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 Stäger
MSc. Materials Science, ETH Zurich
born 18.06.1989
citizen of Glarus Süd 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. Araújo 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 construction 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
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
3.6
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
Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.4
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. Darüber hinaus entwickeln wir neue
Konstruktionsmethoden für raumfüllenden Kugelpackungen.
Zuerst studieren wir die Rotationsdynamik von aneinanderliegenden Kugeln, sogenannten Kugelclustern. Dabei nehmen wir an, dass die Kugeln
an ihrer Position fixiert sind, jedoch frei rotieren können. Weiter befassen
wir uns nur mit bipartiten Kugelclustern. Ein Kugelcluster ist bipartit, falls
man es mit nur zwei Farben so einfärben kann, sodass sich keine Kugeln
gleicher Farbe berühren. Jedes bipartite Kugelcluster hat schlupffreie Rotationszustände, in welchen sich alle Kugeln gleichzeitig drehen können, ohne
Schlupf zwischen berührenden Kugeln. Mit einem Modell, welches lediglich
Gleitreibung berücksichtigt, untersuchen wir, wie sich bipartite Kugelcluster von einem Anfangszustand mit frei wählbaren Winkelgeschwindigkeiten
zu einem schlupffreien Endzustand bewegen. Dabei stellen wir fest, dass gewisse Summen von Variablen, welche die Massen, Radien, Winkelgeschwindigkeiten und Positionen der Kugeln einbeziehen, zeitlich unveränderlich
sind. Das bedeutet, dass diese vom Anfangszustand bis hin zum schlupffreien Endzustand konstant bleiben. Für gewisse bipartite Kugelcluster ist
der schlupffreie Zustand eindeutig bestimmt für gegebene Werte der zeitlich konstanten Summen. In diesem Fall kann man den Endzustand für
jeden beliebigen Anfangszustand exakt vorhersagen. Diese vorhersagbaren
Kugelcluster müssen genau vier Freiheitsgrade im schlupffreien Zustand haben. Glücklicherweise sind solche einfach zu konstruieren und können aus
i
nur zwei, aber auch aus viel mehr Kugeln bestehen, theoretisch aus unbegrenzt vielen. Überraschenderweise ist der schlupffreie Endzustand solcher
Kugelcluster unabhängig von der Stärke der Gleitreibung zwischen den Kugeln. Desweiteren lässt sich der schlupffreie Zustand kontrollieren. Durch die
externe Kontrolle von lediglich zwei beliebigen Kugeln kann jeder mögliche
schlupffreie Rotationszustand des Kugelclusters kontrolliert werden, was wir
auch experimentell demonstrieren. Im schlupffreien Zustand eines jeden bipartiten zweidimensionalen Clusters gleich grosser Scheiben sind alle Rotationsgeschwindigkeiten identisch. Nicht so in einem dreidimensionalen Kugelcluster, wo Kugeln gleicher Grsse unterschiedliche Rotationsgeschwindigkeiten haben knnen. Deshalb ist es möglich in einem kontrollierbaren Kugelcluster die Kugeln entlang einer frei wählbaren Richtung zu beschleunigen, was eine zuvor unbekannte mechanische Funktionalität darstellt.
Desweiteren befassen wir uns mit raumfüllenden Kugelpackungen. Im Detail
behandeln wir nur Packungen, welche mittels Kugelinversionen erzeugt werden können, was immer exakt selbstähnliche Packungen sind. Diese Packungen sind fraktal und die Grössenverteilung ihrer Kugeln folgt asymptotisch
einem Potenzgesetz, woraus man die fraktale Dimension abschätzen kann.
