## 61 Citations

Configuration spaces of equal spheres touching a given sphere: The twelve spheres problem

- Mathematics
- 2018

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…

The Strong Thirteen Spheres Problem

- MathematicsDiscret. Comput. Geom.
- 2012

A computer-assisted proof is given based on an enumeration of irreducible graphs of 13 equal-size non-overlapping spheres touching the unit sphere of the thirteen spheres problem.

Historical Overview of the Kepler Conjecture

- PhysicsDiscret. Comput. Geom.
- 2006

AbstractThis paper is the first in a series of six papers devoted to the
proof of the Kepler conjecture, which asserts that no packing of
congruent balls in three dimensions has density greater than…

Kepler’s conjecture: How some of the greatest minds in history helped solve one of the oldest math problems in the world

- Physics
- 2004

If you pour unit spheres randomly into a large container, you will fill only some 55 to 60 percent of the space. If you shake the box while you are filling it, you will get a denser packing -…

The Kissing Problem in Three Dimensions

- MathematicsDiscret. Comput. Geom.
- 2006

A new solution of the Newton--Gregory problem is presented that uses the extension of the Delsarte method and relies on basic calculus and simple spherical geometry.

Sphere Packings Ii Section 1. Introduction

- Mathematics
- 2007

An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of R 3 into…

Sphere Packings, II

- MathematicsDiscret. Comput. Geom.
- 1997

The second step of a program to prove the Kepler conjecture on sphere packings leads to a decomposition of R3 into polyhedra, which has density at most that of a regular tetrahedron.

The kissing number in four dimensions

- Mathematics
- 2003

The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three…

A Rejoinder to Hales’s Article

- Mathematics
- 1995

ConclusionThis then is an item-by-item reality check on the mathematical content of their objections to my article [2], on which they have based their assessment on the status of the Kepler…

2 1 A pr 2 00 5 THE KISSING NUMBER IN FOUR DIMENSIONS

- Mathematics
- 2005

The kissing number problem asks for the maximal number of equal size nonoverlapping spheres that can touch another sphere of the same size in n-dimensional space. This problem in dimension three was…

## References

SHOWING 1-10 OF 12 REFERENCES

A new bound on the local density of sphere packings

- Computer ScienceDiscret. Comput. Geom.
- 1993

Borders are proved by cutting a Voronoi polyhedron into cones, one for each of its faces, and the sum of all the cone volume bounds is minimized when there are 13 faces each of solid angle 4π/13.

Sphere Packings, Lattices and Groups

- MathematicsGrundlehren der mathematischen Wissenschaften
- 1988

The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to…

The Problem of the Thirteen Spheres

- Philosophy, MathematicsThe Mathematical Gazette
- 1956

An independent proof of this impossibility of David Gregory's conjecture that a sphere can touch thirteen non-overlapping spheres equal to it is outlined.

A mesolithic camp in Denmark

- Environmental Science
- 1987

The excavation of Vaenget Nord has yielded clues to the rich foraging culture that flourished on the coasts of northern Europe during the Mesolithic. Decapage was used for the reconstruction of the…

Music of the spheres.

- EducationScience
- 1991

Pythagoras combined within one person the attributes of a mystic, philosopher, and scientist, as one laying down the foundations of science, as well as influencing all European ethics not directly inherited from the East.