Thomas C. Hales
The honeycomb problem on the sphere asks for the perimeter-minimizing partition of the sphere into N equal areas. This article solves the problem when N=12. The unique minimizer is a tiling of 12 regular pentagons in the dodecahedral arrangement.
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We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter's regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m>6, star-honeycombs {m, m/2} are embeddable while {m/2, m} are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representi…
The classical honeycomb conjecture asserts that any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. Pappus discusses this problem in his preface to Book V. This paper gives the first general proof of the conjecture. The revision is the published version, which allows disconnected honeycomb cells and gaps between cells.
We prove a stability version of the isodiametric inequality on the sphere and in the hyperbolic space.
We prove the isodiametric inequality in the spherical and in the hyperbolic space
In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and $d\geq 2$ we show that solutions to the container problem can not have a ``simple structure'' for large $n$. By this we in particular find that there exist arbitrary small $r>0$, such that packings in a smooth, 3-dimensional convex body, with a maximum number of spheres of radius $r$, are necessarily not hexagonal close packings. This contradicts Kepler's famous statement that the cubic or hexagonal close packing ``will…
The Honeycomb Conjecture in normed planes and an alpha-convex variant of a theorem of Dowker
本文研究了normed平面中的蜂窝猜想，证明了正凸平铺的平均平方周长最小，并探讨了与Dowker定理α-凸变种相关的联系，进一步给出了关于平铺单元等周比的问题解答。
We algorithmically characterize the maximal contact number problem for finite congruent lattice sphere packings in $\mathbb{R}^d$ and show that in $\mathbb{R}^3$ this problem is equivalent to determining the maximal coordination of a monatomic crystal.
The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standin…
We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for $\ell_2$ in the context of the Urysohn space $\Ur$. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.
Recently, we enumerate up to isometry, all locally rigid circle packings on the unit sphere with number of circles N<12. This problem is equivalent to the enumeration of irreducible contact graphs. In this paper we show that by using the list of irreducible graphs can solve various problems of extreme packings such as the Tammes problem for the sphere and the projective plane, the maximal contacts problem, Danzer's and other problems on irreducible contact graphs.
The problem of twelve spheres is to understand, as a function of $r \in (0,r_{max}(12)]$, the configuration space of $12$ non-overlapping equal spheres of radius $r$ touching a central unit sphere. It considers to what extent, and in what fashion, touching spheres can be varied, subject to the constraint of always touching the central sphere. Such constrained motion problems are of interest in physics and materials science, and the problem involves topology and geometry. This paper reviews the history of work on this problem, presents some new results, and formulates some conjectures. It also …
In this paper we prove that no packing of unit balls in Euclidean space $\mathbb{R}^8$ has density greater than that of the $E_8$-lattice packing.
The main goal of the paper is to solve some problems about shadow for the sphere generalized on the case of the ellipsoid. Here, the essence of the problem is to find the the minimal number of non-overlapping balls with centers on the sphere which are not holding the center of the sphere and such that every line passing through the center of the sphere would intersect at least one of the balls. Another method of solving the shadow problem for the sphere is also proposed.
On the Inscribed Sphere and Concurrent Lines through the Centers of Apollonius Spheres in $\mathbb{R}^n$
本文研究了在R^n中由共同家族球体构成的阿波罗尼奥斯问题解之间的关系，证明了通过解中心的直线交于一点，并推广了Morita的定理到n维空间。
The densest local packing (DLP) problem in d-dimensional Euclidean space Rd involves the placement of N nonoverlapping spheres of unit diameter near an additional fixed unit-diameter sphere such that the greatest distance from the center of the fixed sphere to the centers of any of the N surrounding spheres is minimized. Solutions to the DLP problem are relevant to the realizability of pair correlation functions for packings of nonoverlapping spheres and might prove useful in improving upon the best known upper bounds on the maximum packing fraction of sphere packings in dimensions greater tha…
We consider the sets of dimensions for which there is an optimal sphere packing with special regularity properties (respectively, a lattice, or a periodic set with a given bound on the number of translations, or an arbitrary periodic set). We show that all these sets are oracle-computable, given an oracle that orders an associated set of spherical codes by increasing Kolmogorov complexity.
We present an efficient Monte Carlo method for the lattice sphere packing problem in d dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond previous methods not only in exploring higher dimensions but also in shedding light on the statistical mechanics underlying the problem in question. We observe evidence of a phase transition in the thermodynamic limit $d\to\infty$. In the dimensions explored in the present work, the results are consistent with a first-order crystallization transition, but leave …
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