Apollonian Ball Packings and Stacked Polytopes

Chen, Hao

doi:10.1007/s00454-016-9777-3

Cited by 9 publications

(12 citation statements)

References 34 publications

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“…Finally we note that for s = −1 and d = 3 the growing simplicial complex model presented here belongs to the class of stacked polytopes that are equivalent to Apollonian packings, whose discrete Lorentzian geometry is raising recent interest in the mathematical community 48 49 50 51 . In fact, these stacked polytopes in d = 3 have a symmetry group G that is a noncompact discrete subgroup of the Lorentz group SO(3, 1) = SL(2, )/ .…”

Section: Resultsmentioning

confidence: 99%

Emergent Hyperbolic Network Geometry

Bianconi¹,

Rahmede²

2017

Sci Rep

137

179

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A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explore whether this geometry is hyperbolic. Specifically we show that an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes that are purely combinatorial. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display the major universal properties of real complex networks (scale-free degree distribution, small-world and communities) at the same time. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by relevant changes in the network geometry.

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Section: Resultsmentioning

confidence: 99%

Emergent Hyperbolic Network Geometry

Bianconi¹,

Rahmede²

2017

Sci Rep

137

179

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“…For every point x ∈ E \ B, the part of S visible from x is a spherical cap on S. For an edge-scribed polytope, the caps corresponding to the vertices have disjoint interiors. After a stereographic projection, they form a ball packing in Euclidean space whose tangency graph is isomorphic to the 1-skeleton of the polytope; see [Che16]. The dual version of Proposition 5.5 says that every truncated polytope is edge-scribable.…”

Section: Ridge-scribabilitymentioning

confidence: 99%

“…In fact, every 3-polytope has a realization with all its edges tangent to a sphere. This follows from Koebe-Andreev-Thurston's remarkable Circle Packing Theorem [Koe36, And71a, And71b, Thu] because edge-scribed 3-polytopes are strongly related to circle packings; see [Zie07] for a nice exposition, and [Che16] for a discussion in higher dimensions. This was later generalized by Schramm [Sch92], who showed that an edge-tangent realization exists even if the sphere is replaced by an arbitrary strictly convex body with smooth boundary.…”

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confidence: 99%

Scribability problems for polytopes

Chen

Padrol

2017

European Journal of Combinatorics

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In this paper we study various scribability problems for polytopes. We begin with the classical k-scribability problem proposed by Steiner and generalized by Schulte, which asks about the existence of d-polytopes that cannot be realized with all k-faces tangent to a sphere. We answer this problem for stacked and cyclic polytopes for all values of d and k. We then continue with the weak scribability problem proposed by Grünbaum and Shephard, for which we complete the work of Schulte by presenting non weakly circumscribable 3-polytopes. Finally, we propose new (i, j)-scribability problems, in a strong and a weak version, which generalize the classical ones. They ask about the existence of d-polytopes that can not be realized with all their i-faces "avoiding" the sphere and all their j-faces "cutting" the sphere. We provide such examples for all the cases where j − i ≤ d − 3. Lorentzian view of polytopesClassical scribability problems only consider bounded convex polytopes in Euclidean space. To define polarity properly in this setup, one must assume that the polytope contains the origin in its interior. This presents the major difficulty in Schulte's work on weak k-scribability, and also leads to a minor flaw in his proof regarding strong k-scribability (see Remark 3.3).We find it more natural and convenient to work with spherical polytopes, which arise from pointed polyhedral cones in Lorentzian space. This section is dedicated to the introduction of this setup. The main advantage is that, for spherical polytopes, polarity is always well-defined and well-behaved. This facilitates the study of weak scribability and enables us to obtain Theorem 4. At the same time, as we will see in Lemma 3.2, strong scribability in spherical space and in Euclidean space are equivalent, so the new setting is compatible with previous studies. In fact, the presence of spherical geometry is necessary only in few occasions (e.g. Example 3.5).2.1. Convex polyhedral cones in Lorentzian space. A (closed) non-empty subset of R d+1 is a convex cone if it is closed under positive linear combinations. A convex cone is pointed if it does not contain any linear subspace of R d+1 . A convex cone is polyhedral if it is the conical hull of finitely many vectors in R d+1 , i.e. a set K of the form

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“…For Lorentzian Coxeter systems of level 2, we also study the tangency graphs of Boyd-Maxwell ball packings. In [Che13], the tangency graphs of Apollonian ball packings are compared to the 1-skeleton of stacked polytopes. In Theorem 3.9, we describe the tangency graph of Boyd-Maxwell ball packing in terms of the corresponding Coxeter complex.…”

Section: Introductionmentioning

confidence: 99%

Lorentzian Coxeter systems and Boyd–Maxwell ball packings

Chen

Labbé

2014

Geom Dedicata

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In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In this paper, we show that the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: Given a geometric representation of a Coxeter group in a Lorentz space, the set of limit directions of weights equals the set of limit roots. Additionally, we use Coxeter complexes to describe tangency graphs of the corresponding Boyd-Maxwell ball packings. Finally, we enumerate all the Coxeter systems that generate Boyd-Maxwell ball packings.2010 Mathematics Subject Classification. Primary 52C17, 20F55; Secondary 05C30.

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Apollonian Ball Packings and Stacked Polytopes

Cited by 9 publications

References 34 publications

Emergent Hyperbolic Network Geometry

Emergent Hyperbolic Network Geometry

Scribability problems for polytopes

Lorentzian Coxeter systems and Boyd–Maxwell ball packings

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