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Doctoral Thesis
Sphere assemblies: control of rotation states and construction of space-filling packings
Author(s): Stäger, Dominik V.
Publication Date: 2016
Permanent Link: https://doi.org/10.3929/ethz-a-010780463
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ETH Library Diss. ETH No. 23864
Sphere Assemblies: Control of Rotation States and Construction of Space-Filling Packings
A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich (Dr. sc. ETH Zurich)
presented by Dominik Valerian Stager¨ MSc. Materials Science, ETH Zurich
born 18.06.1989 citizen of Glarus S¨ud GL, Switzerland
accepted on the recommendation of Prof. Dr. Hans J. Herrmann, examiner Prof. Dr. Tomaso Aste, co-examiner Prof. Dr. Ronald Peikert, co-examiner
2016
Acknowledgment
I would like to thank Prof. Dr. Hans J. Herrmann for proposing me the topic for my PhD, which fascinated me from the beginning to the end, and which led to fruitful results. Furthermore, I would like to thank him for the supervision of this thesis, especially for the numerous constructive criticism. Special thanks goes to Prof. Dr. Nuno A. M. Ara´ujo for his supervision through a large period of my PhD. I highly appreciated the support of his sharp and incredibly fast mind. I would like to thank Dr. Falk Wittel for his support regarding the con- struction of experiments, and Sergio Solorzano Rocha for fruitful discussions about my research. I am grateful to Prof. Dr. Tomaso Aste and Prof. Dr. Ronald Peikert for accepting the request to co-examine this thesis and for showing interest in my work. Special thanks goes to my coworkers for the amazing company throughout my PhD. After all this time, I feel like being part of a big family with a lot of awesome brothers. To see so many friends at work every single day is an enormous motivation and an incredible source of energy. I do not want to have missed out on any of the great lunch breaks, and neither any of the numerous laughs we shared every day. I am very grateful for all the good moments that we could share apart from work and the unforgettable stories that resulted from them. Last but not least, I want to thank my family and all my friends with whom I could share plenty of good times that let me enjoy life to its fullest during this special period.
Contents
Kurzfassung i
Abstract iii
Related Publications v
1 Introduction 1 1.1 Overview ...... 1 1.2 Background ...... 3 1.3 Motivation ...... 5 1.3.1 Understanding of Rotational Dynamics ...... 5 1.3.2 Search for Further 3D Space-Filling Bearings ..... 6
2 Prediction and Control of Slip-Free Rotation States in Sphere Assemblies 7 2.1 Introduction ...... 7 2.2 Kinetics of Bipartite Assemblies ...... 9 2.3 Construction of 4DOF Assemblies ...... 11 2.4 Prediction of the Final Slip-Free State ...... 16 2.5 Control of the Slip-Free State ...... 19 2.5.1 Control Within the Model ...... 20 2.5.2 Control in Experiment ...... 22 2.6 Experimental Details ...... 26 2.7 Final Remarks ...... 27 3 Construction of Self-Similar Space-Filling Sphere Packings in Three and Four Dimensions 29 3.1 Introduction ...... 29 3.2 Basic Idea of Generating a Packing ...... 31 3.3 Circle Inversion ...... 31 3.3.1 Basic Properties ...... 31 3.3.2 Multiple Inversion Circles ...... 33 3.3.3 Mathematics of Circle Inversion ...... 35 3.4 Constraints on the Generating Setup ...... 37 3.5 How to Construct Generating Setups ...... 41 3.5.1 Construction of 2D Generating Setups ...... 41 3.5.2 Generalization to Higher Dimensions ...... 42 3.5.3 Determine Positions and Radii of Setup Elements .. 47 3.6 Discovered Packings ...... 53 3.6.1 Fractal Dimension ...... 53 3.6.2 Contact Network ...... 55 3.6.3 Topological Comparison ...... 67 3.7 Modified Packings ...... 71 3.8 Final Remarks ...... 71
4 Cutting Self-Similar Space-Filling Sphere Packings 75 4.1 Introduction ...... 75 4.2 Properties of Packings ...... 77 4.3 Cutting ...... 77 4.3.1 Random Cuts ...... 79 4.3.2 Special Cuts ...... 84 4.4 Final Remarks ...... 92
5 Conclusion 95
References 97 Kurzfassung
Wir widmen uns zwei verschiedenen Themenbereichen. Wir untersuchen die Rotationsdynamik von aneinanderliegenden, individuell rotierenden Kugeln, welche sich gegenseitig beeinflussen. Daruber¨ hinaus entwickeln wir neue Konstruktionsmethoden fur¨ raumfullenden¨ Kugelpackungen. Zuerst studieren wir die Rotationsdynamik von aneinanderliegenden Ku- geln, sogenannten Kugelclustern. Dabei nehmen wir an, dass die Kugeln an ihrer Position fixiert sind, jedoch frei rotieren k¨onnen. Weiter befassen wir uns nur mit bipartiten Kugelclustern. Ein Kugelcluster ist bipartit, falls man es mit nur zwei Farben so einf¨arben kann, sodass sich keine Kugeln gleicher Farbe beruhren.¨ Jedes bipartite Kugelcluster hat schlupffreie Rota- tionszust¨ande, in welchen sich alle Kugeln gleichzeitig drehen k¨onnen, ohne Schlupf zwischen beruhrenden¨ Kugeln. Mit einem Modell, welches lediglich Gleitreibung berucksichtigt,¨ untersuchen wir, wie sich bipartite Kugelclus- ter von einem Anfangszustand mit frei w¨ahlbaren Winkelgeschwindigkeiten zu einem schlupffreien Endzustand bewegen. Dabei stellen wir fest, dass ge- wisse Summen von Variablen, welche die Massen, Radien, Winkelgeschwin- digkeiten und Positionen der Kugeln einbeziehen, zeitlich unver¨anderlich sind. Das bedeutet, dass diese vom Anfangszustand bis hin zum schlupf- freien Endzustand konstant bleiben. Fur¨ gewisse bipartite Kugelcluster ist der schlupffreie Zustand eindeutig bestimmt fur¨ gegebene Werte der zeit- lich konstanten Summen. In diesem Fall kann man den Endzustand fur¨ jeden beliebigen Anfangszustand exakt vorhersagen. Diese vorhersagbaren Kugelcluster mussen¨ genau vier Freiheitsgrade im schlupffreien Zustand ha- ben. Glucklicherweise¨ sind solche einfach zu konstruieren und k¨onnen aus
i nur zwei, aber auch aus viel mehr Kugeln bestehen, theoretisch aus unbe- grenzt vielen. Uberraschenderweise¨ ist der schlupffreie Endzustand solcher Kugelcluster unabh¨angig von der St¨arke der Gleitreibung zwischen den Ku- geln. Desweiteren l¨asst sich der schlupffreie Zustand kontrollieren. Durch die externe Kontrolle von lediglich zwei beliebigen Kugeln kann jeder m¨ogliche schlupffreie Rotationszustand des Kugelclusters kontrolliert werden, was wir auch experimentell demonstrieren. Im schlupffreien Zustand eines jeden bi- partiten zweidimensionalen Clusters gleich grosser Scheiben sind alle Rota- tionsgeschwindigkeiten identisch. Nicht so in einem dreidimensionalen Ku- gelcluster, wo Kugeln gleicher Grsse unterschiedliche Rotationsgeschwindig- keiten haben knnen. Deshalb ist es m¨oglich in einem kontrollierbaren Ku- gelcluster die Kugeln entlang einer frei w¨ahlbaren Richtung zu beschleuni- gen, was eine zuvor unbekannte mechanische Funktionalit¨at darstellt. Desweiteren befassen wir uns mit raumfullenden¨ Kugelpackungen. Im Detail behandeln wir nur Packungen, welche mittels Kugelinversionen erzeugt wer- den k¨onnen, was immer exakt selbst¨ahnliche Packungen sind. Diese Packun- gen sind fraktal und die Gr¨ossenverteilung ihrer Kugeln folgt asymptotisch einem Potenzgesetz, woraus man die fraktale Dimension absch¨atzen kann. Inspiriert durch vorhergehende Arbeiten entwickeln wir eine Konstruktions- methode in zwei Dimensionen, welche wir fur¨ jede h¨ohere Dimension verall- gemeinern. Mittels der neuen Konstruktionsmethode finden wir zahlreiche neue Topologien in drei und vier Dimensionen. Zus¨atzlich stellen wir eine Strategie vor um neue niedrigerdimensionale Topologien aus bereits ent- deckten zu schneiden. Im Ganzen erm¨oglicht uns dies weitere raumfullende¨ Topologien in beliebigen Dimensionen zu finden. Die zahlreichen Topologien die wir finden weisen eine grosse Bandbreite an fraktalen Dimensionen auf und k¨onnen als eine Auswahl fur¨ ideal dichte Packungen mit verschiedenen Gr¨ossenverteilungen gesehen werden.
ii Abstract
We study two different areas. We investigate the rotational dynamics of assemblies of touching, individually rotating spheres, which influence each other’s rotation. Beyond that, we develop new methods to construct space- filling sphere packings. First, we study the rotational dynamics of an assembly of contacting spheres. We are interested in the case where the spheres can not move in space, but are allowed to rotate. We only deal with bipartite assemblies, i.e., assemblies where one can color the spheres using only two colors such that no spheres of same color touch. Any bipartite assembly of spheres has slip-free rotation states, i.e., it is possible that all spheres rotate without any slip between contacting spheres. With a model that only considers sliding friction, we investigate how bipartite assemblies drive from an initial rotation state with arbitrary angular velocities toward a slip-free state. We find that certain sums of variables are time-invariant, i.e., they stay constant during the dy- namics toward the slip-free state. These sums involve the individual spheres’ masses, radii, angular velocities, and positions. For certain bipartite assem- blies, the final slip-free state uniquely corresponds to specific values of the time-invariant sums, such that one can directly predict the final state from the initial one. These predictable assemblies need to have exactly four de- grees of freedom in their slip-free state. Luckily, those assemblies can easily be constructed and the number of spheres can range from two to, theoreti- cally, infinity. The final slip-free state of those assemblies surprisingly does not depend on the strength of sliding friction between spheres. Furthermore, by only controlling two arbitrary spheres externally, the slip-free state can
iii be controlled, what we demonstrate experimentally. In any slip-free state of any two-dimensional bipartite assembly of equally sized disks, all disks have the same rotational speed. Not so in three dimensions, such that in a con- trollable assembly of equally sized spheres, spheres can have different speeds of rotations. Therefore, one can accelerate the rotation of spheres along such an assembly, what is a previously unknown mechanical functionality. Second, we study space-filling sphere packings. In particular, packings that are constructed using inversive geometry, i.e., sphere inversion. These pack- ings are exactly self-similar and fractal, and their size distribution follows asymptotically a power law, from which one can estimate the fractal di- mension. Inspired by previous works, we develop a construction method in two dimensions which we generalize to any higher dimensions. We use it to find new topologies in three and four dimensions. Additionally, we present a strategy to find new sub-dimensional topologies from existing ones by cutting packings. Altogether, this provides a framework to find various space-filling topologies of spheres in arbitrary dimensions. The various topologies we find show a broad range of fractal dimensions and can be seen as a selection for ideally dense packings with different size distributions.
iv Related Publications
This thesis contains content of the following published or for peer-review submitted articles:
D. V. St¨ager, N. A. M. Ara´ujo, and H. J. Herrmann, “Prediction and • control of slip-free rotation states in sphere assemblies,” Phys. Rev. Lett., vol. 116, p. 254301, 2016. (arXiv:1505.07348)
D. V. St¨ager and H. J. Herrmann, “Self-similar space-filling sphere • packings in three and four dimensions,” submitted to Fractals, 2016. (arXiv:1607.08391)
D. V. St¨ager and H. J. Herrmann, “Cutting self-similar space-filling • sphere packings,” submitted to Fractals, 2016. (arXiv:1609.03811)
For each of these three articles, the author contributed the most effort, in- cluding: design, implementation, and execution of all simulations; analysis and interpretation of all data generated from simulations; all conceptual and mathematical derivations; design, construction, and execution of all experiments; analysis and interpretation of all data generated from exper- iments; execution of all numerical analyses; literature research; writing of the manuscript; and design and creation of all figures.
v
Chapter 1
Introduction
1.1 Overview
Two aspects of assemblies of spheres are studied. We investigate the control of the rotational dynamics of touching, individually rotating spheres that influence each other’s rotation. And we develop new construction methods for space-filling packings. In the following, we give a brief overview of the content of this thesis. Section 1.2 provides the relevant background information. In Sec. 1.3, we describe the lack of knowledge that was the motivation for this work. The main part is organized in the following way: First, we investigate the dynamics of rotation of spheres in Chapter 2. We study contacting spheres that are fixed at their position but can rotate freely, as shown in Fig. 1.1(a). We investigate how they influence each other’s rotation through sliding friction. We theoretically discover the possibility to control the rotation state of an assembly by externally controlling only two spheres. Since this is a previously unknown mechanical functionality, we demonstrate its feasibility experimentally. Second, we deal with exactly self-similar space-filling packings of spheres, as the three-dimensional Apollonian Gasket shown in Fig. 1.1(b). They are called space-filling because when considering up to infinitesimally small spheres, there is no porosity left in these packings. An object is exactly self-similar if it is exactly similar to itself, i.e., the whole has the same shape
1 CHAPTER 1. INTRODUCTION a b
Figure 1.1: (a) Assembly of contacting spheres that individually rotate with random angular velocities. (b) The three-dimensional Apollonian Gasket is an example of a self-similar space-filling sphere packing. The construction principle of the Apollonian Gasket is to start with four tangent spheres inside a sphere and iteratively fill the largest pore with the largest possible sphere. The packing is named after the Greek mathematician Apollonius of Perga, who lived in the years 262 - 190 BC. Here, the surrounding sphere is visualized as an open shell. Some spheres are removed to allow looking inside the packing. as one or more of the individual parts. For an exactly self-similar packing, this means that one can zoom into the packing arbitrarily often and one will find the same spatial arrangement of spheres over and over again. Even though such perfect packings are difficult to realize experimentally, their size distribution and spatial arrangement serve as idealized references for dense packings of nearly spherical particles. Dense packings are needed in various industrial applications, such that they have been subject of many experimental and theoretical studies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Here, we discover new topologies of self-similar space-filling packings. Different topologies can differ in their fractal dimension, which defines the asymptotic behavior of the size distribution of the spheres [15, 16, 17]. By
2 1.2. BACKGROUND developing a two-dimensional construction method that we generalize to any higher dimensions, we find new topologies in three and four dimensions in Chapter 3. Additionally we develop a strategy to find even further sub- dimensional topologies as cuts of other’s in Chapter 4. Altogether, this provides a framework to search for space-filling packings with a broad range of fractal dimensions that serve as references for ideally dense packings. Space-filling packings such as the Apollonian Gasket are studied in relation to various topics, for instance: packing construction [18, 19, 20, 21, 22, 23, 24, 25, 26, 27], packing modeling [10, 11], packing optimization [12], granular materials [9, 13, 14], bearings [28, 29, 30, 31], fractality [32, 15, 16, 33, 34, 35, 36, 37, 38], packings of integral curvature [39, 40, 41, 42, 43, 44, 45], group theory [46, 47, 48], M¨obius mappings [49], hyperbolic geometry [50, 51], synchronization [52], mathematical artwork [53, 54], tilings [55], networks [56, 57, 58, 59, 60, 61, 62], random walks [63, 60], fluid flow [64], and percolation [65, 66]. Detailed conclusions and outlooks of the individual studies in Chapters 2, 3, and 4 can be found at the end of the Chapters in the Final Remarks sections. An overall conclusion of this thesis can be found in Chapter 5.