Inspiriert durch vorhergehende Arbeiten entwickeln wir eine Konstruktionsmethode in zwei Dimensionen, welche wir für jede höhere Dimension verallgemeinern. Mittels der neuen Konstruktionsmethode finden wir zahlreiche
neue Topologien in drei und vier Dimensionen. Zusätzlich stellen wir eine
Strategie vor um neue niedrigerdimensionale Topologien aus bereits entdeckten zu schneiden. Im Ganzen ermöglicht uns dies weitere raumfüllende
Topologien in beliebigen Dimensionen zu finden. Die zahlreichen Topologien
die wir finden weisen eine grosse Bandbreite an fraktalen Dimensionen auf
und können als eine Auswahl für ideal dichte Packungen mit verschiedenen
Grössenverteilungen 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 spacefilling 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 dynamics toward the slip-free state. These sums involve the individual spheres’
masses, radii, angular velocities, and positions. For certain bipartite assemblies, 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 degrees of freedom in their slip-free state. Luckily, those assemblies can easily
be constructed and the number of spheres can range from two to, theoretically, 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 controllable 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 packings are exactly self-similar and fractal, and their size distribution follows
asymptotically a power law, from which one can estimate the fractal dimension. 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. Stäger, N. A. M. Araújo, 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. Stäger and H. J. Herrmann, “Self-similar space-filling sphere
packings in three and four dimensions,” submitted to Fractals, 2016.
(arXiv:1607.08391)
• D. V. Stäger 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, including: 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 experiments; 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 subdimensional 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], Möbius 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 spacefilling 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 various 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. Twodimensional space-filling bearings have been studied with respect to their
synchronizability [52], i.e., the stability of the slip-free state. We were interested 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 bearings 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 dynamics. This motivated us to search for construction methods producing further
three-dimensional bearings, which resulted in the studies presented in Chapters 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, sliding friction at the contacts between spheres will generally slow down and
finally stop the rotation of all spheres. However, if the assembly is bipartite (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 rotational 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
equal tangential velocities
different 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
~1
~2
I2 α
I1 α
=
,
r1
r2
(2.1)
where I1 , I2 , α
~ 1, α
~ 2 , 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 multiple contacts and multiple simultaneously acting sliding forces. Each sliding
force contributes to the angular acceleration of the sphere. We define α
~ ik
as the contribution to the angular acceleration α
~ i due to the sliding force
P k
at contact k, such that α
~i = k α
~ i , the sum running over all contacts. We
write Eq. (2.1) analogously for a contact k between spheres i and j with the
~ jk . For simplicity, we consider the moment of inertia
contributions α
~ ik and α
of a sphere i proportional to its mass mi and its radius squared, as, for
example, for homogeneous solid spheres. We find
m i ri α
~ ik − mj rj α
~ jk = ~0.
(2.2)
Summing this equation over all contacts k, we obtain
X
s i m i ri α
~ i = ~0,
i
10
(2.3)
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
X
~ :=
A
si m i r i ω
~ i,
(2.4)
i
where ω
~ i is the angular velocity of the sphere i. Equation (2.3) shows that
~
~ is time invariant.
∂ A/∂t = ~0 , i.e., A
We derive one further time invariant quantity. Let ~xi be the position vector
~ of the entire
of the center of sphere i, where we choose the center of mass M
~ = ~0. We multiply
assembly to be the origin of our coordinate system, i.e., M
(dot product) Eq. (2.2) by ~xi and obtain mi ri α
~ ik · ~xi − mj rj α
~ jk · ~xi = 0. Since
~ jk , mj rj α
~ jk · ~xij vanishes. We
the vector ~xij = ~xj − ~xi is perpendicular to α
~ ik · ~xi − mj rj α
~ jk · ~xj = 0.
subtract it from the previous equation and get mi ri α
~ we thereby define the time invariant
Analogously to the derivation of A,
quantity B as
B :=
X
i
2.3
si m i r i ω
~ i · ~xi .
(2.5)
Construction of 4DOF Assemblies
~ ∈ R3 and B ∈ R to predict the slip-free
Later we use the quantities A
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
as ω
~ isf × ri x̂ij = ω
~ jsf × rj x̂ji , where ω
~ isf is the angular velocity of sphere i in
the slip-free state (sf), and x̂ij is the unit vector pointing from sphere i to
sphere j. We rewrite this condition in analogy to Ref. [30] as
sj r j ω
~ jsf − si ri ω
~ isf = cij ~xij ,
(2.6)
~ isf to ω
~ jsf . The slip-free state
which defines cij ∈ R that uniquely relates ω
is uniquely defined by the angular velocity ω
~ isf of a single sphere i and the
parameters cij of all contacting spheres i and j. Since ω
~ isf ∈ R3 , the number
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 + . . . + cN 1~xN 1 = ~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 · · · ~xN 1 ). 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
sj r j ω
~ jsf − si ri ω
~ isf = c~xij .
