## A proof of the dodecahedral conjecture (1998)

Citations: | 5 - 2 self |

### BibTeX

@MISC{Hales98aproof,

author = {Thomas C. Hales and Sean Mclaughlin},

title = {A proof of the dodecahedral conjecture},

year = {1998}

}

### OpenURL

### Abstract

This article gives a proof of Fejes Tóth’s Dodecahedral conjecture: the volume of a Voronoi polyhedron in a three-dimensional packing of balls of unit radius is at least the volume of a regular dodecahedron of unit inradius. 1

### Citations

1790 |
The Analysis of Linear Partial Differential Operators III
- Hörmander
- 1983
(Show Context)
Citation Context ... decidable. Thus, the truth of the Dodecahedral conjecture can be decided in theory by standard algorithms such as Collin’s cylindrical algebraic decomposition [7], 8or the Cohen-Hörmander algorithm =-=[21]-=-. However, in practice, these decision procedures take exponential time in the number of quantifiers, and thus are far too slow to be of practical value for this conjecture. To formulate the Dodecahed... |

488 |
Introduction to Interval Computations
- Alefeid, Herzberger
- 1983
(Show Context)
Citation Context ...ic to control for floating-point rounding errors. Every real number x is represented on the computer as an interval [a, b] containing x, where a and b are exactly representable floating point numbers =-=[1, 32]-=-. The calculations conform to IEEE-754 standards [23]. Approximations to inverse trigonometric functions are based on published approximations [20]. 143.7 Nonlinear optimization Previous subsections ... |

388 |
Quantifier elimination for real closed fields by cylindrical algebraic decomposition
- Collins
- 1975
(Show Context)
Citation Context ...elementary theory of the real numbers is decidable. Thus, the truth of the Dodecahedral conjecture can be decided in theory by standard algorithms such as Collin’s cylindrical algebraic decomposition =-=[7]-=-, 8or the Cohen-Hörmander algorithm [21]. However, in practice, these decision procedures take exponential time in the number of quantifiers, and thus are far too slow to be of practical value for th... |

311 |
Rigorous Global Search: Continuous Problems
- Kearfott
- 1996
(Show Context)
Citation Context ...R m , for some m. The computer program verifies that ( f1(x) > 0) ∨ ( f2(x) > 0) ∨ · · · ∨ ( fr(x) > 0), (3) for every point x ∈ R. The approach is similar to the approach described in R. B. Kearfott =-=[24]-=-, based on interval arithmetic. Our methods are similar to algorithms in widespread use for rigorous global optimization. Closely related algorithms are also described in [37]. The method is based on ... |

175 |
Isabelle: a generic theorem prover, volume 828 of LNCS
- Paulson
- 1994
(Show Context)
Citation Context ...ubject of G. Bauer’s dissertation in computer science at the Technical University of Munich [2]. This 172page dissertation translates the Java code into the formal theorem proving system Isabelle/HOL =-=[31]-=- and gives a detailed mathematical treatment of the graph theory underpinning the computer code. The dissertation analyzes every line of code. Building on the work of this thesis, B. Bauer and T. Nipk... |

117 | The Kepler conjecture
- Hales
- 1998
(Show Context)
Citation Context ...edral conjecture [22]. However, the proof did not hold up to careful analysis. “As of this writing, Kepler’s conjecture as well as the dodecahedral conjecture are still unproven” [4, p761]. See also, =-=[12]-=-. An alternative approach to the Dodecahedral conjecture is described in [4]. Unfortunately, a counterexample has been found to both parts of the third conjecture of that article. The counterexample i... |

95 |
Version Control with Subversion. O’Reilly & Associates
- Collins-Sussman, Fitzpatrick, et al.
- 2004
(Show Context)
Citation Context ...hive is under version control by Google Code [10]. The site consists of a download area where one may obtain the source code and supporting documents to this paper. Additionally there is a subversion =-=[8]-=- repository. This means that the snapshot of the code and documents in the exact form they took at the time of creation is permanently available. It also means that any changes (for instance, a bug fi... |

68 |
A computer-checked proof of the Four Colour Theorem
- Gonthier
- 2005
(Show Context)
Citation Context ...d edge.. See Figure 5. Figure 5: A planar graph as hypermap, with faces and nodes Hypermaps are the primary combinatorial object used by Gonthier in the formalization of the Four-Color theorem in Coq =-=[9]-=-. Hypermaps, by being purely combinatorial, are more convenient to represent on a computer than planar graphs. 34Not all hypermaps arise from a planar graph in this way. Those that do have two specia... |