1.2 Background
In 1990, Ref. [28] discovered a method to construct two-dimensional space- filling bearings, as the one shown in Fig. 1.2. They were called bearings because they allow the rotation of all disks without slip between contacting disks. Therefore, this slip-free state is free of sliding friction. The study was motivated by the observation of seismic gaps [67, 68], which are regions between two tectonic plates that show unexpectedly low seismic activity and development of heat. In samples of granular material in a seismic gap, one found that the size distributions of grains follows approximately a power law [69], i.e., shows self-similarity. The same holds for the space-filling bearings such that because of their slip-free rotation state, they were proposed as simple models for seismic gaps, which could be responsible for the lack of
3
1.3. MOTIVATION consecutive contacts. In a bipartite packing, every loop size is even. In 2000, Reference [29] generalized these space-filling bearings to an arbitrary size of smallest loops. In 2004, Ref. [30] presented the first three-dimensional space-filling bearing, which is a bipartite sphere packing with smallest loop size four. They proved that any bipartite sphere packing has slip-free rotation states. The presented bearing in Ref. [30] was found by a generalization [25] of a construction method for the three-dimensional Apollonian Gasket shown in Ref. [35]. Apart from this bearing, this generalized method also produces three other previously unknown space-filling sphere packings which are not bipartite. All these packings are fractal packings, each with a distinct fractal dimension, what indicates that they are different topologies. These topologies can be seen as ideal references for highly dense granular packings, which have vari- ous applications and therefore are studied experimentally and theoretically [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. In 2005, a method to construct random space-filling bearings in two and three dimensions was presented [31], which depending on a model parameter have slightly different fractal dimensions inside a narrow range.
1.3 Motivation
Two open questions originally motivated this work, which we elaborate in the two following sections.
1.3.1 Understanding of Rotational Dynamics
It was proven that any bipartite sphere packing has slip-free rotation states [30], but the dynamics of such systems was not understood yet. Two- dimensional space-filling bearings have been studied with respect to their synchronizability [52], i.e., the stability of the slip-free state. We were inter- ested in how three-dimensional bearings drive toward a slip-free state from an initially chosen rotation state with slip. From the understanding of this dynamics, we expected insight into the behavior of such bearings that might
5 CHAPTER 1. INTRODUCTION offer applications in designing mechanical bearings that allow the transfer, damping, dissipation, and filtering of rotational energy. This motivated us for the study of the rotational dynamics of bipartite sphere assemblies in Chapter 2.
1.3.2 Search for Further 3D Space-Filling Bearings
The generalized construction method for two-dimensional space-filling bear- ings of Ref. [29] allows producing an infinite amount of discrete bipartite topologies with arbitrary smallest sizes of loops. In three dimensions, one previously only knew the highly ordered space-filling bearing of Ref. [30] and the random ones of Ref. [31], where all these bearings exhibit a smallest size of loops of four. It was an open question if one can also construct three-dimensional bearings with smallest sizes of loops larger than four. Such bearings might show specific, previously unknown rotational dynam- ics. This motivated us to search for construction methods producing further three-dimensional bearings, which resulted in the studies presented in Chap- ters 3 and 4.
6 Chapter 2
Prediction and Control of Slip-Free Rotation States in Sphere Assemblies
2.1 Introduction
In a fixed assembly of touching spheres that can rotate individually, slid- ing friction at the contacts between spheres will generally slow down and finally stop the rotation of all spheres. However, if the assembly is bipar- tite (compare Fig. 1.2), it will instead drive toward a stationary slip-free rotation state which is free of sliding friction, as discussed in Sec. 1.2. This state happens to have at least four degrees of freedom. Here we introduce time invariant quantities and with them show, for assemblies with exactly four degrees of freedom, that no matter how many spheres belong to the assembly, one can always predict from the initial state of rotation the final state, which we prove to be independent of the type and strength of sliding friction. This allows imposing any slip-free state by only controlling two spheres, providing a method to control the collective state of rotation of an assembly. With this work, we contribute to the understanding of the rota- tional dynamics of dense packings which are studied in the context of shear bands [70, 71, 72, 73, 74] and seismic gaps [70, 71, 75, 74, 28, 30, 31]. Since we propose a new way of controlling the rotation of spheres, mechanics and
7 CHAPTER 2. PREDICTION AND CONTROL OF SLIP-FREE ROTATION STATES IN SPHERE ASSEMBLIES
a b
driving wheels
Figure 2.1: Operation of an assembly of spheres in a slip-free state. (a) Three pairs of plastic spheres in contact were stacked onto each other with an alternating stacking angle of 35 degrees and their positions were fixed using ball rollers. When forcing the bottom pair of spheres to rotate according to the scheme shown in (b) using external driving wheels, the other spheres adjust their rotation due to friction. In the stationary state, the top pair of spheres rotates more than three times faster than the bottom pair. robotics is another area of applicability of our results. Consider three pairs of plastic spheres in contact alternatively displaced (at a stacking angle) on top of each other, with their positions fixed by ball rollers as shown in Fig. 2.1(a). Forcing the two bottom spheres to rotate as indicated in Fig. 2.1(b), we find a stationary state of rotation in which the top pair of spheres rotates more than three times faster than the bottom pair. Since any two touching spheres have equal tangential velocities at their contact point, this state of rotation is slip-free and, therefore, free of sliding
8 2.2. KINETICS OF BIPARTITE ASSEMBLIES
relaxation
slip-free state different tangential velocities equal tangential velocities
Figure 2.2: Relaxation of a single pair of spheres toward the slip-free state. Rotating spheres with sliding forces at their contact (left) and in the final slip-free state (right), in which the contact moves along circles (dashed).
Due to Newton’s third law of motion F~2 = F~1, where F~1 and F~2 are the − sliding forces acting on the first and second sphere, respectively. friction. This assembly is bipartite and as we show later, its slip-free state has four degrees of freedom, such that we call it a 4DOF assembly. Here, we first explore the kinetics of bipartite assemblies in general in Sec. 2.2. We then show how to construct 4DOF assemblies in Sec. 2.3, how to predict their final slip-free state from an arbitrary initial one in Sec. 2.4, and how to control the rotation state theoretically and experimentally in Sec. 2.5. We provide experimental details in Sec. 2.6 and draw conclusions in Sec. 2.7. The discovery of the ability to control the rotation state of an assembly was unexpected. Our work is motivated by previous studies on bipartite assemblies of disks [70, 72, 73, 71, 34, 56, 62, 75, 28, 52, 74] and spheres [30, 31].
2.2 Kinetics of Bipartite Assemblies
Let us first consider two single spheres in contact that relax from an arbitrary initial rotation state toward a slip-free state, as shown in Fig. 2.2. In the following, we color any sphere either red or yellow such that no spheres of the same color touch, what is only possible in bipartite assemblies. We
9 CHAPTER 2. PREDICTION AND CONTROL OF SLIP-FREE ROTATION STATES IN SPHERE ASSEMBLIES assume spheres are perfectly rigid and we only consider sliding friction. As long as the slip-free state is not reached yet, the tangential velocities of the spheres at the contact differ and the two sliding forces at the contact (one on each sphere) tend to reduce this velocity difference. The two forces are opposite to each other and each force F~ produces a torque T~ = ~r F~ , × where ~r points from the center of the corresponding sphere to the contact. The two torques are parallel and their magnitude is proportional to the radius of the corresponding sphere. Using the law of motion T~ = I~α, we find independently of the type and strength of sliding friction I ~α I ~α 1 1 = 2 2 , (2.1) r1 r2 where I1, I2, ~α1, ~α2, r1, and r2 are the moments of inertia, the angular accelerations, and the radii of the first and the second sphere, respectively. Note that to enforce our assumption that the centers of the spheres remain fixed, we apply two constraint forces, one on the center of each sphere. Each force is opposite to the sliding force acting on the corresponding sphere. Because of the exerted torque by the constraint forces, angular momentum is not conserved. In bipartite assemblies with many spheres, a single sphere might have multi- ple contacts and multiple simultaneously acting sliding forces. Each sliding k force contributes to the angular acceleration of the sphere. We define ~αi as the contribution to the angular acceleration ~αi due to the sliding force k at contact k, such that ~αi = k ~αi , the sum running over all contacts. We write Eq. (2.1) analogously forP a contact k between spheres i and j with the k k contributions ~αi and ~αj . For simplicity, we consider the moment of inertia of a sphere i proportional to its mass mi and its radius squared, as, for example, for homogeneous solid spheres. We find
k k miri~α mjrj ~α = ~0. (2.2) i − j Summing this equation over all contacts k, we obtain
simiri~αi = ~0, (2.3) i X 10 2.3. CONSTRUCTION OF 4DOF ASSEMBLIES
where the sum runs over all spheres and si is +1 if sphere i is red and 1 if − it is yellow. Using Eq. (2.3) we define
A~ := simiri~ωi, (2.4) i X where ~ωi is the angular velocity of the sphere i. Equation (2.3) shows that ∂A/∂t~ = ~0 , i.e., A~ is time invariant.
We derive one further time invariant quantity. Let ~xi be the position vector of the center of sphere i, where we choose the center of mass M~ of the entire assembly to be the origin of our coordinate system, i.e., M~ = ~0. We multiply k k (dot product) Eq. (2.2) by ~xi and obtain miri~αi ~xi mjrj ~αj ~xi = 0. Since k· − k · the vector ~xij = ~xj ~xi is perpendicular to ~αj , mjrj ~αj ~xij vanishes. We − k · k subtract it from the previous equation and get miri~α ~xi mjrj ~α ~xj = 0. i · − j · Analogously to the derivation of A~, we thereby define the time invariant quantity B as
B := simiri~ωi ~xi. (2.5) · i X 2.3 Construction of 4DOF Assemblies
Later we use the quantities A~ R3 and B R to predict the slip-free ∈ ∈ state of 4DOF assemblies. Now we identify how 4DOF assemblies can be constructed. The condition for the slip-free state of two spheres i and j is that their tangential velocities at their contact are equal. We formulate this sf sf sf as ~ω rixˆij = ~ω rjxˆji, where ~ω is the angular velocity of sphere i in i × j × i the slip-free state (sf), andx ˆij is the unit vector pointing from sphere i to sphere j. We rewrite this condition in analogy to Ref. [30] as
sf sf sjrj~ω siri~ω = cij~xij, (2.6) j − i R sf sf which defines cij that uniquely relates ~ωi to ~ωj . The slip-free state ∈ sf is uniquely defined by the angular velocity ~ωi of a single sphere i and the sf 3 parameters cij of all contacting spheres i and j. Since ~ω R , the number i ∈ of DOFs of the slip-free state is equal to three plus the number of DOFs of the set of cij’s, such that we only have a 4DOF assembly, if the set of cij’s is
11 CHAPTER 2. PREDICTION AND CONTROL OF SLIP-FREE ROTATION STATES IN SPHERE ASSEMBLIES a b
independent
Figure 2.3: Parameters describing the slip-free state. The cij parameters that uniquely relate the angular velocities of contacting spheres i and j in the slip-free state are independent for an open chain (a) and are all equal to one single c for a noncoplanar-4 loop (b), which is a 4DOF assembly. For cij = 0, the angular velocities of spheres i and j are antiparallel. restricted to a single DOF. Therefore a single pair of touching spheres with a single cij is the simplest 4DOF assembly. For any longer open chain of spheres, we have an independent cij for each contact as shown in Fig. 2.3(a), not resulting in a 4DOF assembly. Let us consider a bipartite loop with an even number of spheres N 4. ≥ Since for every contact Eq. (2.6) needs to hold, one finds the constraint
c12~x12 + c23~x23 + ... + cN1~xN1 = ~0, (2.7) where the indexes 1 to N are given to the spheres in consecutive order. The set of cij’s has N R DOFs, where R is the rank of the 3 N matrix − × (~x12 ~x23 ~xN1). R is equal to the number of linear independent columns ··· of the matrix and is either two, when all centers of the spheres are coplanar, or three otherwise. Thus, the cij’s of the contacts have N 2 DOFs in case all − the centers of the spheres are coplanar, and N 3 otherwise. The only loop − that is a 4DOF assembly is therefore a loop of four spheres whose centers are not coplanar (see e.g. Fig. 2.3(b)), which we denote as a noncoplanar-4 loop, where all cij’s have to be equal to a single parameter c. Starting with a pair of spheres or any other 4DOF assembly, one can construct more complex
12 2.3. CONSTRUCTION OF 4DOF ASSEMBLIES
4DOF assemblies by iterative extension in the following ways, as long as the assembly is bipartite, as explained in the following.
To extend a 4DOF assembly, one needs to ensure that all cij’s are equal to a single c. Starting from any 4DOF assembly, one finds for any pair of contacting spheres i and j that
sf sf sjrj~ω siri~ω = c~xij. (2.8) j − i Let us first relate the angular velocities of two spheres i and k which are not in contact, but are both touching a third sphere j. We sum Eq. (2.8) and an analogous equation for the contact between sphere j and k to find
sf sf fs sf sjrj~ωj siri~ωi + skrk~ω sjrj~ωj = c~xij + c~xjk, − k − (2.9) fs sf skrk~ω siri~ω = c~xik, k − i which shows the same relation as Eq. (2.8) for the non-contacting spheres i and k. Therefore, Eq. (2.8) is valid for any pair of spheres in the assembly. There are two options of how to extend a 4DOF assembly. One way is to connect two 4DOF assemblies A and B. Before they are connected, each of them has an independent parameter c which we call cA and cB, that describe the slip-free state according to Eq. (2.8), which holds for any two spheres i and j in the assembly. If we want to form another 4DOF assembly by connecting A and B, we need to make sure that the way of connecting enforces cA = cB for the slip-free state. This can be done by involving two spheres of each assembly to couple the two parameters cA and cB. To involve two spheres of each assembly we need to establish at least two contacts. If we connect sphere i and j from assembly A to sphere k and l from assembly B as shown in Fig. 2.4(a), and consider the fact that A and B are 4DOF assemblies with parameters cA and cB, respectively, we can establish the constraints
sf sf sjrj~ω siri~ω = cA~xij, (2.10) j − i sf sf slrl~ω sjrj~ω = cjl~xjl, (2.11) l − j sf sf skrk~ω slrl~ω = cB~xlk, (2.12) k − l sf sf siri~ω skrk~ω = cki~xki, (2.13) i − k
13 CHAPTER 2. PREDICTION AND CONTROL OF SLIP-FREE ROTATION STATES IN SPHERE ASSEMBLIES
a b
Figure 2.4: Ways of extending 4DOF assemblies. A 4DOF assembly can be iteratively extended using two rules of connecting as minimal requirements. Either connect two 4DOF assemblies A and B by establishing two contacts involving two spheres of each assembly such that the centers of the four spheres i, j, k, and l involved are not coplanar (a). Or integrate a single sphere k in a 4DOF assembly A by establishing two contacts between them such that the centers of involved spheres are not collinear (b).
