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### The Thirteen Spheres: A New Proof

Kurt M. Anstreicher
2004 Discrete & Computational Geometry
The "thirteen spheres problem," also known as the "Gregory-Newton problem," is to determine the maximum number of three-dimensional spheres that can simultaneously touch a given sphere, where all the spheres  ...  In this paper we describe a new proof that uses linear programming bounds and properties of spherical Delaunay triangulations.  ...  The Thirteen Spheres: A New Proof 625 easy to show that no such graph can exist.  ...

### The Strong Thirteen Spheres Problem

Oleg R. Musin, Alexey S. Tarasov
2012 Discrete & Computational Geometry
In the paper we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on a enumeration of the so-called irreducible graphs.  ...  The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size.  ...  Supported by the Russian government project 11.G34.31.0053.  ...

### The problem of thirteen spheres — a proof for undergraduates

H. Maehara
2007 European journal of combinatorics (Print)
The purpose of this note is to present an elementary proof of the fact that no more than twelve unit balls can simultaneously touch a unit ball in 3-space, a proof that is accessible for undergraduates  ...  Applying a similar method, Musin  also gave a new proof of the problem of thirteen spheres.  ...  Then, can thirteen unit balls touch a unit ball simultaneously? This is the problem of thirteen spheres, and it is said that this problem was discussed between A. Newton and D. Gregory in 1694.  ...

### A variant of the problem of the thirteen spheres

L. Fejes Tóth, A. Heppes
1967 Canadian Journal of Mathematics - Journal Canadien de Mathematiques
Hoppe (see (1)) proved that 7V 3 = 12, settling thereby a famous point of controversy between Newton and David Gregory, known as the problem of the thirteen spheres (see (3) ).  ...  Consider now a bunch of balls all touching a ball. We enlarge this bunch by adding new balls touching at least one ball of the original bunch.  ...  In the spherical (n -1)-space consisting of the boundary of 0 draw spheres with radius 15° about the projections. According to our theorem these spheres form a packing.  ...

### Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem [article]

Rob Kusner, Wöden Kusner, Jeffrey C. Lagarias, Senya Shlosman
2017 arXiv   pre-print
The problem of twelve spheres is to understand, as a function of r ∈ (0,r_max(12)], the configuration space of 12 non-overlapping equal spheres of radius r touching a central unit sphere.  ...  This paper reviews the history of work on this problem, presents some new results, and formulates some conjectures.  ...  Acknowledgments The authors were each supported by ICERM in the Spring 2015 program on "Phase Transitions and Emergent Properties." R.  ...

### Sphere packings, I

T. C. Hales
1997 Discrete & Computational Geometry
We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program.  ...  Yt) to be the (ordered) simplex whose i th edge has length yi. If S is a Delaunay simplex in a fixed Delaunay star, then it has a distinguished vertex, the vertex common to all simplices in the star.  ...  The constraints Ivi+lvii = Ivi -Vi-l[ = 2.51 force us to drag vi to a new position on the sphere of radius 2.  ...

### An Attitude of Complexity: Thirteen Essays on the Nature and Construction of Reality Under the Challenge of Zeno's Paradox

Scott A Albers
2014 Social Science Research Network
Conceptually this article is the first of a three-part series.  ...  They exist whether the piano is played by a pianist or simply banged on by a child for joy of making racket. The physical instrument of the piano forms The Plane of Definition for piano performance.  ...  Part Three: Describes a new nomenclature for the Rings based upon this factorization of the Rings themselves. Part Four: Speculations as to further potential proof, and a Conclusion.  ...

### A new class of infinite sphere packings

David Boyd
1974 Pacific Journal of Mathematics
The packings considered in this paper are packings of a unit sphere in iV-dimensional Euclidean space by an infinite number of unequal spheres.  ...  More specifically, we are interested in complete packings, those which exhaust the volume of the packed sphere.  ...  That is, we begin with a 'cluster' of (N + 2) disjoint spheres and by successive inversions replace the spheres one at a time with new spheres in such a way that the separations between the spheres in  ...

### An Attitude of Complexity: Thirteen Essays on the Nature and Construction of Reality Under the Challenge of Zeno's Paradox

Scott A. Albers
2015 Social Science Research Network
Conceptually this article is the first of a three-part series.  ...  This paper was entitled "The Golden Mean, the Arab Spring and a 10-Step Analysis of American Economic History."  ...  If we add to this new point a relationship in space between all the other points in the sphere the dimension of time is added as a fourth dimension, i.e. the time it takes to arrive at a one point vs.  ...

### Platonic solids, Archimedean solids and semi-equivelar maps on the sphere [article]

Basudeb Datta, Dipendu Maity
2021 arXiv   pre-print
In the course of the proof of our main result, we present a combinatorial characterization in terms of an inequality of all the types of semi-equivelar maps on 𝕊^2.  ...  A vertex-transitive map X is a map on a surface on which the automorphism group of X acts transitively on the set of vertices of X.  ...  These imply that the new square xuyv in X is of type 2 (as in the proof of Lemma 4.7).  ...

### Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space

Károly Bezdek
2012 Discrete & Computational Geometry
Namely, let us imagine that we are given a lattice unit sphere packing with the center points forming the lattice L in Euclidean 3-space (and with certain pairs of unit balls touching each other) and then  ...  Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings.  ...  Acknowledgements The author wishes to thank an anonymous referee for a number of helpful comments and suggestions.  ...

### A Proof of Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve [article]

Thomas C. Hales
2012 arXiv   pre-print
Proof. By Lemma 3, we may replace V with a new set in V if necessary so that (V, E + (V)) is a biconnected fan. We show that the smaller fan (V, E 2 (V)) is also biconnected.  ...  Proof. We take the spherical Delaunay triangulation of the sphere S 2 (2) induced by V ′ . Triangles correspond to triangulated faces of the polyhedron obtained as the convex hull of V ′ ⊂ R 3 .  ...  Every hypermap with tame contact is isomorphic to a hypermap in the given list of eight hypermaps, or is isomorphic to the opposite of a hypermap in the list. Proof.  ...

### Contact Numbers for Congruent Sphere Packings Via Voronoi Diagrams

K'roly Bezdek
2012 2012 Ninth International Symposium on Voronoi Diagrams in Science and Engineering
Continuing the investigations of Harborth (1974) and the author (2002) we study the following basic problem on sphere packings.  ...  sphere packings.  ...  ACKNOWLEDGMENT Partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.  ...

### Phylogenies constrained by the crossover process as illustrated by human hemoglobins and a thirteen-cycle, eleven-amino-acid repeat in human apolipoprotein A-I

W M Fitch
1977 Genetics
The pattern is shown to be suitable to the formation of alpha helices with an amphipathic character consistent with the formation of a micellar structure, a process entirely appropriate to the protein's  ...  known function in the blood stream as a lipid carrier.  ...  to form a polyhedron approximating a cage just big enough to enclose the surface of the sphere.  ...

### Fluid Resistance to Moving Spheres

R. G. Lunnon
1926 Proceedings of the Royal Society A
One good proof of their value is the following.  ...  Thirteen points on the standard R graph result from the present experi ments (fig. 7 ).  ...
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