(2.8)
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
~ jsf − si ri ω
~ isf + sk rk ω
~ kf s − sj rj ω
~ jsf = c~xij + c~xjk ,
sj rj ω
sk rk ω
~ kf s − si ri ω
~ isf = c~xik ,
(2.9)
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
~ jsf − si ri ω
~ isf = cA~xij ,
sj rj ω
s l rl ω
~ lsf − sj rj ω
~ jsf = cjl ~xjl ,
sk r k ω
~ ksf − sl rl ω
~ lsf = cB ~xlk ,
si r i ω
~ isf − sk rk ω
~ ksf = cki~xki ,
13
(2.10)
(2.11)
(2.12)
(2.13)
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
~ jsf − si ri ω
~ isf = cA~xij ,
sj rj ω
sk rk ω
~ ksf − sj rj ω
~ jsf = cjl ~xjk ,
(2.15)
(2.16)
si r i ω
~ isf − sk rk ω
~ ksf = cik ~xki ,
(2.17)
cA~xij + cjk ~xjk + cik ~xki = ~0.
(2.18)
can be added to obtain
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 assembly. 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 minimal 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
ω
~ isf =
si
sj rj ω
~ jsf + c~xji .
ri
15
(2.19)
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
~
assembly, but an imaginary reference sphere with angular velocity ω
~ sf = Ω,
j
sj = rj = 1, and ~xj = ~0 to obtain the general expression
si ~
sf
Ω + c~xi ,
ω
~i =
ri
(2.20)
~ ∈ R3 is a vectorial reference quantity.
where Ω
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~ and B defined in Eq. (2.4) and (2.5). We write Eq. (2.20)
variant quantities A
for a sphere i, multiply it by si mi ri , and sum over all spheres to end up with
~ on the left hand side and to eliminate c from the right hand side, beA
P
~ = ~0, since we defined the center of mass M
~ of the
xi = c M
cause
i cmi ~
~ = ~0). We
entire assembly to be the origin of our coordinate system (M
P
~ = A/M
~
find Ω
, with the total mass M =
mi . Second, we multiply (dot
i
product) Eq. (2.20) written for sphere i by si mi ri~xi and sum over all spheres
~ on the right hand
to end up with B on the left hand side and eliminate Ω
P
side to find c = H/I, where I = i mi |~xi |2 . We then formulate the angular
velocities of the slip-free state as a function of time invariant quantities only
as
si
ω
~ isf =
ri
~
B
A
+ ~xi
M
I
!
.
(2.21)
Equation (2.21) shows that the final slip-free state can be predicted using
~ and B defined in Eqs. (2.4) and (2.5). Surthe time invariant quantities A
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
magnitudes of 100
sliding forces:
10⁻2
prop. to
⁻4
start
4
10
constant 10⁻6
3
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 chosen initial angular velocities toward a slip-free state with sliding forces with
magnitudes proportional to the difference in tangential velocities (solid) and
P
with constant magnitudes (dashed). ∆ = i |~ωi − ω
~ isf | measures the deviation from the slip-free state. Inset: Exponential 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
seconds with a time step of 10−6 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
~ 6= ~0 and B = 0 all axes of rotation are parallel
different slip-free states. For A
~ For B 6= 0 all axes of rotation of the spheres meet at the position
to A.
~
~ = ~0 they meet at the center of mass.
~x = I A/(BM
), and in particular for A
acting on sphere j due to contact with sphere i points in the direction opposite 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 = −σv̂ij ,
where the magnitude is constant, with σ = 0.05 and with v̂ij being the unit
vector along ~vij .
~ and B regarding the slip-free state predicted
Figure 2.6 pictures the role of A
~ For B 6= 0
by Eq. (2.21). For B = 0 all axes of rotation are parallel to A.
~
they all intersect at ~x = I A/(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
~ (A
~ new ) and B (Bnew ) from the
(2.5), and (2.21), one can derive the new A
~ old and Bold ) when blocking sphere i as described in
previous (old) values (A
the following. By applying a perturbation ∆~ωi to the angular velocity of a
sphere i one can impose the changes
~ = si mi ri ∆~ωi
∆i A
(2.22)
∆i B = si mi ri ∆~ωi · ~xi .