14 2.3. CONSTRUCTION OF 4DOF ASSEMBLIES which if combined lead to a single constraint
cA~xij + cjl~xjl + cB~xlk + cki~xki = ~0. (2.14)
Only if one finds three linear independent vectors among ~xij, ~xjl, ~xlk, and
~xki, i.e., if the centers of the four spheres i, j, k, and l are not coplanar,
Eq. (2.14) enforces cA = cjl = cB = cki, such that the resulting assembly is a 4DOF assembly. Note that spheres i and j of the bipartite assembly A can be of any color, as long as the final assembly is also bipartite. Another way to extend a 4DOF assembly is to integrate a single sphere k by establishing contacts to two spheres i and j of a assembly A, as shown in Fig. 2.4(b). The constraints
sf sf sjrj~ω siri~ω = cA~xij, (2.15) j − i sf sf skrk~ω sjrj~ω = cjl~xjk, (2.16) k − j sf sf siri~ω skrk~ω = cik~xki, (2.17) i − k can be added to obtain
cA~xij + cjk~xjk + cik~xki = ~0. (2.18)
Only if the centers of the spheres i, j, and k are not collinear, Eq. (2.18) enforces cA = cjk = cik. In that case, the resulting assembly is a 4DOF as- sembly. Starting from any 4DOF assembly, one can construct more complex 4DOF assemblies iteratively using the two presented rules for connecting as minimal requirements. The formation of additional contacts during the process does not change the fact that the resulting assembly is a 4DOF assembly as long as it is bipartite. With these extension options as mini- mal requirements, one can even form the space-filling assembly presented in Ref. [30]. Let us now describe the slip-free state for 4DOF assemblies in a general way. Since Eq. (2.8) holds for any pair of spheres i and j in the assembly, we can express the angular velocity of any sphere i as a function of the angular velocity of a reference sphere j as
sf si sf ~ωi = sjrj~ωj + c~xji . (2.19) ri 15 CHAPTER 2. PREDICTION AND CONTROL OF SLIP-FREE ROTATION STATES IN SPHERE ASSEMBLIES
We now choose the reference sphere j not to be an actual sphere of the sf ~ assembly, but an imaginary reference sphere with angular velocity ~ωj = Ω, sj = rj = 1, and ~xj = ~0 to obtain the general expression
sf si ~ ~ωi = Ω+ c~xi , (2.20) ri where Ω~ R3 is a vectorial reference quantity. ∈
2.4 Prediction of the Final Slip-Free State
The slip-free state as written in Eq. (2.20) can be predicted using the time in- variant quantities A~ and B defined in Eq. (2.4)and(2.5). We write Eq. (2.20) for a sphere i, multiply it by simiri, and sum over all spheres to end up with A~ on the left hand side and to eliminate c from the right hand side, be- ~ ~ ~ cause i cmi~xi = cM = 0, since we defined the center of mass M of the ~ ~ entireP assembly to be the origin of our coordinate system (M = 0). We ~ ~ find Ω= A/M, with the total mass M = i mi. Second, we multiply (dot product) Eq. (2.20) written for sphere i byPsimiri~xi and sum over all spheres to end up with B on the left hand side and eliminate Ω~ on the right hand 2 side to find c = H/I, where I = mi ~xi . We then formulate the angular i | | velocities of the slip-free state asP a function of time invariant quantities only as ~ sf si A B ~ωi = + ~xi . (2.21) ri M I ! Equation (2.21) shows that the final slip-free state can be predicted using the time invariant quantities A~ and B defined in Eqs. (2.4) and (2.5). Sur- prisingly, the final state is independent of the type and strength of sliding friction. Different sliding forces merely lead to different kinetic pathways toward the slip-free state as as illustrated in Fig. 2.5, which in general also depend on the geometry of the assembly and the moments of inertia of the spheres. We show a noncoplanar-4 loop, a simple 4DOF assembly, relaxing from a random initial configuration toward the slip-free state predicted by
Eq. (2.21) for two different types of sliding forces. Any sliding force F~ij
16 2.4. PREDICTION OF THE FINAL SLIP-FREE STATE
start magnitudes of 100 4 sliding forces: 10⁻2 prop. to 10⁻4 3 constant 10⁻6 0 1 2 3 4 5
2
1 slip-free state 0 0 1 2 time 3 4 5
Figure 2.5: Different sliding forces merely lead to different kinetic pathways toward the slip-free state. 4DOF assembly that relaxes from randomly cho- sen initial angular velocities toward a slip-free state with sliding forces with magnitudes proportional to the difference in tangential velocities (solid) and sf with constant magnitudes (dashed). ∆ = ~ωi ~ω measures the devia- i | − i | tion from the slip-free state. Inset: ExponentialP decay of ∆ found for the proportional force. The final slip-free state is independent of the sliding forces in contrast to intermediate states with same ∆. Simulation time in 6 seconds with a time step of 10− s.
17 CHAPTER 2. PREDICTION AND CONTROL OF SLIP-FREE ROTATION STATES IN SPHERE ASSEMBLIES
Figure 2.6: Description of different slip-free states. 4DOF assembly in different slip-free states. For A~ = ~0 and B = 0 all axes of rotation are parallel 6 to A~. For B = 0 all axes of rotation of the spheres meet at the position 6 ~x = IA/~ (BM), and in particular for A~ = ~0 they meet at the center of mass. acting on sphere j due to contact with sphere i points in the direction oppo- site to the relative velocity ~vij = ~vj ~vi at the contact. We considered two − cases. First, F~ij = σ~vij, where the magnitude of each force is proportional − to the difference in tangential velocities, with σ = 3 and second, F~ij = σvˆij, − where the magnitude is constant, with σ =0.05 and withv ˆij being the unit vector along ~vij. Figure 2.6 pictures the role of A~ and B regarding the slip-free state predicted by Eq. (2.21). For B = 0 all axes of rotation are parallel to A~. For B =0 6 they all intersect at ~x = IA/~ (BM), and a sphere located at ~x would be at rest and would rotate faster the larger its distance to ~x is. Remarkably, if one blocks one sphere, generally not all spheres will stop rotating but the assembly will instead relax toward a slip-free state where all axes of rotation intersect at the center of the blocked sphere, changing the slip-free state as explained in detail in the following. From Eqs. (2.4),
18 2.5. CONTROL OF THE SLIP-FREE STATE
(2.5), and (2.21), one can derive the new A~ (A~new) and B (Bnew) from the previous (old) values (A~old and Bold) when blocking sphere i as described in the following. By applying a perturbation ∆~ωi to the angular velocity of a sphere i one can impose the changes
∆iA~ = simiri∆~ωi (2.22) and
∆iB = simiri∆~ωi ~xi. (2.23) · Remember that the origin of the position vector ~xi of the center of any sphere i is the center of mass of the entire assembly. Blocking a single sphere permanently and letting the assembly relax to a slip-free state has the same effect on A~ and B as applying a perturbation ∆~ωi that leads to sf ~ the sphere i being at rest in the slip-free state, i.e., ∆~ωi = 0. To ensure sf ~ ∆~ωi = 0 we find from ~ sf si Anew Bnew ~ ~ωi = + ~xi = 0, (2.24) ri M I ! that
A~new = MBnew~xi/I. (2.25) − From the fact that ∆iA~ ~xi =∆iB (compare Eqs. (2.22) and (2.23)), we find · using ∆iA~ = A~new A~old and ∆iB = Bnew Bold, that − − Bold A~old ~xi Bnew = − 2· . (2.26) 1+ M ~xi /I | | In contrast to blocking a single sphere, blocking two or more spheres at the same time will force every sphere to stop. Assemblies with more than four DOFs allow more than one sphere to be blocked without stopping all spheres.
2.5 Control of the Slip-Free State
First, we will explore the possibilities of controlling the slip-free state within our model in Sec. 2.5.1, and afterward consider a real system experimentally in Sec. 2.5.2.
19 CHAPTER 2. PREDICTION AND CONTROL OF SLIP-FREE ROTATION STATES IN SPHERE ASSEMBLIES
2.5.1 Control Within the Model
To impose any desired slip-free state of a 4DOF assembly within our model, one can apply instantaneous changes to two arbitrarily chosen angular ve- locities as we show here. We want to impose a desired change ∆A~ and ∆B by applying external changes ∆~ωi and ∆~ωj to the angular velocity of sphere i and j, respectively. Eqs. (2.22) and (2.23) show the changes ∆iA~ and
∆iB imposed by the change ∆~ωi. The individually imposed changes in A~ and B need to sum up to the desired change, i.e., ∆A~ =∆iA~ +∆jA~ and
∆B =∆iB +∆jB. We define ∆~ωi = ωi∆ˆωi, whereω ˆi is the unit vector of the external change ∆~ωi. From Eqs. (2.22) and (2.23) we find
∆A~ =∆iA~ +∆jA~ = simiriωi∆ˆωi + sjmjrj∆~ωj (2.27) and
∆B =∆iB +∆jB = simiriωi∆ˆωi ~xi + sjmjrj∆~ωj ~xj. (2.28) · ·
We multiply (dot product) Eq. (2.27) with the position vector ~xj pointing from the center of mass of the assembly to the center of sphere j and subtract Eq. (2.28) from it to find
∆A~ ~xj ∆B ωi = · − , (2.29) simiri∆ˆωi ~xij · such that we can choose any ∆ˆωi as long as ∆ˆωi ~xij = 0. From Eq. (2.29) · 6 we know ∆~ωi since ∆~ωi = ωi∆ˆωi and we obtain using Eq. (2.27) that
∆~ωj = (∆A~ simiri∆~ωi)/(sjmjrj). (2.30) − Furthermore, it is possible to determine all global quantities relevant for the final slip-free state by accessing not more than two spheres of a 4DOF assembly regardless its size. Compared to Eq. (2.21), relevant are the total mass M, the center of mass, i.e., the origin of the position vectors ~xi, and the parameter I. Starting from any slip-free state, one can separately apply two changes ∆1~ωi (first) and ∆2~ωi (second) to an accessible sphere i. One needs to wait after each change till the slip-free state is reached, and determine the
20 2.5. CONTROL OF THE SLIP-FREE STATE
sf sf corresponding changes ∆1~ωi and ∆2~ωi in angular velocity from the previous to the new slip-free state. Applying a change ∆~ωi to a sphere i leads to a sf change ∆~ωi of the angular velocity of sphere i between the previous to the new slip-free state. We use
~ sf si ∆iA ∆iB ∆~ωi = + ~xi (2.31) ri M I ! and Eqs. (2.22) and (2.23) to find
sf ∆~ω = mi(∆~ωi/M + (∆~ωi ~xi)~xi/I). (2.32) i · Let us derive two scalar equations from Eq. (2.32) by first squaring it (dot product) to find
sf 2 2 2 2 2 ∆~ωi /mi = ∆~ωi /M + 2(∆~ωi ~xi) /(MI)+ | | | | · (2.33) 2 2 2 (∆~ωi ~xi) ~xi /I · | | and second multiplying it (dot product) with ∆~ωi to find
sf 2 2 ∆~ω ∆~ωi/mi = ∆~ωi /M + (∆~ωi ~xi) /I. (2.34) i · | | ·
2 Combining the two we can eliminate the term (∆~ωi ~xi) and after some · rearrangements we find
sf 2 2 2 ∆~ωi /(mi ∆~ωi )= | | | | (2.35) 2 2 sf 2 1/M + (2/M + ~xi /I)(∆~ω ∆~ωi/(mi ∆~ωi ) 1/M). | | i | | − sf For a single change ∆~ωi and its induced change ∆~ωi we see in Eq. (2.35) that 2 we have the two unknown quantities M and ~xi /I. So with two changes | sf | sf ∆1~ωi and ∆2~ωi and their induced changes ∆1~ωi and ∆2~ωi one can find
M =(b + √b2 4ac)/(2a), − a = f2g1 f1g2 , b = f2 f1 , c = g2 g1 , | − | | − | | − | (2.36) sf 2 2 2 fn = ∆n~ω /(m ∆n~ωi ), | i | i | | sf gn =∆n~ω ∆n~ωi/(mi ∆n~ωi ). i · | |
21 CHAPTER 2. PREDICTION AND CONTROL OF SLIP-FREE ROTATION STATES IN SPHERE ASSEMBLIES
To locate the center of mass of the assembly we can use Eqs. (2.4), (2.5), and (2.21) to find a vector
sf ~x′ =∆~ω mi∆~ωi/M, (2.37) i i − that is parallel to the vector ~xi pointing from the center of mass to the center of sphere i. By applying an additional change ∆~ωj to a second accessible sphere j, one can use Eq. (2.37) to obtain a vector ~xj′ parallel to ~xj. The center of mass is the only point in space that can be reached both going along ~xi′ from the center of sphere i and going along ~xj′ from the center of sphere j, in the general case where ~xi′ is not parallel to ~xj′ . At last, we can solve Eq. (2.34) for I to find
2 sf 2 I = (∆~ωi ~xi) / ∆~ω ∆~ωi/mi ∆~ωi /M . (2.38) · i · −| | 2.5.2 Control in Experiment
Compare to our model as discussed in Sec. 2.5.1, a real system that is subject to e.g., rolling, torsion, and air friction would always come to rest from any initial rotation state. Nevertheless, one can preserve any desired slip-free state of Eq. (2.21) (compare Fig. 2.6) by preserving two angular velocities accordingly. How close the real stationary state will reach the desired state depends on various details of the assembly such as spatial arrangement, ma- terial, size, contact forces between spheres, and the fixing structure. We conducted a simple experiment as a first demonstration on how the reached stationary state might depend on the spatial arrangement. Two horizon- tally contacting pairs of hollow TPU spheres were stacked onto each other as shown in Fig. 2.7(a) with varying stacking angle α (compare Fig. 2.7(c)), forming a noncoplanar-4 loop (4DOF assembly) for 0◦ <α< 90◦, and a coplanar-4 loop (5DOF assembly) for α = 0◦. We fixed the positions of the spheres using ball rollers, allowing the spheres to rotate. The two lower spheres were forced to rotate with equal absolute angular velocity around the axis connecting their centers. When we force them to rotate in opposite directions as shown in Fig. 2.7(c), we impose a state where all rotation axes
22 2.5. CONTROL OF THE SLIP-FREE STATE
a b
amplifying mode
constant mode
c d
driving wheels constant mode amplifying mode
Figure 2.7: Tabletop experiment to compare with the model. (a) Two pairs of contacting TPU spheres were stacked onto each other and their positions were fixed using ball rollers. The two lower spheres were forced to rotate with equal absolute angular velocity using driving wheels attached to motors. (b) Ratio of the angular velocities of an upper and a lower sphere (experiment (symbols) in comparison with theory (lines)) versus different stacking angles α (compare (c)), for two different modes of rotation indicated in (c) (constant mode) and (d) (amplifying mode).
23 CHAPTER 2. PREDICTION AND CONTROL OF SLIP-FREE ROTATION STATES IN SPHERE ASSEMBLIES are parallel and the absolute angular velocities are equal, such that we call this the constant mode. When forcing both to rotate in the same direc- tion as shown in Fig. 2.7(d), we impose for α > 0◦ a state where all axes of rotation meet at the contact between the two lower spheres. Then the two upper spheres will rotate at a higher angular velocity as we have shown in Fig. 2.1; therefore, we call it the amplifying mode. Figure 2.7(b) shows the measured ratio of the angular velocities of an upper and a lower sphere (symbols) as well as the prediction by our model (lines). In the constant mode the experiment agrees well with our prediction for all angles α. In the amplifying mode, we find excellent agreement for α 60◦, 84◦ . For ≈ { } α 12◦, 24◦, 36◦ , the angular velocities of the upper spheres were on av- ≈ { } erage smaller than the predicted ones, and we did not observe smooth but partially jerky rotations becoming jerkier for decreasing α. Inthe amplifying mode, the spatial arrangement for low α triggers this deviation which orig- inates from the elasticity of the spheres. In our experimental setup, elastic deformation can lead to jerky motions by causing displacements of spheres and by the formation of non-negligible contact areas between spheres that can lead to torsional forces, as explained in the next paragraph. We give details on the construction of the experiment in Sec. 2.6.
As in the amplifying mode of operation shown in Fig. 2.8(a), elastic spheres can lead to torsional forces due to the formation of finite contact areas between spheres as shown in Fig. 2.8(b). Let us assume that the velocity of each contact point is proportional to its distance to the rotation axis of the sphere. If not all points of the contact area have the same distance to the rotation axis, the contact area shows an inhomogeneous velocity profile. This is the case for the contact areas between lower and upper spheres in the amplifying mode as shown in Fig. 2.8(c). The angular velocities are assumed to be the ones predicted by our theory and in this example, we have α = 36◦. To get a better view on the inhomogeneity of the velocity profiles, we plot the difference to the velocity in the center of the contact areas as shown in Fig. 2.8(d). There one can see that the stronger inhomogeneity of the upper contact area will lead to forces on both contact areas as shown in Fig. 2.8(e),
24 2.5. CONTROL OF THE SLIP-FREE STATE
a b
driving wheels
c d e
min max min max
Figure 2.8: Elasticity can lead to torsional forces. (a) Amplifying mode of operation: upper spheres (3,4) rotate faster than lower spheres (1,2). (b) the elasticity of spheres leads to a contact area (striped) between spheres instead of point-like contacts assumed in our theory. (c) Velocity profiles of the contact areas between sphere 4 (top) and sphere 1 (bottom). (d) Profile of the velocities after subtracting the velocity at the center of the contact areas. (e) Resulting forces acting on the contact areas lead to a torsional net force on the spheres (length of arrows are proportional to the velocity difference between touching points).
that sum up to a torsional net force acting on the spheres.
