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The Thirteen Spheres: A New Proof

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

2012
*
Discrete & Computational Geometry
*

In

doi:10.1007/s00454-011-9392-2
fatcat:unnmkweugrbmzaxct3syljnybu
*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

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 [13] 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

1967
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

Hoppe (see (1)) proved that 7V 3 = 12, settling thereby

doi:10.4153/cjm-1967-100-x
fatcat:jfbyn4fotfgzbdzs7e7a7zdtve
*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]

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

1997
*
Discrete & Computational Geometry
*

We describe

doi:10.1007/bf02770863
fatcat:6du2crv7v5hvdayynr4wv76m5e
*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

2014
*
Social Science Research Network
*

Conceptually this article is

doi:10.2139/ssrn.2436252
fatcat:zqnzcysiynga3k7hx2u6mjb5pq
*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

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

2015
*
Social Science Research Network
*

Conceptually this article is

doi:10.2139/ssrn.2610900
fatcat:h7bb72222felhjr6ho2fesc6zy
*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]

2021
*
arXiv
*
pre-print

In

arXiv:1804.06692v4
fatcat:js7uce2pqzeofbgups7t4n4glq
*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

2012
*
Discrete & Computational Geometry
*

Namely, let us imagine that we are given

doi:10.1007/s00454-012-9405-9
fatcat:eyqcwyb7mzexll2mj6g2yecjj4
*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]

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

2012
*
2012 Ninth International Symposium on Voronoi Diagrams in Science and Engineering
*

Continuing

doi:10.1109/isvd.2012.19
dblp:conf/isvd/Bezdek12a
fatcat:263ugwxvkvef5kuqh2os3behsq
*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

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

1926
*
Proceedings of the Royal Society A
*

One good

doi:10.1098/rspa.1926.0017
fatcat:mejrnop6nrg2dflf4ejpol4q6u
*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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