51 | R.: KNITRO: An integrated package for nonlinear optimization
- Byrd, Nocedal, et al.
- 2006
(Show Context)
Citation Context ... of f1 on the domain {x ∈ [a1, b1] × · · · × [am, bm] : f2(x) ≤ 0, . . . , fr(x) ≤ 0} is positive. Nonlinear optimization libraries have been used to test all the inequalities in the collection [25], =-=[6]-=-. The code generates a large random set X of points in the domain and runs the algorithm for each initial point x ∈ X to find a local minimum to the objective function f1. If X is sufficiently large a... |

43 |
Lagerungen in der Ebene, auf der Kugel und im Raum, Vol 65
- Toth, L
- 1953
(Show Context)
Citation Context ...regarded as being nearly as difficult as the Dodecahedral conjecture itself. L. Fejes Tóth returned to the Dodecahedral conjecture in a number of publications. It is a prominent part of his two books =-=[36]-=-, [35]. According to the strategy of [36], the Dodecahedral conjecture forms a step towards the solution of the sphere packing problem (discussed below). In [35] , he proved that the Dodecahedral conj... |

36 |
The packing of equal spheres
- Rogers
- 1958
(Show Context)
Citation Context ...odecahedral conjecture gives an upper bound on density of 0.755. Upper bounds on the density based on lower bounds on the volume of a Voronoi cell in the literature include Rogers’ upper bound 0.7797 =-=[33]-=-, and Muder’s upper bounds 0.77836 [27] and 0.7731 [28]. In 1993, Hsiang published what he claimed to be proofs of the Kepler conjecture and the Dodecahedral conjecture [22]. However, the proof did no... |

35 |
Computer Approximations
- Hart, Cheney, et al.
- 1968
(Show Context)
Citation Context ...< 0. To reduce the calculations to rational expressions r(x0), rational approximations to the functions √ x, arctan(x), and arccos(x) with explicit error bounds are required. These were obtained from =-=[2]-=-. Reliable approximations to various constants (such as π and √ 2) with explicit error bounds are also required. These were obtained in Mathematica and were double checked against Maple. 59E Inequali... |

31 |
Tóth, Regular Figures
- Fejes
- 1964
(Show Context)
Citation Context ...t to prove than expected. In the years since the statement of the conjecture, there have been a number of developments. L. Fejes Tóth himself made considerable progress. In his book “Regular Figures” =-=[7]-=-, he proved that if m is the total number of spheres surrounding a central sphere S0 and n is the number of spheres whose center to center distance from S0 is less than 2 √ 3tan(π/5) ≈ 2.516817, then ... |

28 |
Regular figures
- Tóth, L
- 1964
(Show Context)
Citation Context ...ed as being nearly as difficult as the Dodecahedral conjecture itself. L. Fejes Tóth returned to the Dodecahedral conjecture in a number of publications. It is a prominent part of his two books [36], =-=[35]-=-. According to the strategy of [36], the Dodecahedral conjecture forms a step towards the solution of the sphere packing problem (discussed below). In [35] , he proved that the Dodecahedral conjecture... |

23 | Sphere packings - Hales - 1997 |

20 |
On the sphere packing problem and the proof of Kepler's conjecture
- Hsiang
- 1993
(Show Context)
Citation Context ...Rogers’ upper bound 0.7797 [33], and Muder’s upper bounds 0.77836 [27] and 0.7731 [28]. In 1993, Hsiang published what he claimed to be proofs of the Kepler conjecture and the Dodecahedral conjecture =-=[22]-=-. However, the proof did not hold up to careful analysis. “As of this writing, Kepler’s conjecture as well as the dodecahedral conjecture are still unproven” [4, p761]. See also, [12]. An alternative ... |

17 | Introduction to the Flyspeck project
- Hales
- 2006
(Show Context)
Citation Context ...full version of the proof. The computer code has also been entirely rewritten. A formalization project, called Flyspeck, aims to provide a complete formalization of the proof of the Kepler conjecture =-=[15, 11]-=-. (A formal proof is one in which every logical inference of the proof has been independently checked by computer, all the way to the primitive axioms at the foundations of mathematics.) A parallel pr... |

17 |
et al., Computer Approximations
- Hart
- 1968
(Show Context)
Citation Context ... are exactly representable floating point numbers [1, 32]. The calculations conform to IEEE-754 standards [23]. Approximations to inverse trigonometric functions are based on published approximations =-=[20]-=-. 143.7 Nonlinear optimization Previous subsections describe the three main pieces of computer code used in the proof of the Dodecahedral conjecture: graph generation, linear programming, and interva... |