25 CHAPTER 2. PREDICTION AND CONTROL OF SLIP-FREE ROTATION STATES IN SPHERE ASSEMBLIES 2.6 Experimental Details
As spheres for the assembly, we used hollow TPU spheres with the trade name Super High Bounce Jump Ball with a 10 cm diameter. The positions of the spheres were fixed using bolt fixing ball transfer units (ball rollers) with 19 mm carbon steel balls manufactured by Alwayse Engineering Limited. For each sphere, two to three ball rollers were placed such that with all spheres inserted, all positions stay locked in operation. On the thread of the ball rollers, we placed springs with 37.4 N spring rate followed by a short brass pipe and a screw nut, such that the ball rollers could be fixed at the pipe, using boss head clamps and rods, and their positions could be slightly adjusted using the screw nut. In the case of a too large contact force between the ball roller and a sphere, the spring gives in. Due to the usual application of ball rollers in cargo transfer, they come greased. Since we want to prevent the spheres from slipping on another, we degreased the ball rollers using petroleum. They were one by one dipped into a cup of petroleum such that the ball compartment is covered. Each ball roller was gently patted and rolled on the bottom of the cup for about one minute. Then, before they were dried on air, the petroleum was shaken off and absorbed with household paper. All ball rollers were cleaned in this way four times, where the petroleum was replaced after having washed every ball roller once in it. To control the rotation of the two lower spheres, we used for each of them a 31.7 mm large foam rubber tire that we fixed, using a propeller hub, on a gear motor with nine turns per minute and a peak torque of 1.8 N when operating at 12 V. The positions of the wheels were fixed such that the contact forces between the wheels and the spheres were large enough to prevent slip during operation and small enough to prevent strong deformation of the spheres. Every operating setup was caught on video over five minutes. The angular velocities were measured by manual video analysis. Signs were drawn onto the spheres in advance to ease video analysis. We measured the angular velocity as the number of times a sphere turned during the five minutes divided by the exact number of passed seconds. We assumed the rotation axes of the spheres to be fixed over time, which turned out to be not always
26 2.7. FINAL REMARKS exactly the case especially for low values of the stacking angle α, but a reasonable assumption. To stack two or three pairs of spheres for a rather simple demonstration, the hollow TPU spheres and the ball rollers fixed in springs worked well. To build assemblies with more spheres, the TPU spheres will turn out to be too soft at some point, and due to compression in different directions depending on the operation mode, they will not suit anymore. For optimal slip-free operation, we suggest spheres to be as light and stiff as possible while having as much grip between spheres as possible.
2.7 Final Remarks
This study advances the understanding of the rotational degrees of freedom of assemblies and thus of the internal dynamics of shear deformation and seismic gaps. In a technological perspective, the ability to control the rota- tion state of an assembly of rotating spheres in contact is a newly discovered functionality with a general but yet unexplored potential. It is likely to find use in mechanics and robotics to control the orientation and rotation of spheres. The possibility to amplify the angular velocities of spheres along a certain contact network could be employed as an alternative to power trans- mission gears. Since the ability to control the rotation state is robust against changes in the spatial arrangement, as long as contacts are conserved, as- semblies allow for desired or inevitable displacements during operation.
27
Chapter 3
Construction of Self-Similar Space-Filling Sphere Packings in Three and Four Dimensions
3.1 Introduction
We generate space-filling packings using inversive geometry which coercively leads to exactly self-similar fractal packings. Note that there are further methods to construct space-filling fractal packings such as those produc- ing the Apollonian Gasket (compare Fig. 1.2), the Kleinian circle packings [24, 23], the random bipartite packings of Ref. [31], or generalizations of Apollonian packings [21, 22]. This work is motivated by previous studies on exactly self-similar space-filling bearings, as elaborated in the next para- graph. First, two families, F1 and F2, of two-dimensional disk bearings were found [28], having smallest loops of size four. Later, these two families were gen- eralized to smallest loops of any even size [29]. In both Refs. [28, 29], the packings are constructed in a strip geometry by iteratively applying con- formal transformations, namely reflections, translations, and inversions, to some initially placed disks. In contrast, Ref. [35] presents a way to con- struct a self-similar space-filling packing using inversive geometry only. In particular, they construct the 3D Apollonian Gasket which is based on the
29 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS geometry of a tetrahedron. The packing is constructed inside a sphere, un- like the space-filling disk packings constructed on a strip in Refs. [28, 29]. Later, this construction technique was generalized to other Platonic solids than the tetrahedron [25], leading to a total of five packings, of which one is a bearing. When applied in two dimensions, this approach turns out to generate the topologies of family F1 presented in Ref. [28], i.e., bipartite packings with smallest loops of size four. Note that any of these packings can be inverted as a whole to switch from the strip configuration to a packing enclosed by a circle and vice versa.
For this work, we carefully studied both of the families F1 and F2 presented in Ref. [28] for smallest loop size four and for arbitrary even smallest loop size in Ref. [29], to find a general method to construct all of them enclosed by a circle and using inversive geometry only, inspired by the construction methods in Refs. [35, 25]. In comparison to the construction on a strip, where different configurations generally differ in the length of the periodic unit cell, the construction inside a circle leads to configurations which are based on different regular polygons. This method can be straightforwardly extended to any higher dimension and we used it to find further packings in three and four dimensions. We care about four-dimensional packings because by cutting them with a three-dimensional hyperplane, a three-dimensional packing can be obtained. In that way, any 4D packing serves as a source for further 3D packings.
The order of content in this Chapter is the following. In Sec. 3.2, we give a basic idea of how we use inversive geometry to generate space-filling pack- ings. In Sec. 3.3, we provide the necessary knowledge about inversive ge- ometry that is needed to understand our work. In Sec. 3.4, we show the constraints which our construction method needs to fulfill to lead to a space- filing packing. In Sec. 3.5, we describe how to construct all two-dimensional packings of both families F1 and F2 enclosed by a circle and generalize our method to higher dimensions. In Sec. 3.6, we present the newly discov- ered packings and characterize them by estimating their fractal dimensions and analyzing their contact networks. Furthermore, we discuss in Sec. 3.7
30 3.2. BASIC IDEA OF GENERATING A PACKING how one can generate further variations of packings. At last, we give final remarks in Sec. 3.8.
3.2 Basic Idea of Generating a Packing
In two dimensions, each packing is constructed inside the unit circle, which can be seen as a hole that we aim to fill. We start by placing initial disks inside it. These disks and the unit circle hole itself act as seeds of the packing, out of which new disks will be generated by inversion. For that, a group of inversion circles is used. They form, together with the seeds, the generating setup of the packing as shown in Fig. 3.1a. The seeds are inverted at the inversion circles to generate new disks as shown in Fig. 3.1b. The newly generated disks can again be inverted to obtain further disks. This can be repeated infinitely many times till all space is filled as shown in Fig. 3.1c. Along the same line, one can generate space-filling packings in any higher dimension. To end up with a space-filling packing, the generating setup needs to fulfill certain constraints on how disks and inversion circles are placed. Before we discuss these constraints in Sec. 3.4, we will provide the necessary under- standing of inversive geometry in the following section.
3.3 Circle Inversion
We explain important properties of circle inversion considering a single in- version circle in Sec. 3.3.1 and multiple inversion circles in Sec. 3.3.2. We fur- thermore show how to simplify circle inversion mathematically in Sec. 3.3.3. Note that all properties explained in the following hold analogously for sphere inversions in 3D and in any higher dimension.
3.3.1 Basic Properties
Figure 3.2 shows the basic properties of circle inversion. If we invert a single point P at an inversion circle I as shown in Fig. 3.2a, the image P ′ will
31
3.3. CIRCLE INVERSION
The center of I is inverted to the point at infinity. Therefore, if C touches the center of I, it is mapped onto a line, i.e., onto a circle with infinite radius as shown in Figs. 3.2f and 3.2g. Finally, we treat every circle either as a disk or a hole, referring to the area inside or outside the circle, respectively. To distinguish between disks and holes mathematically, we assign a positive radius r > 0 to disks and a negative radius r < 0 to holes, such that the surface of both disks and holes is the circle with radius r . This convention is meaningful, since if | | the center of an inversion circle I lies inside a disk with radius r > 0, the inversion (mathematical details in Sec. 3.3.3) turns the disk inside out into a hole, i.e., the area inside the surface of the disk is mapped onto the area outside the surface of the hole (Fig. 3.2h).
3.3.2 Multiple Inversion Circles
A single inversion circle divides space into two sections that are mapped onto each other, the inside and the outside. The situation is more complex if multiple inversion circles are present. Let us deal with two inversion circles
I1 and I2.
If I1 and I2 do not overlap, any disk lying outside of both of them can be iteratively inverted at I1 and I2 an infinite number of times leading to in- finitely many new disks as shown in Fig. 3.3a. Like the initial disk itself, all its images will lie both inside the circle Tin and outside the circle Tout which are the only two circles that are tangent to the initial disk and perpendicu- lar to both I1 and I2. We can see that I1 and I2 together divide space into an infinity of non-overlapping areas, which we will call “sections”, that are mapped onto each other. I1 and I2 themselves are section borders. Besides, the other section borders that divide space are obtained by iteratively in- verting I1 and I2 at one another (dotted circles in Fig. 3.3a). Each of these section borders lies inside I1 or I2, such that the space outside both of them remains a single section. As shown in Fig. 3.3b, in the case where I2 lies inside I1 or vice versa, space is also divided into infinite sections, but some of the section borders lie outside of both inversion circles.
33 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS
a b c
d e f
g h
Figure 3.2: Examples of circle inversion. (a) Inversion of a single point P at an inversion circle I (dashed). The distance to the center of I is d for 2 point P and d′ = R /d for its inverse P ′, where R is the radius of I. (b) A circle C lying outside I leads to the inverse circle C′ lying inside and vice versa. (c) C touching I from the outside results in C′ touching I from the inside. (d) C intersecting I with an angle α leads to C′ intersecting I with an angle π α. (e) If C is perpendicular to I, C′ is identical to C. (f,g) C − touching the center of I results in C′ being a line, i.e., a circle with infinite radius. (h) If a circle includes the center of I, the inversion turns the circle inside out, i.e., the area inside the circle is mapped onto the area outside its inverse, as indicated by the arrow.
34 3.3. CIRCLE INVERSION
If I1 and I2 intersect, the intersecting angle α determines in how many sections space is divided. If α = nπ/m, where n and m are integers without common prime factors and 1 n 3.3.3 Mathematics of Circle Inversion Circle inversion is simplified by using inversion coordinates for circles and inversion circles. We generalize in the following to any dimension higher or equal to two. Therefore, we refer to n-spheres with n 1, where a circle ≥ and a sphere are a 1-sphere and a 2-sphere, respectively. The inversion coordinates (a1, a2, . . . , an+3) of an n-sphere are defined by its center ~x = (x1,...,xn+1) and its radius r as xi ai = , for i = 1,...,n +1 r { } 2 2 2 x1 + ... + xn+1 r 1 an+2 = − − , (3.1) 2r x2 + ... + x2 r2 +1 a = 1 n+1 − . n+3 2r They satisfy the relation a2 + ... + a2 a2 = 1 and one can express the 1 n+2 − n+3 usual parameters as 1 r = , a a n+3 n+2 (3.2) a−i xi = , for i = 1,...,n +1 . an+3 an+2 { } − 35 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS a b c d e f Figure 3.3: Examples of two inversion circles I1 and I2 (dashed). (a) If I1 and I2 do not overlap, they divide space into an infinity of non-overlapping sections, where the inversion circles are section borders themselves. The other section borders (dotted) are obtained by iteratively inverting I1 and I2 at one another. A disk D placed in the area both outside I1 and I2, which is a single section, leads to infinitely many nonoverlapping disks, one in each section. All disks lie outside Tout and inside Tin, which are the two circles tangent to D and perpendicular to both I1 and I2. (b) If an inversion circle lies inside another, the space inside and outside both of them is divided into infinite sections. (c) If I1 and I2 intersect with an angle α = nπ/m, where n and m are integers without common prime factors and 1 n I2. A disk placed inside one of the sections only, will result in a total of 2m disks, one in each section. (d) If a disk is placed on a section border but not perpendicular to it, overlapping disks will be generated. (e) A disk placed perpendicular to a section border, will result in a total of m disks. (f) α = 1: If α/π is not a rational number, space is divided into infinite infinitesimally small sections. Thus, any initially placed disk results in completely filling the space which is both inside the circle Tin and outside the circle Tout with infinitely many partially overlapping disks. 36 3.4. CONSTRAINTS ON THE GENERATING SETUP By using the inversion coordinates for an n-sphere, the inversion of a sphere (with coordinates ai) leads to the image (ai′ ) by (a′ , . . . , a′ )⊤ = M (a1, . . . , an+3)⊤, (3.3) 1 n+3 · where the superscript denotes the transpose, and the (n + 3) (n + 3) ⊤ × matrix M is defined by the coordinates Ai of the inversion sphere at which we invert at as M = I 2(A1,...,An+3)⊤ (A1,...,An+2, An+3), (3.4) − · − where I is the identity matrix. A more detailed derivation of the matrix M can be found in Ref. [76]. Note that we consider a sphere with r > 0 as filled, referring to the space inside the sphere, and a sphere with radius r < 0 as a hole, referring to the space outside the sphere of radius r . In | | the special case of an+2 = an+3, we have a half-space defined by the plane with normal vector (a1, ..., an+1) and distance an+2 from the origin, covering the space in the direction of the normal vector. 3.4 Constraints on the Generating Setup As we have seen in Fig. 3.1, a generating setup consists of seeds and inversion circles. To lead to a non-overlapping space-filling packing, the setup needs to fulfill two constraints, as we explain in the following two paragraphs. First, we need to avoid the partial overlap of disks (compare Fig. 3.3d). Partial overlap is avoided if every placed seed is perpendicular to all inter- secting section borders created by the inversion circles (compare Fig. 3.3e). We achieve this by the following strategy. As shown in Fig. 3.3c, if two inver- sion circles intersect with an angle α = nπ/m, where n and m are integers without common prime factors and 1 n 37 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS circles and at the same time on the outside of all others. This ensures two things. On one hand, as indicated in Fig. 3.4a, the seed will be perpen- dicular to all the section borders created by the inversion circles that it is perpendicular to, since these borders are a mapping of the inversion circles themselves. On the other hand, the combination of all the inversion circles that are outside of the seed will never create a section border that will inter- sect the seed as indicated in Fig. 3.4b. In addition to that, in the special case where by construction we have a mirror symmetry between two intersecting inversion circles, we allow intersecting angles of α =2π/m with any integer m 3. For even m, the area outside both inversion circles remains a single ≥ section. And for odd m, the area outside both inversion circles is divided into just two sections by a line along a mirror symmetry of the setup, which by default is perpendicular to all intersecting seeds, as shown in Fig. 3.4c. Second, to guarantee that the resulting packing is space-filling, the seeds and inversion circles together need to cover all space. This is a conjecture from previous studies [30, 76], which we will prove here with a detailed explanation. We have seen in Fig. 3.3 how multiple inversion circles divide space into sections. Figure 3.5 shows a generating setup with all sections created by its inversion circles. One can group these sections such that each section of a certain group can be mapped onto any other section of the same group by a sequence of inversions. In Fig. 3.5, we find two groups. If we cover a single section of a certain group, the whole group will eventually be covered by iterative inversions. In every generating setup, every group contains at least one section that is outside all inversion circles. This follows from the fact that any section that is inside an inversion circle, can be inverted at this inversion circle to be mapped onto a larger section outside of it. If this inverse of the section lies again inside any of the inversion circles, we can invert it again to map it outside of that inversion circle onto an even larger section. This can be repeated till we end up with a section that lies outside all inversion circles. Therefore if we cover all space outside all inversion circles with seeds, all space will eventually be covered by iterative inversions. 