(2.23)
and
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
~ and B as applying a perturbation ∆~ωi that leads to
the same effect on A
the sphere i being at rest in the slip-free state, i.e., ∆~ωisf = ~0. To ensure
∆~ωisf = ~0 we find from
si
ω
~ isf =
ri
~ new Bnew
A
+
~xi
M
I
!
= ~0,
(2.24)
that
~ new = −M Bnew ~xi /I.
A
(2.25)
~ · ~xi = ∆i B (compare Eqs. (2.22) and (2.23)), we find
From the fact that ∆i A
~ old and ∆i B = Bnew − Bold , that
~=A
~ new − A
using ∆i A
Bnew =
~ old · ~xi
Bold − A
.
1 + M |~xi |2 /I
(2.26)
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~ and ∆B
locities as we show here. We want to impose a desired change ∆A
by applying external changes ∆~ωi and ∆~ωj to the angular velocity of sphere
~ and
i and j, respectively. Eqs. (2.22) and (2.23) show the changes ∆i A
~
∆i B imposed by the change ∆~ωi . The individually imposed changes in A
~ = ∆i A
~ + ∆j A
~ and
and B need to sum up to the desired change, i.e., ∆A
∆B = ∆i B + ∆j B. We define ∆~ωi = ωi ∆ω̂i , where ω̂i is the unit vector of
the external change ∆~ωi . From Eqs. (2.22) and (2.23) we find
~ + ∆j A
~ = si mi ri ωi ∆ω̂i + sj mj rj ∆~ωj
~ = ∆i A
∆A
(2.27)
and
∆B = ∆i B + ∆j B = si mi ri ωi ∆ω̂i · ~xi + sj mj rj ∆~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
ωi =
~ · ~xj − ∆B
∆A
,
si mi ri ∆ω̂i · ~xij
(2.29)
such that we can choose any ∆ω̂i as long as ∆ω̂i · ~xij 6= 0. From Eq. (2.29)
we know ∆~ωi since ∆~ωi = ωi ∆ω̂i and we obtain using Eq. (2.27) that
~ − si mi ri ∆~ωi )/(sj mj rj ).
∆~ωj = (∆A
(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 ω
~ i (first) and ∆2 ω
~ i (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
corresponding changes ∆1 ω
~ isf and ∆2 ω
~ isf in angular velocity from the previous
to the new slip-free state. Applying a change ∆~ωi to a sphere i leads to a
change ∆~ωisf of the angular velocity of sphere i between the previous to the
new slip-free state. We use
∆~ωisf
~ ∆i B
∆i A
+
~xi
M
I
si
=
ri
!
(2.31)
and Eqs. (2.22) and (2.23) to find
∆~ωisf = mi (∆~ωi /M + (∆~ωi · ~xi )~xi /I).
(2.32)
Let us derive two scalar equations from Eq. (2.32) by first squaring it (dot
product) to find
|∆~ωisf |2 /m2i = |∆~ωi |2 /M 2 + 2(∆~ωi · ~xi )2 /(M I)+
(∆~ωi · ~xi )2 |~xi |2 /I 2
(2.33)
and second multiplying it (dot product) with ∆~ωi to find
∆~ωisf · ∆~ωi /mi = |∆~ωi |2 /M + (∆~ωi · ~xi )2 /I.
(2.34)
Combining the two we can eliminate the term (∆~ωi · ~xi )2 and after some
rearrangements we find
|∆~ωisf |2 /(m2i |∆~ωi |2 ) =
1/M 2 + (2/M + |~xi |2 /I)(∆~ωisf ∆~ωi /(mi |∆~ωi |2 ) − 1/M ).
(2.35)
For a single change ∆~ωi and its induced change ∆~ωisf we see in Eq. (2.35) that
we have the two unknown quantities M and |~xi |2 /I. So with two changes
~ i and ∆2 ω
~ i and their induced changes ∆1 ω
~ isf and ∆2 ω
~ isf one can find
∆1 ω
M = (b +
√
b2 − 4ac)/(2a),
a = |f2 g1 − f1 g2 |, b = |f2 − f1 |, c = |g2 − g1 |,
fn = |∆n ω
~ isf |2 /(m2i |∆n ω
~ i |2 ),
gn = ∆n ω
~ isf · ∆n ω
~ i /(mi |∆n ω
~ i |).