16 |
The Sphere Packing Problem
- Hales
- 1992
(Show Context)
Citation Context ... Set Formula C.2. a(y1, y2, . . . , y6) = y1y2y3 + 1 2 y1(y 2 2 + y2 3 − y2 1 4 ) + 2 y2(y 2 1 + y2 3 − y2 1 5 ) + 2 y3(y 2 1 + y2 2 − y2 6 ). (7) Lemma C.3. sol(S) = 2 arccot( 2a ∆ 1/2). Proof. (See =-=[3]-=-, p.64). We use the branch of arccot taking values in [0, π]. C.2.19 U Define u : [4, 16] 3 → R by u(x1, x2, x6) = (y1 + y2 + y6)(y1 + y2 − y6)(y1 − y2 + y6)(−y1 + y2 + y6) C.2.20 vol = −x 2 1 − x 2 2... |

13 |
A new bound on the local density of sphere packing, Discrete and comp. geom
- Muder
(Show Context)
Citation Context ...of 0.755. Upper bounds on the density based on lower bounds on the volume of a Voronoi cell in the literature include Rogers’ upper bound 0.7797 [33], and Muder’s upper bounds 0.77836 [27] and 0.7731 =-=[28]-=-. In 1993, Hsiang published what he claimed to be proofs of the Kepler conjecture and the Dodecahedral conjecture [22]. However, the proof did not hold up to careful analysis. “As of this writing, Kep... |

13 |
The problem of the thirteen spheres
- Leech
- 1956
(Show Context)
Citation Context ...phere in a packing of equal spheres is at least the volume of a regular dodecahedron with inradius 1. 1 Introduction and History It is known that at most 12 congruent spheres can be tangent to a 13th =-=[12]-=-. In 1943, L. Fejes Tóth conjectured that a lower bound on the volume of a Voronoi polyhedron of a sphere, S0, in a packing of unit spheres is obtained by arranging 12 spheres around S0 such that its ... |

11 | Flyspeck I: Tame graphs
- Nipkow, Bauer, et al.
(Show Context)
Citation Context ...of the Dodecahedral conjecture. These long-term projects will take many years to complete. Nevertheless, significant progress has already been made toward the formal verification of the computer code =-=[29, 30]-=-. The revisions in this article incorporate the parts of Flyspeck Light that have already been completed. 1.5 Truncation The distance from the center of the regular dodecahedron to a vertex is tdod = ... |

9 |
Some algorithms arising in the proof of the Kepler conjecture
- Hales
- 2003
(Show Context)
Citation Context ... the subregion P is concave at the corner v, so that the arc from p(v) to p(w) begins in P, then crosses out at e(v1, v2). Geometric considerations show that |v1 − v2| ≥ 2.91. In fact, Problem 2.3 of =-=[5]-=- shows that the shortest possible distance for |v1 − v2| under the condition that |v − w| ≤ 2 is the length of the segment passing through the triangle of sides 2, 2T, 2T with both endpoints at distan... |

7 |
Formal Global Optimisation with Taylor Models,” in IJCAR, ser
- Zumkeller
- 2006
(Show Context)
Citation Context ...cribed in R. B. Kearfott [24], based on interval arithmetic. Our methods are similar to algorithms in widespread use for rigorous global optimization. Closely related algorithms are also described in =-=[37]-=-. The method is based on an iteratively refined partition of the domain R into a finite number of smaller and smaller rectangles that cover R. Start with X = {R}, then repeat the following procedure. ... |

7 |
Tóth, Über die dichteste Kugellagerung
- Fejes
- 1943
(Show Context)
Citation Context ...ron of a sphere in a packing of congruent spheres in E 3 means the set of the points that are not further from the center of the given sphere than from any other sphere center of the packing” [1]. In =-=[6]-=-, the dodecahedral conjecture is stated and proved with an assumption about the closest a 13th sphere could get. However, this seemingly benign assumption ended up being more difficult to prove than e... |

6 |
Putting the best face on a Voronoi polyhedron
- Muder
- 1988
(Show Context)
Citation Context ...n ≤ 12. We rely heavily on this result in our method of proof. Upper bounds on the density of packings have been improved gradually over the years. In [16], Rogers proves an upper bound of 0.7797. In =-=[13]-=- and 1[14], Muder makes significant improvements on Rogers’ bound with bounds of 0.77836 and 0.7731 respectively. The dodecahedral conjecture implies a bound of 0.754697 . . .. In 1993, Hsiang publis... |

5 |
Flyspeck II: The Basic Linear Programs
- Obua
(Show Context)
Citation Context ...of the Dodecahedral conjecture. These long-term projects will take many years to complete. Nevertheless, significant progress has already been made toward the formal verification of the computer code =-=[29, 30]-=-. The revisions in this article incorporate the parts of Flyspeck Light that have already been completed. 1.5 Truncation The distance from the center of the regular dodecahedron to a vertex is tdod = ... |