38 3.4. CONSTRAINTS ON THE GENERATING SETUP a b c Figure 3.4: Constraints on intersecting angles in the generating setup. (a) A seed (filled) is always placed perpendicular to some inversion circles (dashed) and therefore will be perpendicular to all section borders created by these inversion circles, since they are a mapping of the inversion circles themselves. (b) For any two intersecting inversion circles, we allow in general only intersecting angles of α = π/m with m 2, such that the area outside ≥ both inversion circles remains a single section. (c) For intersecting inversion circles with a mirror symmetry of the generating setup between them, we allow intersecting angles α =2π/m with any integer m 3, such that even ≥ in the case of odd m, the section border outside both inversion circles is a mirror line of the setup and is therefore by default perpendicular to all intersecting seeds. 39 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS Figure 3.5: The inversion circles (dashed) of the generating setup divide space into sections (section borders with radius r > 0.01 shown in white). The sections form groups, here two, colored in blue (dark gray) and yellow (light gray), where sections of the same group are mapped onto each other by a certain sequence of inversions. Each group contains at least one section (striped area) which is outside all inversion circles. Therefore, covering the space outside all inversion circles guarantees all sections to be eventually covered, leading to a space-filling packing. 40 3.5. HOW TO CONSTRUCT GENERATING SETUPS 3.5 How to Construct Generating Setups We first describe how we construct generating setups for all two-dimensional packings of families F1 and F2 in the Sec. 3.5.1 before we generalize to higher dimensions in Sec. 3.5.2. How to obtain the exact positions and radii of the setup elements is shown in detail in Sec. 3.5.3. 3.5.1 Construction of 2D Generating Setups Every setup is based on the geometry of a regular polygon, as the one shown in Fig. 3.6a. In addition to the unit circle hole as a first seed, we always have another set of seeds tangent to it, which we call primary seeds (blue colored in all figures). They lie in the direction of the vertices of the polygon. Furthermore, there is a set of inversion circles which lie in the direction of the faces of the polygon, and which have their centers outside the unit circle, such that we call them the outer inversion circles. As shown in Fig. 3.6a, the outer inversion circles are perpendicular to the unit circle and the nearest primary seeds, which already defines their size and positions. For a given regular polygon, they are identical for both packing families F1 and F2. In addition to the outer inversion circles, there are the inner inversion circles, that lie completely inside the unit circle. The size of the primary seeds de- pends on the inner inversion circles. The inner inversion circles are different for the two families. For F1, as shown in Fig. 3.6b, the inner inversion circles lie in the directions of the primary seeds. Figure 3.6c indicates the allowed intersecting angles between different inversion circles. The inner inversion circles are perpen- dicular to the nearest primary seeds and intersect the outer inversion circles with an angle β = π/(3+b) with an integer b 0. Nearest neighboring inner ≥ inversion circles intersect each other with an angle γ = 2π/(2 + c) with an integer c 0. This never leads to overlapping disks since there is a mirror ≥ symmetry between any two nearest neighbors of the inner inversion circles (compare Fig. 3.4c). In the special case of c = 0, inner inversion circles that intersect each other with the angle γ = π are actually identical circles, i.e., 41 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS they collapse to a single inversion circle. For F2, as shown in Fig. 3.6d, the inner inversion circles lie in directions between the primary seeds and the outer inversion circles. As shown in Fig. 3.6e, they are always perpendicular to the outer inversion circles. Inner inversion circles nearest to a certain primary seed intersect each other with an angle β =2π/(3+ b) with an integer b 0. The ones nearest to a certain ≥ outer inversion circle intersect each other with an angle γ =2π/(2+ c) with an integer c 0, collapsing to a single circle for c = 0. ≥ The number of vertices N of the polygon, the choice of family, and the parameters b and c together define the packing. Our integer parameters can be expressed by the ones used in Ref. [29], such that N =3+ n1, b = n2, and c = (l 4)/2, where l is the size of the smallest loops of the packing. − For some parameters, the generating setup does not cover the whole space, but one can add additional seeds to still end up with a space-filling packing. These seeds need to be perpendicular to all inversion circles surrounding the uncovered area. Such additional seeds can only be needed in the center of the packing or in the direction of the edges of the polygon as shown in Fig. 3.6f. Some more examples of generating setups are shown in Fig. 3.7. Note that both increasing b (compare Figs. 3.7c and 3.7d) and increasing c (compare Figs. 3.7b and 3.7c) can lead to more seeds being necessary to cover all space. 3.5.2 Generalization to Higher Dimensions In two dimensions, one can create an infinite number of distinct space-filling topologies for both families F1 and F2. In higher dimensions, we only find a finite number, because for some choices of basic shapes and parameters we are unable to generate a space-filling packing while avoiding overlapping spheres. Therefore the generalized method for higher dimensions only acts as a tool to search for possible generating setups. In analogy to the choice of a regular convex polygon as a base, one can choose a regular convex n-polytope when considering the n-dimensional space. In 3D, there are the Platonic Solids consisting of the tetrahedron, cube, octa- 42 3.5. HOW TO CONSTRUCT GENERATING SETUPS a b c d e f Figure 3.6: Construction of 2D generating setups. (a) Every setup is based on a regular polygon. The primary seeds are tangent to the unit circle and lie in the direction of the vertices of the polygon. The outer inversion circles lie in the direction of its edges. For F1, the inner inversion circles lie in the direction of the primary seeds (b) with intersecting angles as indicated in (c). For F2, the inner inversion circles lie in directions between the primary seeds and the outer inversion circles (d) with intersecting angles as indicated in (e). (f) Some packings need additional seeds, which can only lie in the direction of the edges of the polygon or in its center. 43 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS a b c d e f Figure 3.7: Different generating setups in the format (b,c) for F1 based on the square from (a) to (d): (0,0), (0,1), (0,3), (1,3), and for F2 based on the triangle in (e) (0,0) and (f) (1,1). 44 3.5. HOW TO CONSTRUCT GENERATING SETUPS hedron, dodecahedron, and icosahedron. In 4D, we have the 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, and 600-cell. In five and higher dimensions, there are only the three shapes that exist in any dimension n, i.e., the n-simplex (triangle (n=2), tetrahedron (n=3), 5-cell (n=4)), the n-cube (square (n=2), cube (n=3), 8-cell (n=4)), and the n-orthoplex (square (n=2), octahedron (n=3), 16-cell (n=4)). In 2D, we placed the outer inversion circles in the direction of the edges of the chosen polygon and the primary seeds in the direction of the vertices. In higher dimensions, we have more options to posi- tion the corresponding elements. In 3D, one can position them on either the vertices, edges, or faces. Note that some shapes are dual to each other, such as the cube and the octahedron in 3D. Therefore the positions of the faces of a cube are identical to the vertices of an octahedron, and vice versa. Placing the outer inversion spheres on the faces of a cube and the primary seeds on its vertices, is equal to placing the outer inversion spheres on the vertices of an octahedron and the primary seeds on its faces. Thus, to avoid finding each generating setup twice because of shape dualities, we consider every shape but place the primary seeds always on lower dimensional elements than the outer inversion spheres. The vertices are the 0-dimensional ele- ments, followed by one-dimensional edges, two-dimensional faces, etc. This way, in 3D and 4D, it happens that all possible generating setups have the primary seeds positioned at the vertices of the chosen shape, and the outer inversion spheres therefore at edges, faces, or cells (only for 4D). In every generating setup in three or more dimensions, the outer inversion spheres will intersect each other. Therefore, in contrast to 2D, one addition- ally needs to check the intersecting angles of all intersecting outer inversion spheres. All intersecting angles α =2π/m with an integer m 3 are allowed ≥ (compare Fig. 3.4c), since we have a mirror symmetry by default between any pair of outer inversion spheres. Figure 3.8 shows examples of the in- tersecting angles of outer inversion spheres. If a forbidden angle exists, no generating setup can be derived. After the choice of positions for the primary seeds and outer inversion spheres, one can for both F1 and F2, choose a combination of b and c what 45 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS Figure 3.8: Outer inversion spheres are only allowed to intersect each other with angles α = 2π/m with m 3 being an integer (check-mark). For ≥ non-integer m (cross), no packing can be constructed. From left to right: Outer inversion spheres and unit sphere hole (central) of setups based on the cube, octahedron, and icosahedron, with outer inversion spheres at the faces of the shape and primary seeds at the vertices. 46 3.5. HOW TO CONSTRUCT GENERATING SETUPS defines the inner inversion spheres, the primary seeds, and all additional seeds, such that the whole setup is defined. A complete 3D generating setup can be seen in Fig. 3.9a, with Fig. 3.9b showing only the inversion spheres and Fig. 3.9c only the seeds. The possible positions for additional seeds, which are needed to cover potentially uncovered space, are defined by the positions of the primary seeds. For symmetric reasons, they can only lie in the center or in the directions of the edges, faces, etc., of the convex polytope that has the primary seeds as vertices, as indicated in Fig. 3.9d. A central seed needs to be perpendicular to the inner inversion spheres (Fig. 3.9e), whereas the other additional seeds need to be perpendicular to the inner and outer inversion spheres (Figs. 3.9f and 3.9g). After calculating the sizes and positions of all elements in the generating setup, one needs to check if all the constraints, that ensure that a space-filling packing can be generated, are fulfilled: Seeds are not allowed to intersect, what in contrast to 2D can happen in higher dimensions. Furthermore, one needs to check all intersecting angles between inversion spheres, since in addition to the defined angles by b and c, intersections of not nearest neighbors of inversion spheres with a forbidden angle might exist. If all these constraints are fulfilled, a space-filling packing can be generated as the one shown in Fig. 3.9h. The search for generating setups for any choice of positions of outer inversion spheres and primary seeds can be started with parameters b = c =0. Ifa valid setup is found, one can continue the search by increasing either b or c. At some point, additional seeds are needed (compare Fig. 3.7). While increasing b or c further, some seeds will finally overlap each other, and one knows that no further setups can be found by further increasing the parameters. 3.5.3 Determine Positions and Radii of Setup Elements Apart from simple trigonometry we will in many calculations use the relation for two intersecting spheres, which is valid in any dimension, that states 2 2 2 d = r1 + r2 +2r1r2 cos α, (3.5) 47 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS a b c d e f g h Figure 3.9: (a) Complete 3D generating setup based on the octahedron with outer inversion spheres at its faces with b = 0 and c = 4. (b) Inversion spheres only. (c) Seeds only. (d) The convex polytope that has the primary seeds as vertices defines possible positions for additional seeds. They can only be at the center or in the direction of the edges, faces, etc., of the poly- tope. (e) An additional seed in the center would need to be perpendicular to the inner inversion spheres. Other additional seeds in the direction of the edges (f) or faces (g) of the polytope in (d) need to be perpendicular to both inner and outer inversion spheres. (h) Resulting packing with lowest considered radius rmin =0.005. Unit sphere visualized as an open shell and some spheres removed to allow looking inside the packing. 48 3.5. HOW TO CONSTRUCT GENERATING SETUPS Figure 3.10: Solving for the details of the outer inversion spheres: In the right triangle shown, we find that the distance dout =1/ cos α =1/(ˆxout xˆs) · 2 and the radius rout = d 1 out − p where d is the distance between the centers of two spheres with radii r1 and r2 that intersect with an angle α. For spheres perpendicular to each other Eq. (3.5) simplifies to 2 2 2 d = r1 + r2. (3.6) Every element of the setup is defined by the unit vector of its position, the distance of its center to the center of the unit sphere, and its radius. By choosing a regular polygon on which we base our generating setup, we define the unit vectors of the outer inversion spheres and the primary seeds. The distance of the outer inversion spheres dout and their radius rout is defined, given the fact that they need to be perpendicular to the unit sphere and the nearest primary seeds. In the right triangle shown in Fig. 3.10, we find dout = 1/ cos α = 1/(ˆxout xˆs), wherex ˆout andx ˆs are the unit vectors of an · outer inversion sphere and a nearest neighboring primary seed, respectively. Because the outer inversion sphere is perpendicular to the unit sphere, we 2 find rout = d 1. out − Next, we wantp to solve for the inner inversion spheres, which are different for the two families F1 and F2. For F1, the unit vectors of the inner inversion spheres are identical to the 49 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS a b Figure 3.11: Solving for the details of the inner inversion spheres for F1: 2 2 2 In (a) we find for d1 that d1 =2rin(1+cos γ)= din(1 xˆin1 xˆin2 ). In (b) we 2 2 2 2− 2· 2 2 find for d2 that d = d + d 2doutdinxˆin xˆout = r + r +2r r cos β. 2 out in − · out in out in ones of the primary seeds. One can express the distance d1 between two nearest inner inversion spheres, as shown in Fig. 3.11a, in two different ways to get the equation 2 2 2 (d =) 2r (1 + cos γ)= d (1 xˆin xˆin ), (3.7) 1 in in − 1 · 2 wherex ˆin1 andx ˆin2 are the unit vectors of two nearest inner inversion spheres. From the fact that an inner inversion sphere intersects its closest outer in- version spheres with an angle β, one can express the distance d2 between them shown in Fig. 3.11b in two different ways to get the equation 2 2 2 2 2 2 2 (d =) d + d 2doutdinxˆin xˆout = r + r +2r r cos β, (3.8) 2 out in − · out in out in wherex ˆin andx ˆout are the unit vectors of a closest pair of inner and outer inversion spheres. One can find rin and din from Eqs. (3.7) and (3.8). For F2, the unit vector of an inner inversion spherex ˆin is a combination of the one of its nearest outer inversion spherex ˆout and the one of its nearest primary seedx ˆs such that we can write xˆin = pxˆout + qxˆs. (3.9) 50 3.5. HOW TO CONSTRUCT GENERATING SETUPS The condition for this vector to be a unit vector gives us the equation 2 2 2 (ˆx )= p + q +2pqxˆout xˆs =1. (3.10) in · Every inner inversion sphere is perpendicular to the nearest outer inversion sphere such that 2 2 2 (doutxˆout dinxˆin) = r + r , (3.11) − out in wherex ˆin andx ˆout are the unit vectors of a closest pair of inner and outer inversion spheres. Using Eq. (3.9), this gives us 2 2 2 2 d + d 2doutdin(p + qxˆout xˆin)= r + r . (3.12) out in − · out in From the fact that two inner inversion spheres that are nearest neighbors of an outer inversion sphere intersect with an angle γ, we can express the distance d3 shown in Fig. 3.12a in two different ways as 2 2 2 (d =) (~xin,s ~xin,s ) =2r (1 + cos γ), (3.13) 3 1 − 2 in with ~xin,s1 and ~xin,s2 being the unit vectors of two inner inversion spheres that are closest to an outer inversion sphere. Since ~xin,s1 = din(pxˆout + qxˆs1 ) and ~xin,s2 = din(pxˆout + qxˆs2 ), wherex ˆs1 andx ˆs2 are the unit vectors of two primary seeds that are nearest to an outer inversion sphere with the unit vectorx ˆout, we find from Eq. (3.13) that 2 2 2 d q (2 2ˆxs xˆs )=2r (1 + cos γ). (3.14) in − 1 · 2 in In analogy to the derivation of Eq. (3.14), we can derive a similar equation for the angle β by expressing the the distance d4 as shown in Fig. 3.12b in two different ways to finally find 2 2 2 d p (2 2ˆxout xˆout )=2r (1 + cos β). (3.15) in − 1 · 2 in One can now find p, q, rin, and din from the four Eqs. (3.10), (3.12), (3.14), and (3.15). The unit vectorx ˆin can then be found from Eq. (3.9). After solving for the inner inversion spheres, we can find the radius for the primary seeds rs. For both F1 and F2, we get from the fact that every primary seed is perpendicular to a closest inner inversion sphere the equation 2 2 2 2 2 r + r =(dsxˆs dinxˆin) = d + d 2dsdinxˆs xˆin, (3.16) s in − s in − · 51 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS a b Figure 3.12: Solving for the details of the inner inversion spheres for F2: 2 2 2 (a) For d3 we find d3 =(~xin,s1 ~xin,s2 ) =2rin(1+cos γ). (b) For d4 we find 2 2 2− d =(~xin,out ~xin,out ) =2r (1 + cos β). 