21
(2.36)
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
~x′i = ∆~ωisf − mi ∆~ωi /M,
(2.37)
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 ~x′j parallel to ~xj . The
center of mass is the only point in space that can be reached both going
along ~x′i from the center of sphere i and going along ~x′j from the center of
sphere j, in the general case where ~x′i is not parallel to ~x′j . At last, we can
solve Eq. (2.34) for I to find
I = (∆~ωi · ~xi )2 / ∆~ωisf · ∆~ωi /mi − |∆~ωi |2 /M .
2.5.2
(2.38)
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, material, 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 horizontally 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 direction 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 average smaller than the predicted ones, and we did not observe smooth but
partially jerky rotations becoming jerkier for decreasing α. In the amplifying
mode, the spatial arrangement for low α triggers this deviation which originates 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
min
d
max
e
max
min
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 rotation 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 transmission gears. Since the ability to control the rotation state is robust against
changes in the spatial arrangement, as long as contacts are conserved, assemblies 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 producing 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 paragraph.
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 generalized to smallest loops of any even size [29]. In both Refs. [28, 29], the
packings are constructed in a strip geometry by iteratively applying conformal transformations, namely reflections, translations, and inversions, to
some initially placed disks. In contrast, Ref. [35] presents a way to construct 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, unlike 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 packings. In Sec. 3.3, we provide the necessary knowledge about inversive geometry 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 spacefiling 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 discovered 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 understanding of inversive geometry in the following section.
3.3
Circle Inversion
We explain important properties of circle inversion considering a single inversion circle in Sec. 3.3.1 and multiple inversion circles in Sec. 3.3.2. We furthermore 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 infinitely 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 perpendicular 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 inverting 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
e
d
g
f
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
point P and d′ = R2 /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 < m, space is divided into 2m sections as
shown in Fig. 3.3c. n of these sections lie outside both I1 and I2 , as well as
in their overlapping region. m − n sections lie exclusively in I1 , and another
m − n sections exclusively in I2 . A disk that is placed inside one of the
sections will by iterative inversions at I1 and I2 lead to a disk in every section.
If a disk is placed on a section border but not perpendicular to it, overlapping
disks will be generated (Fig. 3.3d). To avoid overlapping, one should place
disks only inside sections or, as shown in Fig. 3.3e, perpendicular to section
borders. If α/π is not a rational number, space is divided into infinite
infinitesimally small sections, such that any initially placed disk would lead
to infinitely many partially overlapping images as shown in Fig. 3.3f for
α = 1.
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
x2 + . . . + x2n+1 − r2 − 1
an+2 = 1
,
(3.1)
2r
x2 + . . . + x2n+1 − r2 + 1
.
an+3 = 1
2r
They satisfy the relation a21 + . . . + a2n+2 − a2n+3 = 1 and one can express the
usual parameters as
1
,
an+3 − an+2
ai
, for i = {1, . . . , n + 1}.
xi =
an+3 − an+2
r=
35
(3.2)
CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING
SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS
b
a
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 < m, space
is divided into 2m sections. n of these sections lie outside of both I1 and
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 (a′i ) by
(a′1 , . . . , a′n+3 )⊤ = M · (a1 , . . . , an+3 )⊤ ,
(3.3)
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 intersecting 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 inversion circles intersect with an angle α = nπ/m, where n and m are integers
without common prime factors and 1 ≤ n < m, space is divided into 2m
sections. Importantly, n of these sections are outside both inversion circles.
Because of that, we generally only allow intersecting angles α = π/m with
m ≥ 2. This ensures that the area outside both inversion circles remains a
single section. Given that, we place seeds perpendicular to some inversion
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 perpendicular 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 intersect 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
Figure 3.4:
b
c
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 depends 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 perpendicular 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, octa42
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 position 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 elements, 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 additionally 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 intersecting 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. If a
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
d2 = r12 + r22 + 2r1 r2 cos α,
47
(3.5)
CHAPTER 3. CONSTRUCTION OF SELF-SIMILAR SPACE-FILLING
SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS
a
e
Figure 3.9:
b
f
c
g
d
h
(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 polytope. (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/(x̂out · x̂s )
p
and the radius rout = d2out − 1
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
d2 = r12 + r22 .
(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/(x̂out · x̂s ), where x̂out and x̂s 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
p
find rout = d2out − 1.
