes Hao Chen1 Received: 28 December 2013 / Revised: 4 March 2016 / Accepted: 9 March 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We investigate in this paper the relation between Apollonian d-ball packings and stacked (d + 1)-polytopes for dimension d ≥ 3. For d = 3, the relation is fully described: we prove that the 1-skeleton of a stacked 4-polytope is the tangency graph of an Apollonian 3-ball packing if and only if there is no six 4-cliques sharing a 3-clique. For higher dimension, we have some partial results. Keywords Apollonian ball packing · Stacked polytope · k-Tree · Forbidden subgraph Mathematics Subject Classification 52C17 · 52B11 · 20F55 1 Introduction A ball packing is a collection of balls with disjoint interiors. A graph is said to be ball packable if it can be realized by the tangency relations of a ball packing. The combi- natorics of disk packings (2-dimensional ball packings) is well understood thanks to the Koebe–Andreev–Thurston’s disk packing theorem, which asserts that every pla- Editor in Charge: János Pach The work was done when the author was a PhD candidate at Institut für Mathematik of Freie Universität Berlin. An alternative version of the manuscript appeared in the PhD thesis of the author. Hao Chen [email protected] 1 Departement of Mathematics and Computer Science, Technische Universiteit Eindhoven, Eindhoven, The Netherlands 123 Discrete Comput Geom nar graph is disk packable.
    [Show full text]
  • Ball Packings and Lorentzian Discrete Geometry by 陈 浩 CHEN, Hao
    Ball Packings and Lorentzian Discrete Geometry by H i CHEN, Hao A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Mathematics in the Institut f¨urMathematik Fachbereich Mathematik und Informatik Freie Universit¨atBerlin Berlin July 28, 2014 Supervisor and first reviewer: Prof. G¨unter M. Ziegler Second reviewer: Prof. Christophe Hohlweg Date of defense: June 11, 2014 Institut f¨urMathematik Freie Universit¨atBerlin To my wife. Summary While the tangency graphs of disk packings are completely characterised by Koebe{ Andreev{Thurston's disk packing theorem, not so much is known about the combina- torics of ball packings in higher dimensions. This thesis tries to make a contribution by investigating some higher dimensional ball packings. The key idea throughout this thesis is a correspondence between balls in Euclidean space and space-like directions in Lorentz space. This allows us to interpret a ball packing as a set of points in the projective Lorentz space. Our investigation starts with Descartes' configuration, the simplest ball packing studied in this thesis. It serves as the basic element for further constructions. Then we explicitly construct some small packings whose tangency graph can be expressed as graph joins, and identify some graph joins that are not the tangency graph of any ball packing. With the help of these examples, we characterise the tangency graphs of Apollonian ball packings in dimension 3 in terms of the 1-skeletons of stacked polytopes. Partial results are obtained for higher dimensions. Boyd{Maxwell packings form a large class of ball packings that are generated by inversions, generalising Apollonian packings.
    [Show full text]
  • On Isoclinal Sequences of Spheres
    proceedings of the american mathematical society Volume 88. Number 4, August 1983 ON ISOCLINAL SEQUENCES OF SPHERES ASIA IVlt WEISS Abstract. In «-dimensional inversive geometry it is possible to construct an infinite sequence of ( n — 1)-spheres with the property that every n + 2 consecutive numbers are isoclinal. For every such sequence there is a Möbius transformation which advances the spheres of the sequence by one. 1. Introduction. In the beautiful paper [1], Coxeter showed that in «-dimensional inversive space (n > 2) there is an infinite sequence of (n — l)-spheres such that every n + 2 consecutive members are mutually tangent. Given such sequence there is a Möbius transformation mapping the sequence onto itself (see [4]). Mauldon in [3] considered sets of spheres with the same mutual inclination, and Gerber in [2] extended these sets to infinite sequences. Here we give a recursion formula for computing coordinates of the spheres in such sequences. Using this formula we show that such a sequence can, as well as a tangent one, be mapped onto itself by a Möbius transformation. 2. Notation and definitions. In considering «-dimensional inversive geometry we can use vectors with « + 2 coordinates to denote points and spheres. Let 2 be a unit sphere in R"+1 centred at the origin, and II an «-flat through the origin. Stereographic projection establishes one-to-one correspondence between II and 2\{north pole}. To every point X in n we can assign an (« + 2)-tuple (x, 1) where x is the coordinate of the stereographic projection of X. Similarly, to every sphere Y in n we can assign an (« + 2)-tuple (csc 0 • y, cot 0), where y is the coordinate of the centre and 0 the angular radius of the stereographic projection of Y.
    [Show full text]