3 | Formalizing Plane Graph Theory — Towards a Formalized Proof of the Kepler Conjecture
- Bauer
- 2006
(Show Context)
Citation Context ... and Dodecahedral conjecture. They differ only in their input parameters. This computer program became the subject of G. Bauer’s dissertation in computer science at the Technical University of Munich =-=[2]-=-. This 172page dissertation translates the Java code into the formal theorem proving system Isabelle/HOL [31] and gives a detailed mathematical treatment of the graph theory underpinning the computer ... |

3 | Sphere packings in 3-space
- Bezdek
- 2004
(Show Context)
Citation Context ...ures that the surface area of any Voronoi cell in a packing of unit balls is at least that of a regular dodecahedron of inradius 1. This strengthened version of the Dodecahedral problem is still open =-=[5]-=-. 21.2 The sphere packing problem The Kepler conjecture, also known as the sphere packing problem, asserts that no packing of congruent balls in three dimensions has density greater than the density ... |

2 |
Isoperimetric inequalities and the dodecahedral conjecture
- Bezdek
- 1997
(Show Context)
Citation Context ...cture and the Dodecahedral conjecture [22]. However, the proof did not hold up to careful analysis. “As of this writing, Kepler’s conjecture as well as the dodecahedral conjecture are still unproven” =-=[4, p761]-=-. See also, [12]. An alternative approach to the Dodecahedral conjecture is described in [4]. Unfortunately, a counterexample has been found to both parts of the third conjecture of that article. The ... |

2 |
KeplerCode: computer resources for the Kepler and Dodecahedral Conjecutures. http://code.google.com/p/kepler-code
- McLaughlin
(Show Context)
Citation Context ...’s decision procedure for real-closed fields, and formal theorem proving packages. 3.1 Electronic resources A permanent archive has been set up for all of the external resources related to this proof =-=[26]-=-. This archive is under version control by Google Code [10]. The site consists of a download area where one may obtain the source code and supporting documents to this paper. Additionally there is a s... |

1 |
The Flyspeck Fact Sheet, 2003 (revised 2007). http://code.google. com/p/flyspeck/wiki/FlyspeckFactSheet
- Hales
(Show Context)
Citation Context ...full version of the proof. The computer code has also been entirely rewritten. A formalization project, called Flyspeck, aims to provide a complete formalization of the proof of the Kepler conjecture =-=[15, 11]-=-. (A formal proof is one in which every logical inference of the proof has been independently checked by computer, all the way to the primitive axioms at the foundations of mathematics.) A parallel pr... |

1 | A proof of the Dodecahedral conjecture (2002 version). http://arxiv.org/abs/math/9811079v1 - Hales, McLaughlin |

1 |
Tits. A C code for solving (large scale) constrained nonlinear (minimax) optimization problems, generating iterates satisfying all inequality constraints
- Lawrence, Zhou, et al.
- 1997
(Show Context)
Citation Context ...inimum of f1 on the domain {x ∈ [a1, b1] × · · · × [am, bm] : f2(x) ≤ 0, . . . , fr(x) ≤ 0} is positive. Nonlinear optimization libraries have been used to test all the inequalities in the collection =-=[25]-=-, [6]. The code generates a large random set X of points in the domain and runs the algorithm for each initial point x ∈ X to find a local minimum to the objective function f1. If X is sufficiently la... |

1 |
Press et al. Numerical Recipes in C, volume Chapter 20. Less-Numerical Algorithms
- H
- 1992
(Show Context)
Citation Context ...ic to control for floating-point rounding errors. Every real number x is represented on the computer as an interval [a, b] containing x, where a and b are exactly representable floating point numbers =-=[1, 32]-=-. The calculations conform to IEEE-754 standards [23]. Approximations to inverse trigonometric functions are based on published approximations [20]. 143.7 Nonlinear optimization Previous subsections ... |

1 |
Rigorous Global Optimization
- Zumkeller
- 2008
(Show Context)
Citation Context ...d by R. Zumkeller for the theorem proving system Coq [3], although it has not been used to give a formal verification of any of the inequalities that arise in the proof of the Dodecahedral conjecture =-=[38]-=-. These implementations are all independent of one another. (Algorithms were shared among us, but the code was independently implemented.) By comparing the proofs of different inequalities in differen... |

1 |
et al., Numerical Recipes in C, Chapter 20. LessNumerical Algorithms, Second Edition
- Press
- 1992
(Show Context)
Citation Context ...l over round-off errors [10]. These methods may be reliably implemented on machines that allow arithmetic with directed rounding, for example those conforming to the IEEE/ANSI standard 754 [17],[11], =-=[15]-=-. D.2 Interval Arithmetic Interval arithmetic produces an interval in the real line that is guaranteed to contain the result of an arithmetic operation. As the round-off errors accumulate, the interva... |