Next, we want 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
Figure 3.11:
b
Solving for the details of the inner inversion spheres for F1:
2
(1 + cos γ) = d2in (1 − x̂in1 · x̂in2 ). In (b) we
In (a) we find for d1 that d21 = 2rin
2
2
2
2
+ rin
+ 2rout
rin
cos β.
find for d2 that d22 = d2out + d2in − 2dout din x̂in · x̂out = rout
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
(d21 =) 2rin
(1 + cos γ) = d2in (1 − x̂in1 · x̂in2 ),
(3.7)
where x̂in1 and x̂in2 are the unit vectors of two nearest inner inversion spheres.
From the fact that an inner inversion sphere intersects its closest outer inversion 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
(d22 =) d2out + d2in − 2dout din x̂in · x̂out = rout
+ rin
+ 2rout
rin
cos β, (3.8)
where x̂in and x̂out 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 sphere x̂in is a combination of
the one of its nearest outer inversion sphere x̂out and the one of its nearest
primary seed x̂s such that we can write
x̂in = px̂out + qx̂s .
50
(3.9)
3.5. HOW TO CONSTRUCT GENERATING SETUPS
The condition for this vector to be a unit vector gives us the equation
(x̂2in ) = p2 + q 2 + 2pqx̂out · x̂s = 1.
(3.10)
Every inner inversion sphere is perpendicular to the nearest outer inversion
sphere such that
2
2
(dout x̂out − din x̂in )2 = rout
+ rin
,
(3.11)
where x̂in and x̂out are the unit vectors of a closest pair of inner and outer
inversion spheres. Using Eq. (3.9), this gives us
2
2
+ rin
.
d2out + d2in − 2dout din (p + qx̂out · x̂in ) = rout
(3.12)
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
(1 + cos γ),
(d23 =) (~xin,s1 − ~xin,s2 )2 = 2rin
(3.13)
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 (px̂out + qx̂s1 )
and ~xin,s2 = din (px̂out + qx̂s2 ), where x̂s1 and x̂s2 are the unit vectors of two
primary seeds that are nearest to an outer inversion sphere with the unit
vector x̂out , we find from Eq. (3.13) that
2
(1 + cos γ).
d2in q 2 (2 − 2x̂s1 · x̂s2 ) = 2rin
(3.14)
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
(1 + cos β).
d2in p2 (2 − 2x̂out1 · x̂out2 ) = 2rin
(3.15)
One can now find p, q, rin , and din from the four Eqs. (3.10), (3.12), (3.14),
and (3.15). The unit vector x̂in 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
= (ds x̂s − din x̂in )2 = d2s + d2in − 2ds din x̂s · x̂in ,
rs2 + rin
51
(3.16)
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
(a) For d3 we find d23 = (~xin,s1 − ~xin,s2 )2 = 2rin
(1 + cos γ). (b) For d4 we find
2
(1 + cos β).
d24 = (~xin,out1 − ~xin,out2 )2 = 2rin
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
p
2
. Further additional seeds might be needed between
as rcenter = d2in − rin
inner and outer 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 vector x̂as of the position ~xas = das x̂as
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 = din x̂in and a closest outer
inversion sphere at position ~xout = dout x̂out , we find
2
2
ras
+ rin
= (~xas − ~xin )2 = d2as + d2in − das din x̂as · x̂in ,
2
2
+ rout
= (~xas − ~xout )2 = d2as + d2out − das dout x̂as · x̂out .
ras
(3.17)
(3.18)
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 along x̂as .
3.6
Discovered Packings
We searched for generating setups in three and four dimensions. Some generating setups turned out to give the exact same packings and some lead
to different packings but identical topologies. For any two different packings 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 generalized 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 computationally 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 producing 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 smallest 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 inversions 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
N (r) ∼ r−df , s(r) ∼ rD−1−df , p(r) ∼ rD−df ,
(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 ∆dˆfa = dˆfa − df and
∆dˆfb = dˆfb − df on two estimates dˆfa and dˆfb based on a certain interval of
two different functions a(r) ∼ rA−df and b(r) ∼ rB−df , respectively, relate
as
∆dˆfb
∆dˆfa
=
.