4 1 − 2 in where ds is the distance from the center of a primary seed to the center of the unit sphere. Since the primary seeds are tangent to the unit sphere, we have ds =1 rs, which together with Eq. (3.16) can be solved for ds and rs. − Finally there might be uncovered space that can be filled by additional seeds. An additional seed in the center of the packing is needed if the inner inversion spheres do not cover it, i.e., for rin < din. Since this seed would need to be perpendicular to the inner inversion spheres, its radius is defined 2 2 as rcenter = d r . Further additional seeds might be needed between in − in inner and outerp inversions spheres. They can only lie in the directions of the edges, faces, etc., of the convex polytope whose vertices are at the positions of the primary seeds. For a given unit vectorx ˆas of the position ~xas = dasxˆas of such an additional seed, one can find its distance das and its radius ras from the fact that the seed would need to be perpendicular to the closest inner and outer inversion spheres. From the fact that the seed is perpendicular to a closest inner inversion sphere at position ~xin = dinxˆin and a closest outer inversion sphere at position ~xout = doutxˆout, we find 2 2 2 2 2 r + r =(~xas ~xin) = d + d dasdinxˆas xˆin, (3.17) as in − as in − · 2 2 2 2 2 r + r =(~xas ~xout) = d + d dasdoutxˆas xˆout. (3.18) as out − as out − · The system of equations (3.17) and (3.18) has a unique solution for das > 0 52 3.6. DISCOVERED PACKINGS and ras > 0, if there is uncovered space between the inner and outer inversion spheres alongx ˆas. 3.6 Discovered Packings We searched for generating setups in three and four dimensions. Some gen- erating setups turned out to give the exact same packings and some lead to different packings but identical topologies. For any two different pack- ings with equal fractal dimension, i.e., overlapping confidence intervals of the fractal dimension estimate, we checked if they are the same topology or not. To do so, we made a topological comparison of the two corresponding generating setups, as we explain in detail in Sec. 3.6.3. With our general- ized method, we find 54 generating setups in 3D which lead to 34 distinct topologies, of which 29 are new discoveries. In 4D, we find 29 generating setups leading to 13 distinct topologies, none of them reported before. We characterize the packings of all generating setups in different ways to show the topological differences. We determine the fractal dimension as described in Sec. 3.6.1. In Sec. 3.6.2, we analyze the contact network where we check for isolated spheres, count the number of connected clusters, find the smallest loops of each cluster, and check if clusters are bipartite. An overview of all found generating setups with all characterizations can be found in Table 3.1 and 3.2 for 3D and 4D, respectively. 3.6.1 Fractal Dimension First, we describe how to generate packings down to a smallest radius com- putationally efficiently, and then we describe how to numerically estimate the fractal dimension of the generated packing. Starting with the seeds, one can invert every of them at every inversion sphere of the generating setup. One can iteratively repeat that procedure with all newly generated spheres. But one needs to take care of not produc- ing any sphere that is already present. If one inverts a sphere at an inversion sphere that is perpendicular to the sphere, the generated sphere is identical 53 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS to the original sphere. Additionally, different sequences of inversions can lead to the same sphere. Fortunately, there is a simple trick to avoid generating any sphere twice which we adopted from Ref. [35]. One can ensure to generate each sphere only once by only inverting a sphere at an inversion sphere if the inverse has its center inside the target region of the corresponding inversion sphere. The target regions of the inversion spheres are such that they are regions inside the corresponding inversion spheres such that no two target regions overlap but their union equals the union of all inversion spheres. A simple way to define target regions for a set of ordered inversion spheres is to define the target region of the first inversion sphere as the whole inside of it, and the target region of every following inversion sphere as the inside of the inversion sphere minus the overlap with previous inversion spheres. Numerically, one can neglect the surface of the inversion spheres, since only infinitely small spheres could end up having their center on the surface of an inversion sphere, given that spheres can only be perpendicular to inversion spheres. With this method, the generation of all spheres larger than a certain small- est radius becomes a simple branching process where every new inverse of a sphere is smaller than the original, so one can cut a branch if all possible in- versions lead to spheres that are smaller than the considered smallest radius. This allows to calculate the functions to estimate the fractal dimensions as described in Sec. 3.6.1 computationally using very few memory. After generating a packing down to a smallest radius, the fractal dimension can be estimated from the total number of spheres N(r), their cumulative surface s(r), or the remaining porosity p(r) of a packing of spheres with a radius larger r. The unit sphere hole needs to be treated as a sphere of radius one with the corresponding surface but a negative volume, i.e., one needs to add its volume to the remaining porosity. The functions N(r), s(r), and p(r) follow the asymptotic behaviors df D 1 df D df N(r) r− , s(r) r − − , p(r) r − , (3.19) ∼ ∼ ∼ where df is the fractal dimension and D are the dimensions of space [15, 16, 17, 33, 35]. An estimate dˆf for the fractal dimension can be obtained from 54 3.6. DISCOVERED PACKINGS the slope of these functions on a double logarithmic scale as the one shown in Fig. 3.13. We extract estimates on the intervals [r, r√e], which we move toward lower r to see the fluctuations of dˆf (r) to judge its accuracy, as shown in Fig. 3.14. This approach is also used in Ref. [35]. We further use a way to combine different estimates to improve accuracy. From the asymptotic behavior in Eq. (3.19), we assume that the errors ∆dˆf = dˆf df and a a − ∆dˆf = dˆf df on two estimates dˆf and dˆf based on a certain interval of b b − a b A df B df two different functions a(r) r − and b(r) r − , respectively, relate ∼ ∼ as ∆dˆ ∆dˆ fa = fb . (3.20) A df B df − − ˆ We use Eq. (3.20) to define a combined estimate dfa&b based on both func- tions a(r) and b(r) as B dˆ ˆ ˆ ˆ fb dfa&b = xdfa + (1 x)dfb , with x = − , (3.21) − B A + dˆf dˆf − a − b which mostly shows a smoother behavior, as shown in Fig. 3.14. Finally, we take from all estimate functions dˆf (r) the one with the least variability ∆ in the interval [rmin,rmine], and take dˆf (rmin) 5∆ as our ± confidence interval for our best estimation as shown in Fig. 3.15. The lowest 10 7 considered radius was rmin = e− and rmin = e− for 3D and 4D, respectively. Our best estimates for all our 3D and 4D packings can be found in decreasing order in Figs. 3.16 and 3.17, and in the summarizing Tables 3.1 and 3.2. For one of the previously known packings, the 3D Apollonian Gasket, we show the more accurate estimation from Ref. [35], which lies within our determined confidence interval. 3.6.2 Contact Network Depending on the spatial arrangement of the seeds and inversion spheres in the generating setup, the resulting packing forms a single or multiple clusters, where a cluster is a connected network of touching spheres. In particular, noncontacting seeds can lead to isolated spheres that are not in contact with any other spheres. 55 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS A B Figure 3.13: Total number of spheres N, cumulative surface s, and re- maining porosity p as a function of the lowest considered radius r for the previously known 3D packing of Ref. [30]. From the slope in a double loga- rithmic scale one can extract estimates for the fractal dimensions. Estimates from moving intervals from A to B can be found in Fig. 3.14. 56 3.6. DISCOVERED PACKINGS A B Figure 3.14: Estimates dˆf of the fractal dimensions based on the single functions N(r), s(r), and p(r), and combined estimates based on pairs of these functions. Estimates are extracted from moving intervals from A to B shown in Fig. 3.13, for the previously known 3D packing of Ref. [30]. 57 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS r e2 r e r min min min d *+5 f Δ d * f Δ d *-5 f Δ Figure 3.15: As our best estimate (here df∗ ), we take of all estimate functions (compare Fig. 3.14), the one that shows the least variability ∆ in the radius interval [rmin,rmine] (here the function based on the combination of s and p from Fig. 3.14), and take the confidence interval dˆf (rmin) 5∆ as our best ± estimate. This is the previously known 3D packing of Ref. [30]. 58 3.6. DISCOVERED PACKINGS Figure 3.16: Ranked fractal dimension estimates of 3D packings. The calculated confidence intervals are smaller than the symbol sizes. 59 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS Figure 3.17: Ranked fractal dimension estimates of 4D packings. The calculated confidence intervals are smaller than the symbol sizes. 60 3.6. DISCOVERED PACKINGS We characterized the contact network of each packing by counting the num- ber of connected clusters, determine the size of their smallest loops, check if clusters are bipartite, and finally check for isolated spheres, as explained in detail in the following paragraphs. Figure 3.18 shows two packings with different contact networks. All results can be found in the summarizing Ta- bles 3.1 and 3.2. We found that most topologies only have one connected cluster (31 of 34 in 3D and 10 of 13 in 4D). Besides, there are 3 two-cluster topologies in 3D, and 2 two-cluster topologies and even one with three con- nected clusters in 4D. Isolated spheres are present only in topologies with a single bipartite cluster (in 7 of 34 in 3D and in 3 of 13 in 4D). Most clusters are bipartite (62% in 3D and 88% in 4D). Note that in addition to a single previously known exactly self-similar bearing, which here we specify as a topology merely consisting of a single bipartite cluster, we found another 10 and 5 bearings in 3D and 4D, respectively. The number of connected clusters can be derived directly from the generating setup. Each cluster of touching seeds will lead to a connected cluster in the packing. An isolated seed will lead to isolated spheres in the packing only if it is not tangent to any inversion sphere. Otherwise, it will form a cluster with its inversions. To find the smallest loop size of clusters, we generated each packing up to a certain generation of spheres, i.e., up to a certain number of successive inversions starting from the seeds, such that every seed is a part of a closed loop. Every loop can be mapped onto a loop where all pairwise touching spheres are at most one generation apart. Therefore, by iterative inversion of all seeds and their successive images, the smallest loop that a certain seed is part of will be closed first during the generation process. For every seed, we find the first closed and therefore smallest loop that it belongs to, and from that we derive the smallest loops of the clusters. If all smallest loops of different seeds of a cluster are even, the cluster is bipartite, since one can divide the generated spheres into two groups A and B, such that A-spheres only touch B-spheres and vice versa. Then, every contact that appears if more spheres are generated can be mapped onto a 61 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS + = + = Figure 3.18: Two packings with different contact networks. Top: packing consisting of a bipartite cluster with smallest loop size six (left) and isolated spheres (middle). Bottom: packing consisting of two clusters, a bipartite one with smallest loop size twelve (left) and a non-bipartite one with smallest loop size three (middle). contact between a previously generated A-sphere and B-sphere. 62 3.6. DISCOVERED PACKINGS Table 3.1: Summary of 3D generating setups. Base shape of each setup is a Platonic Solid with number of faces shown in column B. The position of the outer inversion spheres is given in column I and can be either the edges (E) or the faces (F) of the base shape. Family number F and parameter b and c in corresponding columns. Smallest loop size of clusters with letter (b) for bipartite clusters in column LS. Presence of isolated spheres indi- cated by cross-marks in column O. Estimate of fractal dimension and rank R in comparison with all packings (R=1 for highest fractal dimension) in corresponding columns. Some setups lead to identical topologies which are assigned a reference number in column T, with numbers in brackets to indi- cate different packings of a single topology. Bearings, i.e., packings merely consisting of a single bipartite cluster, have a cross-mark in column BR. Last column RF identifies previously discovered packings, where AG stands for Apollonian Gasket, BP stands for the bipartite packing of Ref. [30], and 6, 8, and 12 stand for the packings of Ref. [25] based on the cube, octahedron, and dodecahedron, respectively. B I F b c LS IS fractal dim. R T BR RF 4 E 1 0 0 3 2.52638 5e-6 29 1 6 ± 4 E 1 0 1 3 2.53706 2e-5 28 ± 4 E 1 0 2 3 2.51613 2e-5 30 ± 4 E 1 0 3 3 2.50002 6e-5 32 ± 4 E 1 0 4 3 2.4739465 1e-7 34 2 AG ± 4 E 2 0 0 3 2.55863 2e-5 25 3 ± 4 E 2 0 1 3 2.55438 8e-5 26 ± 4 F 1 0 0 3 2.4739465 1e-7 34 2 AG ± 4 F 1 0 1 4b 2.71066 2e-5 13 X ± 4 F 1 0 2 4b 2.76236 5e-5 6 4(1) X ± 4 F 1 0 3 4b X 2.76625 5e-5 4 ± 4 F 1 0 4 4b 2.692627 2e-6 15 X ± 4 F 2 0 0 3 2.4739465 1e-7 34 2 AG ± 4 F 2 0 1 4b 2.73543 1e-4 8 X ± 63 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS B I F b c LS IS fractal dim. R T BR RF 4 F 2 0 2 4b X 2.72307 9e-5 12 5(1) ± 4 F 2 0 3 4b X 2.66723 5e-5 16 6(1) ± 4 F 2 0 4 3 2.61 2e-5 20 ± 6 F 1 0 0 3 2.52638 5e-6 29 1 6 ± 6 F 1 0 1 4b 2.76235 3e-5 6 4(2) X ± 6 F 1 0 2 4b,12b 2.75823 3e-5 7 7(1) ± 6 F 2 0 0 3 2.55863 2e-5 25 3 ± 6 F 2 0 1 4b X 2.723057 3e-6 12 5(2) ± 6 F 2 0 2 4b,6b 2.65707 2e-5 18 8(1) ± 8 E 1 0 0 4b 2.61496 2e-5 19 9(1) X ± 8 E 1 0 1 6b X 2.73397 2e-5 9 10(1) ± 8 E 1 0 2 8b X 2.71055 4e-5 14 11(1) ± 8 E 1 0 3 10b X 2.66319 3e-5 17 12(1) ± 8 E 1 0 4 3,12b 2.60799 6e-5 21 13(1) ± 8 E 1 1 0 4b 2.588191 5e-6 22 14 X BP ± 8 E 1 1 1 4b 2.61496 2e-5 19 9(2) X ± 8 E 1 1 2 3 2.58747 2e-5 23 ± 8 E 2 0 0 4b 2.730156 5e-6 10 15 X ± 8 E 2 0 1 4b,6b 2.65707 2e-5 18 8(2) ± 8 E 2 1 0 4b 2.588191 5e-6 22 14 X BP ± 8 F 1 0 0 4b 2.588191 5e-6 22 14 X BP ± 8 F 1 0 1 6b 2.793143 5e-6 3 X ± 8 F 1 0 2 8b 2.8841 5e-5 1 X ± 8 F 1 0 3 10b X 2.850875 7e-6 2 ± 8 F 1 0 4 4b,12b 2.75824 4e-5 7 7(2) ± 8 F 1 1 0 3 2.488006 8e-6 33 16 8 ± 8 F 1 1 1 4b 2.724834 9e-6 11 X ± 8 F 1 1 2 4b 2.730156 5e-6 10 15 X ± 8 F 2 0 0 4b 2.61496 2e-5 19 9(2) X ± 8 F 2 0 1 6b X 2.73398 5e-5 9 10(2) ± 64 3.6. DISCOVERED PACKINGS B I F b c LS IS fractal dim. R T BR RF 8 F 2 0 2 8b X 2.71055 8e-5 14 11(2) ± 8 F 2 0 3 10b X 2.66319 6e-5 17 12(2) ± 8 F 2 0 4 3,12b 2.608 2e-4 21 13(2) ± 8 F 2 1 0 3 2.488006 8e-6 33 16 8 ± 8 F 2 1 1 4b 2.588191 5e-6 22 14 X BP ± 8 F 2 1 2 3 2.5404 3e-4 27 ± 12 F 1 0 0 3 2.51142 4e-5 31 12 ± 12 F 1 0 1 4b 2.76624 4e-5 5 X ± 12 F 2 0 0 3 2.58594 2e-5 24 ± 12 F 2 0 1 4b X 2.66722 2e-5 16 6(2) ± 65 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS Table 3.2: Summary of 4D generating setups. Base shape of each setup is a convex regular 4-polytope with number of cells shown in column B. The position of the outer inversion spheres is given in column I and can be either the edges (E), faces (F), or cells (C) of the base shape. Family number F and parameter b and c in corresponding columns. Smallest loop size of clusters with letter (b) for bipartite clusters in column LS. Presence of isolated spheres indicated by cross-marks in column O. Estimate of fractal dimension and rank R in comparison with all packings (R=1 for highest fractal dimension) in corresponding columns. Some setups lead to identical topologies which are assigned a reference number in column T, with numbers in brackets to indicate different packings of a single topology. Bearings, i.e., packings