2016
DOI: 10.1007/s00454-016-9777-3
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Apollonian Ball Packings and Stacked Polytopes

Abstract: We investigate in this paper the relation between Apollonian d-ball packings and stacked (d + 1)-polytopes for dimension d ≥ 3. For d = 3, the relation is fully described: we prove that the 1-skeleton of a stacked 4-polytope is the tangency graph of an Apollonian 3-ball packing if and only if there is no six 4-cliques sharing a 3-clique. For higher dimension, we have some partial results.

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Cited by 9 publications
(12 citation statements)
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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%
“…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%
“…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.…”
mentioning
confidence: 99%
“…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%