A − df
B − df
(3.20)
We use Eq. (3.20) to define a combined estimate dˆfa&b based on both functions a(r) and b(r) as
dˆfa&b = xdˆfa + (1 − x)dˆfb , with x =
B − dˆfb
,
B − A + dˆfa − dˆf
(3.21)
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 , rmin e], and take dˆf (rmin ) ± 5∆ as our
confidence interval for our best estimation as shown in Fig. 3.15. The lowest
considered radius was rmin = e−10 and rmin = e−7 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
Figure 3.13:
B
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 logarithmic 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
Figure 3.14:
B
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
rmine2
rmine
rmin
df*+5Δ
df*
Δ
df*-5Δ
Figure 3.15: As our best estimate (here d∗f ), we take of all estimate functions
(compare Fig. 3.14), the one that shows the least variability ∆ in the radius
interval [rmin , rmin e] (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 number 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 Tables 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 connected 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 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. 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
4
E
1
0
0
4
E
1
0
4
E
1
4
E
4
IS
fractal dim.
R
T
3
2.52638±5e-6
29
1
6
1
3
2.53706±2e-5
28
0
2
3
2.51613±2e-5
30
1
0
3
3
2.50002±6e-5
32
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
4
F
1
0
1
4b
2.71066±2e-5
13
4
F
1
0
2
4b
2.76236±5e-5
6
4
F
1
0
3
4b
2.76625±5e-5
4
4
F
1
0
4
4b
2.692627±2e-6
15
4
F
2
0
0
3
2.4739465±1e-7
34
4
F
2
0
1
4b
2.73543±1e-4
8
X
63
BR
2
RF
AG
X
4(1)
X
X
2
AG
X
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
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
F
1
0
1
4b
2.76235±3e-5
6
4(2)
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
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)
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
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
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
F
1
1
1
4b
2.724834±9e-6
11
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
2.73398±5e-5
9
10(2)
X
X
X
64
BR
RF
6
X
X
BP
8
X
3.6. DISCOVERED PACKINGS
B
I
F
b
c
LS
IS
fractal dim.
R
T
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
F
2
1
1
4b
2.588191±5e-6
22
14
8
F
2
1
2
3
2.5404±3e-4
27
12
F
1
0
0
3
2.51142±4e-5
31
12
F
1
0
1
4b
2.76624±4e-5
5
12
F
2
0
0
3
2.58594±2e-5
24
12
F
2
0
1
4b
2.66722±2e-5
16
X
65
BR
RF
8
X
BP
12
X
6(2)
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
5
F
1
0
0
4b
5
F
1
0
1
6b
5
F
1
0
2
5
F
2
0
5
F
2
8
F
8
IS
fractal dimension
R
T
BR
3.6807±0.0009
10
1(1)
X
3.7818±0.0009
6
2(1)
4b,8b
3.6872±0.0003
9
3(1)
0
3
3.59591±0.00002
13
4(1)
0
1
4b
3.70695±0.00002
8
5(1)
X
1
0
0
4b
3.66379±0.00004
11
6
X
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
16
E
2
0
0
3
3.65233±0.00005
12
7
16
F
1
0
0
4b
3.70695±0.00003
8
5
16
F
1
0
1
6b
3.8995±0.001
1
16
F
1
0
2
8b
3.8602±0.0004
4
16
F
1
1
0
3
3.59591±0.00003
13
4
16
F
1
1
1
4b
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
X
X
66
X
X
X
X
3.6. DISCOVERED PACKINGS
B
I
F
b
c
LS
24
F
1
0
0
4b
24
F
1
0
1
6b
24
F
1
0
2
24
F
1
1
24
F
1
24
F
24
fractal dimension
R
T
BR
3.6806±0.0002
10
1(3)
X
3.7816±0.0003
6
2(2)
4b,8b
3.6872±0.0004
9
3(2)
0
4b
3.66379±0.00004
11
6
X
1
1
4b
3.6807±0.0003
10
1(4)
X
2
0
0
4b
3.66379±0.00004
11
6
X
C
1
0
0
4b
3.66379±0.00004
11
6
X
24
C
1
0
1
6b
3.888±0.002
2
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
3.787±0.002
5
8(2)
24
C
2
0
2
4b,4b
3.71673±0.00005
7
3.6.3
IS
X
X
X
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 minimal 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 minimal 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 inversion 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
2
1
tangent
90°
60°
45°
A
4
1
B
2
2
A
4
1
3
B
4
1
3
2
3
Figure 3.20: Two minimal generating setups (top) are topologically equivalent 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 connection, 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 packing 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 < n, to obtain an m-dimensional
packing. Fourth, one can exchange seeds of a generating setup with inver-
sion spheres to increase the fractal dimension of the generated packing. As
long as a single seed remains, the resulting packing will be self-similar and
space-filling. Figure 3.21 shows the fractal dimensions of different packings
resulting from a setup with a different number of primary seeds exchanged
with inversion spheres. And fifth, there might exist an inversion sphere in
the setup which intersects other inversion spheres only perpendicularly. If
such an inversion sphere does not intersect any seed, it can be exchanged
with a seed, what would decrease the fractal dimension, in the opposite way
as shown in Fig. 3.21. In case such an inversion sphere is intersecting seeds,
one can exchange it with a seed and simultaneously exchange the seeds with
inversion spheres. According to what we have seen before, the overall change
in fractal dimension might be positive or negative in that case.