merely consisting of a single bipartite cluster, have a cross-mark in column BR B I F b c LS IS fractal dimension R T BR 5 F 1 0 0 4b 3.6807 0.0009 10 1(1) X ± 5 F 1 0 1 6b X 3.7818 0.0009 6 2(1) ± 5 F 1 0 2 4b,8b 3.6872 0.0003 9 3(1) ± 5 F 2 0 0 3 3.59591 0.00002 13 4(1) ± 5 F 2 0 1 4b 3.70695 0.00002 8 5(1) X ± 8 F 1 0 0 4b 3.66379 0.00004 11 6 X ± 8 F 1 0 1 4b 3.6807 0.001 10 1(2) X ± 8 F 2 0 0 3 3.65233 0.00005 12 7 ± 16 E 1 0 0 4b 3.66379 0.00004 11 6 X ± 16 E 2 0 0 3 3.65233 0.00005 12 7 ± 16 F 1 0 0 4b 3.70695 0.00003 8 5 X ± 16 F 1 0 1 6b 3.8995 0.001 1 X ± 16 F 1 0 2 8b X 3.8602 0.0004 4 ± 16 F 1 1 0 3 3.59591 0.00003 13 4 ± 16 F 1 1 1 4b X 3.7868 0.0006 5 8(1) ± 16 F 2 0 0 3 3.59591 0.00003 13 4 ± 16 F 2 0 1 4b 3.70695 0.00003 8 5 X ± 66 3.6. DISCOVERED PACKINGS B I F b c LS IS fractal dimension R T BR 24 F 1 0 0 4b 3.6806 0.0002 10 1(3) X ± 24 F 1 0 1 6b X 3.7816 0.0003 6 2(2) ± 24 F 1 0 2 4b,8b 3.6872 0.0004 9 3(2) ± 24 F 1 1 0 4b 3.66379 0.00004 11 6 X ± 24 F 1 1 1 4b 3.6807 0.0003 10 1(4) X ± 24 F 2 0 0 4b 3.66379 0.00004 11 6 X ± 24 C 1 0 0 4b 3.66379 0.00004 11 6 X ± 24 C 1 0 1 6b 3.888 0.002 2 X ± 24 C 1 0 2 8b,8b,8b 3.861 0.0006 3 ± 24 C 2 0 0 3 3.65233 0.00005 12 7 ± 24 C 2 0 1 4b X 3.787 0.002 5 8(2) ± 24 C 2 0 2 4b,4b 3.71673 0.00005 7 ± 3.6.3 Topological Comparison To judge if two different packings are the same topology or not, one can carry out a topological comparison. If two generating setups of the different packings are topologically equal, the packings are the same topology. Since a single packing can have different generating setups, one first needs to define a type of setup that is topologically unique for the resulting topology. We define a topologically unique setup as the minimal generating setup of a packing, i.e., the setup with the least amount of seeds and inversion spheres needed. Therefore, we first find from the generating setup of each packing a min- imal setup. We explain this procedure at the two-dimensional example in Fig. 3.19, which can be analogously applied to any higher dimension. We start with the original setup (Fig. 3.19a) and first find all mirror lines, which in a setup can be used as inversion circles of infinite radii. Seeds are only allowed to intersect inversion circles, including mirror lines, perpendicularly, otherwise, they have to lie outside of them. Therefore, we need to define which side of the mirror lines we consider as the outside. We choose an 67 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS a b c d Figure 3.19: Different steps from an original generating setup (a) to a min- imal one (d). (b) reduced setup after considering mirror lines, i.e., infinitely large inversion circles. (c) Potential inversion circle (dashed, highlighted) that can be found from the fact that it maps the two seeds onto each other. 68 3.6. DISCOVERED PACKINGS arbitrary point P in space that we declare to lie outside of all mirror lines, where P should not lie on a mirror line itself. We then neglect every inver- sion circle and seed that lie inside a mirror line. As shown in Fig. 3.19b, this already leads to a reduced setup. From there, we check if any two inversion circles or any two seeds can be mapped onto each other by a new inversion circle. If we find such an inversion circle as shown in Fig. 3.19c, we add it and iteratively invert every seed and inversion circle at inversion circles who they intersect with an angle larger than π/2. We do this to find the largest inverses of each seed and inversion circle which lies outside all mirror lines. In the resulting setup, certain inversion circles and seeds might exist multiple times, such that we only keep a single instance of it. We check if this setup fulfills all constraints as discussed in Sec. 3.4. If it is a valid setup, one needs to proof that it leads to the same packing as the original setup. One can do this by inverting every seed and inversion circle of the original setup iteratively at the inversion circles of the newly proposed setup, till one found the largest inverse of each original seed and inversion circle which lays outside of all mirror lines. If every of these largest inverses is equal to a seed or inversion circle of the newly proposed setup, respectively, we know that the original setup can be generated from the newly proposed one. Therefore, the newly proposed setup leads to the same packing as the original one. One needs to continue to try to reduce every newly accepted setup the same way, till one cannot minimize it any further, to be sure to have found the minimal setup, as the one shown in Fig. 3.19d. After having found two minimal setups for two different packings, one can topologically compare them. The whole topological information lies in the arrangement of the seeds and inversion circles, i.e., in the way the touch and overlap each other. The seeds and inversion circles form a network, where two elements are connected if they intersect or if they are tangent to each other. The kind of connection, i.e., the intersection angle or the fact that they are tangent, can be seen as a weight or label of the connection. If one can find a bijection between the two networks as shown in Fig. 3.20, the two packings are the same topology, otherwise, they are different. 69 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS A A B B 3 4 4 3 1 2 1 2 tangent 90° 60° 45° A B A B 1 4 1 4 2 3 2 3 Figure 3.20: Two minimal generating setups (top) are topologically equiva- lent if one can find a bijection (arrows) between the two connection networks (bottom) of their seeds and inversion circles. Seeds and inversion circles are connected if they are tangent or if they intersect. The details of the con- nection, i.e., the intersection angle or the fact that they are tangent, can be seen as a weight or label of the connection. 70 3.7. MODIFIED PACKINGS 3.7 Modified Packings Packings can be modified in different ways. First of all, any packing can be inverted as a whole at any inversion sphere, which does not change its topology and therefore neither its fractal dimension, but its symmetry and spatial arrangement. Second, any packing can be nested in another pack- ing by exchanging any sphere of a given packing with a packing that is enclosed by a sphere. Third, one can cut any n-dimensional packing with an m-dimensional subspace, given 2 m 3.8 Final Remarks We presented an approach to find setups to generate self-similar space-filling sphere packings in arbitrary dimensions. This allows to generate 34, of which 5 were previously known, and 13 new topologies in 3D and 4D, respectively. We characterized all topologies according to their fractal dimensions and the properties of their contact network. The fractal dimensions range from 2.47 to 2.88 in 3D and from 3.60 to 3.90 in 4D. We explained how the fractal dimension of the generated packing can be increased by exchanging seeds of the generating setups with inversion spheres. Furthermore, each packing 71 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS Figure 3.21: Exchanging seeds by inversion spheres leads to an increased fractal dimension of the resulting packing. At last, the unit sphere hole is the only remaining seed. Confidence intervals are smaller than the size of the symbols. Dashed line is a linear fit. 72 3.8. FINAL REMARKS can be cut in various ways to serve as a source of lower dimensional cuts. The presented topologies together with the possible modifications offer the possibility to obtain space-filling packings with various fractal dimensions and contact-network properties. Reference [25] suggests, that self-similar space-filling packings are inhomo- geneous fractals, such that different cuts of a certain dimension can have different fractal dimensions. A detailed investigation on the fractal dimen- sion of such cuts is subject of Chapter 4. We have seen that exchanging seeds with inversion spheres increases the fractal dimension of the resulting packing. Apart from that, it remains an open question how the specifics of the generating setup influences the fractal dimension. The procedure to find a minimal generating setup as described in Sec. 3.6.3 would be useful for a systematic study which might provide ways that allow a more precise search for a generating setup that leads to a certain desired fractal dimension. Previously, the only known 3D bearings, i.e., single-cluster bipartite space- filling packings without isolated spheres, were the exactly self-similar one from Ref. [30] and the random ones from Ref. [31], which both have a smallest loop size of four. We found additional ones of which two have a smallest loop size of larger than four, namely the packings based on the octahedron with outer inversion spheres at the faces belonging to F1 with b = 0 with smallest loop size six (c = 1) and eight (c = 2). As shown in Sec. 2.3, bipartite space-filling packings with smallest loop size four have slip-free rotation states with four degrees of freedom. When inverted to end up in a packing bounded by two infinite spheres, i.e., bounded by two planes parallel to each other, they allow the simultaneous and synchronized motion of the two planes in any direction. According to Sec. 2.3, a smallest loop size of larger than four might lead to more degrees of freedom and extended frictionless functionality. Regarding applications, it would be valuable to find a packing that allows the independent, simultaneous motion of two planes in a slip-free state. We can already think of a simple sphere assembly that would fulfill this task. Unfortunately, it is not space-filling, nor stable 73 CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS under gravity. To decouple the motion of the two planes, one can put a chain of at least three spheres in between such that each plane touches a sphere on a different end of the chain. Along the derivations in Sec. 2.3, for a given motion of a plane, the touching sphere has one degree of freedom left for its slip-free rotation. It can vary its angular velocity along the vector PA~ pointing from the contact with the plane P to the center of the first sphere A. Let us imagine two further spheres B and C in the chain. If the two vectors AB~ and BC~ , that point from one center of a sphere to the next in the chain, span together with PA~ the three-dimensional space, sphere C has three independent degrees of freedom for its rotational state. Thus, there is always a slip-free rotation state for any independent motion of the two planes. None of the packings discovered here fulfill this task, since one can always find a chain between the planes of only two spheres or one in which the connecting vectors, as explained before, would not span three-dimensional space. It could be part of future studies to find suitable space-filling packings for this task or at least some that are stable under gravity. 74 Chapter 4 Cutting Self-Similar Space-Filling Sphere Packings 4.1 Introduction We investigate the cutting of space-filling packings as the one shown in Fig. 4.1, which are generated using inversive geometry as in Refs. [28, 35, 29, 25] and Chapter 3 and which therefore are exactly self-similar. Refer- ence [25] previously showed that the packing in Fig. 4.1 is not a homogeneous fractal, since two planar cuts have different fractal dimensions; but no fur- ther investigation was carried out. Here, we show that for all the self-similar space-filling packings constructed by inversive geometry of Refs. [28, 35, 29, 25] and Chapter 3, cuts along random hyperplanes generally have a fractal dimension of the one of the uncut packing minus one, what we prove analytically. Nevertheless, these packings are still heterogeneous fractals since cuts along special hyperplanes of a single packing show specific fractal dimensions. We present a strategy to search for such special cuts, which we illustrate on the packing in Fig. 4.1 as well as on a four-dimensional packing of Chapter 3 out of which one can cut, for instance, the packing in Fig. 4.1. This Chapter is organized in the following way. In Sec. 4.2, we introduce some general properties of the considered packings. In Sec. 4.3, we deal with cuts of packings where we consider random cuts in Sec. 4.3.1, and special 75 CHAPTER 4. CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS Figure 4.1: Self-similar space-filling sphere packing, first discovered by Ref. [25], constructed using inversive geometry. This particular packing is bipartite, such that one can color the spheres using two colors such that no spheres of same color touch. The packing is enclosed in the unit sphere, which is visualized as a surrounding shell. Spheres with a radius larger than 0.005 are shown, and some spheres are removed to allow looking inside the packing. cuts, as mentioned before, in Sec. 4.3.2. We draw conclusions in Sec. 4.4. 76 4.2. PROPERTIES OF PACKINGS 4.2 Properties of Packings A detailed explanation of the construction method of the considered pack- ings can be found in Chapter 3, where we show how to generate, using circle inversion, a variety of packings in two, three, and four dimensions, includ- ing all the topologies of Refs. [28, 35, 29, 25]. Since circle inversion can be straightforwardly extended to sphere inversion in any higher dimension, everything we explain here in two or three dimensions holds analogously for higher dimensions. Apart from generating packings, circle inversions can also be used to invert a whole packing. That can change the sizes and spatial arrangement of its disks or spheres, but the topology and fractal dimension are invariant with respect to inversion. Figure 4.2 shows how one can invert a packing in different ways. For example, a highly symmetric packing can be mapped onto an asymmetric one. Furthermore, by inverting a 2D packing with respect to a circle whose center lies on a contact point between two disks, these two disks are mapped onto disks with an infinite radius, i.e., parallel lines that enclose the inverse of the packing. This comes from the fact that an inversion circle maps its own center onto infinity. This kind of configuration is called the strip configuration. The packing in between has a finite unit cell with periodic continuation, i.e., translational symmetry, as shown at the bottom of Fig. 4.2. The unit cell is bounded by two mirror lines which are the inverse of two tangent inversion circles with respect to which the packing is invariant. By construction, at every contact point of touching spheres, one can find two such inversion circles tangent to each other. Each of the packings considered here from Refs. [28, 35, 29, 25] and Chapter 3 can analogously be mapped onto a strip configuration, which for any considered dimensions is enclosed by two hyperplanes. 4.3 Cutting To obtain different cuts of a sphere packing, one can cut along different planes. More generally, one can cut along any sphere, since by inverting the 77 4.3. CUTTING A A B B inversion & rotation A B Figure 4.3: A packing can be cut by any plane and more generally by any sphere. By inverting the whole packing one can map any spherical cut into a planar one and vice versa. whole packing, one can transform every spherical cut into a planar one and vice versa, as shown in Fig. 4.3. 4.3.1 Random Cuts Analogously to Fig. 4.2, one can map a sphere packing onto a periodic strip configuration enclosed by two planes. We use this fact to derive the fractal dimension of random cuts, as explained in the following. 79 CHAPTER 4. CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS Any cut, planar or spherical, can be mapped onto a planar cut in the strip configuration as shown in Fig. 4.4(a,b). The strip configuration has a unit cell with translational symmetry in two dimensions parallel to the two planes that enclose the packing. In this periodic structure, we can look at each unit cell individually. We will generally find that some cells are cut by the cutting plane and some are not. Of the ones that are cut, each individual cell might be cut at a different position by the cutting plane. Let us project all unit cells together with the cutting plane onto a single unit cell as shown in Fig. 4.4(c). For the specifically chosen cut in Fig. 4.4, this projection results in three different unit-cell cuts. Therefore, the cut has a periodic structure since it can be formed out of a sequence of these three unit-cell cuts, infinitely repeating itself. Depending on the orientation of the initial cut, the projection results in a different number of unit-cell cuts as shown in Fig. 4.5. If the cut is chosen randomly, it results generally in an infinite number of different unit-cell cuts, as indicated in Fig. 4.6. In principle, this is the same as using the well known ”cut-and-project” method to obtain a quasiperiodic structure as an ”irrational slice” of a periodic lattice [77, 78, 79, 80, 81, 82, 83]. Important here is that the density of unit-cell cuts is homogeneous across the whole unit cell. Out of a single sphere of the unit cell, infinitely many disks are cut. In detail, one finds the density of disks of radius r that are cut out of spheres of radius R to be 2r √R2 r2 n(R) for0 ∞ ∞ 2r ncut(r)= n˜cut(r, R)dR = n(R)dR. (4.2) 2 2 r r √R r Z Z − 80 4.3. CUTTING a b 3 2 1 cut cut c 3 d 2 1 1 2 3 unit cell translational symmetry Figure 4.4: (a) Any spherical or planar cut of a packing can be inverted at an inversion sphere (transparent) with center (white) at the contact point between two touching cut spheres to map the cut onto a planar cut that is cutting the packing in its strip configuration as shown in (b). Different unit cells (1,2,3) of the strip configuration in (b) are cut at different positions. One can project all unit cells together with the cutting plane onto a single unit cell as shown in (c). In this particular case, this leads to only three different unit-cell cuts. The cut can be formed out of a sequence of the three unit-cell cuts resulting in a periodic strip configuration as shown in (d). Therefore, also the cut itself has a unit cell with translational symmetry. If the resulting cut is periodic or not, depends on the orientation of the cutting plane. 81 CHAPTER 4. CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS Figure 4.5: Different special orientations of cutting planes (left) for which the cut can be formed out of a finite number of different unit-cell cuts (right). In the general case of a random orientation of the cutting plane, infinitely many different unit-cell cuts need to be combined to form the cut as shown in Fig. 4.6. 82 4.3. CUTTING a b ±3 c d ±10 ±∞ Figure 4.6: A randomly oriented cutting plane can in general only be formed out of a combination of infinitely many different unit-cell cuts. (a) Randomly oriented cutting plane cutting a single unit cell. (b,c) Projection of neighboring unit cells together with the cutting plane onto a single unit cell, for different ranges of projection. (d) Projection of all unit cells leading to infinitely many different unit-cell cuts. 83 CHAPTER 4. CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS The density n(R) of spheres follows asymptotically a simple power law df 1 n(R) R− − , (4.3) ∼ where df is the fractal dimension of the packing [15, 16, 17, 33, 35]. From df 1 this we assume n(R)= k R− − , where k > 0 is a constant. Thus, we find · from Eq. (4.2) that df +1 ∞ Γ( ) 2kr df 1 2 df ncut(r)= R− − dR =2πk r− , (4.4) 2 2 df r √R r Γ( ) Z − 2 ∞ t 1 x where Γ denotes the gamma function with Γ(t)= 0 x − e− dx. InEq.(4.4), df we see that ncut(r) r− and we know from Eq. (4.3) that ncut(r) ∼ R ∼ df,cut 1 r− − , where df,cut is the fractal dimension of the cut. Therefore, we find df,cut = df 1, i.e., the fractal dimension of random cuts is always the one − of the uncut packing minus one. 4.3.2 Special Cuts To generate a packing we use seeds and inversion circles which together are called the generating setup, as in Chapter 3. Different topologies originate from different generating setups. Nevertheless, some setups lead to the same packing and some generate different packings but the same topology, as we have seen in Sec. 3.6. In the latter case, the packings can be mapped onto each other through a certain sequence of inversions. However, since we are interested in finding special cuts with distinct fractal dimensions in a single packing, we will look for cuts with different generating setups. Let us first describe how one can find a generating setup of a cut. As discussed in Sec. 3.2 (compare Fig. 3.1), every generating setup consists of seeds and inversion circles. No seed lies completely inside an inversion circle, such that all its inverses are smaller. To lead to a space-filling packing, the seeds and inversion circles together need to cover all space, as proven in Sec. 3.4. In a cut of a sphere packing, every disk is a potential seed for a generating setup. Additionally, we need potential inversion circles, which we find as shown in the following. 84 CHAPTER 4. CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS a b Figure 4.8: Inversion circles (dashed circles) in the cut (dashed plane) are either: (a) (left) Cuts perpendicular to inversion spheres (transparent) what is topologically the same as (right) a cut perpendicular to a mirror plane (transparent); or (b) (left) Cuts containing the intersection of two inver- sion spheres (transparent) that intersect each other perpendicularly what is topologically the same as (right) a cut through the intersection line of two mirror planes (transparent) perpendicular to each other. Disks are just the cuts of spheres. 86 4.3. CUTTING the cut as shown in Fig. 4.8b. That is topologically the same as that the intersection line of two mirror planes perpendicular to each other turns out to be a mirror line in the cut. Apart from the here derived inversion circles which are cuts of inversion spheres, there could in principle also appear inversion circles in a cut which do not lie on the surface of any inversion sphere, but for simplicity, we neglected this more complex scenario. After having found all disks and inversion circles of a cut, one needs to check if they together cover all space. If they do, one can, to end up with a generating setup, neglect every disk and inversion circle whose center is inside another inversion circle. These disks and inversion circles are redun- dant since they can be generated from a larger disk and inversion circle, respectively, by inverting at the inversion circle in which their center lies. For a given smallest radius of spheres and inversion spheres of a packing, one can for a random cut generally not find a generating setup, because in general one finds no inversion circles in a random cut, which one would need to find to be able to cover the empty space between the disks. To find cuts that we can generate, we use the following strategy. We first find all inversion spheres larger than the radius rfind. We want to find cuts in which the cuts of these inversion spheres appear as possible inversion circles (compare Fig. 4.8a). We divide the inversion spheres into the ones that intersect the unit sphere (outer inversion spheres) and that do not (inner inversion spheres). We then find all spherical cuts that are perpendicular to three outer inversion spheres and one inner, and all planar cuts that are perpendicular to three outers. We chose this strategy for its computational efficiency since every special cut needs to contain at least three outer and one inner inversion circle, and there are many more inner than outer inversion spheres. Since some cuts lead to the same topology, we rule some multiple appear- ances out the following way. We only consider cuts that have no inverses larger than themselves and whose centers lie in a chosen area bounded by mirror planes of the packing, where the center of a planar cut lies at infinity in the direction of its normal vector. Some topologies might still appear 87 CHAPTER 4. CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS multiple times, which cannot be mapped onto each other with a single in- version, such that one has to sort them out separately. This comes from the fact that for the particular packing considered here, one can even map the spheres directly touching each other onto each other not by a single in- version but by a sequence of multiple inversions. For all pairs of topologies with overlapping confidence intervals of the fractal dimensions, determined numerically as described later, we therefore made a topological comparison, as explained in detail in Sec. 3.6.3, to judge if they are different topologies or not. Using the described strategy, we searched for special cuts in the packing shown in Fig. 4.1. We generated the packing down to a smallest radius of spheres and inversion spheres with respect to which the packing is in- variant (compare Fig. 4.7) of rmin = 0.005. For the smallest radius of in- version spheres considered to define the cutting sphere or plane, we chose rfind =0.2. We found 32 special cuts resembling different topologies. Their rescaled generating setups can be found in Fig. 4.9, and their fractal dimen- sions in Fig. 4.10. The fractal dimensions were determined as in Sec. 3.6.1, 12 considering the packings with all disks of radius larger than e− . We plot the number of different found topologies versus the search cutoff radius rfind in Fig. 4.11. The number of found topologies N seems to follow a power α law N r− . For cuts for which we did not find a generating setup, we ∝ find cannot be sure if they are special cuts or not, since we only cut spheres and inversion spheres larger than rmin = 0.005. Considering a smaller rmin, one might find a generating setup. We therefore estimate α for different rmin to make a prediction for the limiting case rmin 0. We assume the estimated → α is linearly dependent on rmin as shown in the inset of Fig. 4.11. We con- clude for rmin 0 that α =1.78 0.16. This suggests that one will find an → ± infinite amount of special cuts corresponding to different topologies in the limit of rfind 0 and rmin 0. We assume this to be true for any three → → and higher-dimensional packing. 88 4.3. CUTTING a b c d Figure 4.9: Rescaled generating setups of special cuts out of the packing shown in Fig. 4.1. They are ordered according to their fractal dimension decreasing from left to right and top to bottom. Topologies marked with a letter were already previously discovered by Ref. [28], according to which they have the following parameters in the form (family,m,n): (a) (2,1,1), (b) (2,0,1), (c) (1,0,0), and (d) (1,1,1). 89 CHAPTER 4. CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS Figure 4.10: Ranked fractal dimensions of special cuts shown in Fig. 4.9. The dashed line indicates the fractal dimension of random cuts in general, which is one less than the fractal dimension of the uncut packing. Confidence intervals are smaller than the symbol size. 90 4.3. CUTTING rmin=0.005 N r - ∝ find Figure 4.11: The number of different found topologies N versus the smallest considered radius rfind of inversion spheres to define the cut for smallest considered cut spheres and inversion spheres of radius rmin =0.005. We find α that N can be approximately described by a power law N r− . Inset: ∝ find estimates of the exponent α versus different rmin. We assume the estimated α is linearly dependent on rmin such that we predict for the limiting case of rmin 0 that α =1.78 0.16. This suggests that in the limit of rmin 0, → ± → one can find infinitely many different topologies for rfind 0. → 91 CHAPTER 4. CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS To demonstrate that our cut strategy can analogously be applied to higher- dimensional packings, we also cut a four-dimensional packing of Chapter 3. According to the definition and nomenclature in Chapter 3, this packing can be constructed from a generating setup based on the 16-cell with outer in- version spheres placed in the direction of its faces and seeds in the direction of its vertices, which we choose to lie at ( 1, 0, 0, 0) and its permutations, ± and belonging to family 1 with parameters b = c = 0. Its fractal dimension is 3.70695 0.0003 (see Tab. 3.2), such that for random cuts one would ± find a fractal dimension of one less in the range of 2.70695 0.0003. For ± rfind = 0.5 and rmin = 0.05, we found four special cuts which are shown in Fig. 4.12 together with their generating setups, which we order accord- ing to their fractal dimensions which we found to be 2.780581 0.000003, ± 2.735424 0.000005, 2.70812 0.00002, and 2.588191 0.000005. All are ± ± ± planar cuts through the center of the packing with normal vectors along (1, 1, 1, 1),(1, 1, 1, 0),(2, 1, 1, 1), and (1, 0, 0, 0), respectively. The second and the last cut are topologies discovered previously and the first and third cut are new discoveries. The last cut is exactly the packing of Fig. 4.1. The second one is the topology that according to Chapter 3 can, for instance, be constructed from a generating setup based on the tetrahedron with outer inversion spheres in the direction of its faces belonging to family 2 with pa- rameters b = 0 and c = 1. Due to the high computational effort needed to find special cuts in a four-dimensional packing, we chose a relatively large radius rfind =0.5 for the smallest inversion spheres considered to define our cuts. Even though we only found four special cuts in this case, we expect for any four- and even higher-dimensional packing to find, analogously to the three-dimensional example before, an infinite number of special cuts, which correspond to different topologies, in the limit of rfind 0 and rmin 0. → → 4.4 Final Remarks We have shown that self-similar space-filling sphere packings created by inversive geometry as in Refs. [28, 35, 29, 25] and Chapter 3 are inhomo- 92 4.4. FINAL REMARKS Figure 4.12: Generating setups (top) of special cuts (bottom) out of a four-dimensional packing of Chapter 3. (top) Inversion spheres in light gray and black and seeds in color. (bottom) Some spheres are removed to allow looking inside the three-dimensional cuts. The fractal dimension of the four- dimensional packing is 3.70695 0.0003. The ones of the three-dimensional ± cuts are from left to right 2.780581 0.000003, 2.735424 0.000005, 2.70812 ± ± ± 0.00002, and 2.588191 0.000005, where the first and third cut are newly ± discovered topologies whereas the second is previously known from Chapter 3 and the last one is the packing in Fig. 4.1. geneous fractals but that random cuts generally have a fractal dimension of the one of the packing minus one. We presented a strategy to look for special cuts with distinct fractal dimensions which allows identifying many different topologies out of a single packing. Our numerical analysis suggests that in the limit of a vanishing cutoff of smallest considered radii, one can find infinitely many special cuts corresponding to different topologies. This allows using packings in higher than three dimensions to find new two and three-dimensional topologies, whose direct construction setup is far from being trivial. Reference [25] previously found two planar cuts of the packing in Fig. 4.1 93 CHAPTER 4. CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS with different fractal dimensions, without further investigation. After the detailed analysis here, we know that one of these two cuts is the special cut shown in Fig. 4.9(d), and the other one is a cut with fractal dimension of the uncut packing minus one, which can not be constructed itself by inversive geometry. Bipartite sphere packings like the one in Fig. 4.1 have drawn attention since they allow all spheres to rotate simultaneously in a specific way without any slip between neighboring spheres as shown in Refs. [30, 52], such that some packings even allow the prediction and control of the slip-free rotation state, as shown in Chapter 2. Regarding bipartite packings, it is unknown if some can be used as a bearing to decouple the motion of two parallel planes as mentioned in Sec. 3.8. By cutting four-dimensional bipartite topologies, one might be able to find three-dimensional sphere packings with previously unknown mechanical functionalities regarding their slip-free rotation state. 94 Chapter 5 Conclusion We have studied the rotational dynamics of fixed bipartite assemblies of indi- vidually rotating spheres, where we discovered a way to control the rotation state, and have developed new ways to search for self-similar space-filling packings, as elaborated in the following. We studied how assemblies drive from an arbitrary initial state, due to slid- ing friction between contacting spheres, toward a finally slip-free state. In a model purely based on sliding friction, we have identified special time- invariant sums of variables for the rotational dynamics of bipartite assem- blies. We have shown that for assemblies with exactly four degrees of freedom for their slip-free state, the final state is unique for given values of the found time-invariant sums. Therefore, one can directly predict the final state from the initial one, independent of the strength of sliding friction. Furthermore, this allows to control the slip-free rotation state by only controlling any two spheres, independent of the total number of spheres. We have demonstrated this in experiment, where we find good agreement for an assembly of four spheres for most of the tested spatial arrangements, but also show some lim- itations. In the slip-free state, the rotation axes of all spheres need to meet at a single point, which if this point is at infinity, the rotation axes are par- allel to each other. If this point lies not at infinity, the spheres rotate faster the further away they are from this point. This can be used to accelerate the rotation of spheres along an assembly of equally sized spheres. This is a newly discovered functionality which hopefully will find use in future appli- 95 CHAPTER 5. CONCLUSION cations. It could replace transmission gears where suitable. An interesting feature of these assemblies which is not given in classic transmission gears, is that as long as the contacts between spheres are conserved, the spheres are allowed to be displaced with respect to each other. This might favor such assemblies for certain applications of power transmission. The tabletop ex- periment we built for demonstration serves as a pioneering realization of an assembly whose rotation state can be controlled. Such assemblies could be improved in respect to stable operation in the future. We presented a new construction method for two families of two-dimensional, self-similar, space-filling, previously known disk packings. We generalized this method to arbitrary higher dimensions and used it to search new topolo- gies in three and four dimensions. We found numerous different topologies showing various fractal dimensions. In addition to that, we developed a strategy to search for new sub-dimensional topologies, all showing different fractal dimensions, as cuts of existing ones. The number of found sub- dimensional topologies with respect to a cutoff parameter suggests the ex- istence of infinitely many previously unknown topologies. Altogether, this forms a framework to find space-filling packings in arbitrary dimensions showing a broad range of fractal dimensions. 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