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 inhomogeneous fractals, such that different cuts of a certain dimension can have
different fractal dimensions. A detailed investigation on the fractal dimension 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 spacefilling 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
P~A 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
~ and BC,
~ that point from one center of a sphere to
the two vectors AB
the next in the chain, span together with P~A 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. Reference [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 further 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 packings 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, including 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
ñcut (r, R) =

√
0
2r
n(R)
R2 −r2
for 0 < r ≤ R,
(4.1)
otherwise,
where n(R) is the density of spheres or radius R. When we consider spheres
of all radii, a disk of radius r can be a cut out of any sphere of radius R ≥ r.
Therefore, we can obtain the density ncut (r) of disks in the cut by considering
all spheres of radius R ≥ r. We integrate the density ñcut (r, R) of disks that
are cut out of spheres of radius R, which we defined in Eq. (4.1), over all
R ≥ r to find
ncut (r) =
Z
∞
ñcut (r, R) dR =
r
80
Z
∞
r
√
2r
n(R) dR.
R2 − r 2
(4.2)
4.3. CUTTING
a
b
3
2
1
cut
cut
c
3
2
d
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
n(R) ∼ R−df −1 ,
(4.3)
where df is the fractal dimension of the packing [15, 16, 17, 33, 35]. From
this we assume n(R) = k · R−df −1 , where k > 0 is a constant. Thus, we find
from Eq. (4.2) that
ncut (r) =
Z
∞
r
d +1
Γ( f2 ) −df
2kr
−df −1
√
R
dR = 2πk
r ,
d
R2 − r 2
Γ( 2f )
where Γ denotes the gamma function with Γ(t) =
R∞
0
(4.4)
xt−1 e−x dx. In Eq. (4.4),
we see that ncut (r) ∼ r−df and we know from Eq. (4.3) that ncut (r) ∼
r−df ,cut −1 , 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 inversion 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 redundant 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 appearances 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 inversion, 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 inversion 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 invariant (compare Fig. 4.7) of rmin = 0.005. For the smallest radius of inversion 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 dimensions in Fig. 4.10. The fractal dimensions were determined as in Sec. 3.6.1,
considering the packings with all disks of radius larger than e−12 . 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 ∝ rfind
. For cuts for which we did not find a generating setup, we
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∝rfind
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
. Inset:
that N can be approximately described by a power law N ∝ rfind
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 higherdimensional 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 inversion 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 according 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 parameters 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 inhomo92
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 fourdimensional 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 individually 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 sliding friction between contacting spheres, toward a finally slip-free state. In
a model purely based on sliding friction, we have identified special timeinvariant sums of variables for the rotational dynamics of bipartite assemblies. 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 limitations. 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 parallel 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 appli95
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 experiment 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 topologies 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 subdimensional topologies with respect to a cutoff parameter suggests the existence 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. In the future, an automatized algorithm might be used to continue the search of further topologies
in three dimensions to provide a large selection of space-filling packings, including the specifics of construction and the fractal dimensions. The fractal
dimension is related to the behavior of the size distribution of spheres, which
asymptotically follows a power-law. Therefore, a large collection of topologies could serve as a reference library for highly dense packings with various
size distributions.
Last but not least, self-similar space-filling packings are visually appealing,
such that they already previously found their way into the world of art [84].
The numerous newly discovered topologies serve as a large source for future
artwork.
96
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