• Corpus ID: 13892644

Introduction to the Flyspeck Project

  title={Introduction to the Flyspeck Project},
  author={Thomas C. Hales},
  booktitle={Mathematics, Algorithms, Proofs},
  • T. Hales
  • Published in
    Mathematics, Algorithms…
  • Physics
This article gives an introduction to a long-term project called Flyspeck, whose purpose is to give a formal verification of the Kepler Conjecture. The Kepler Conjecture asserts that the density of a packing of equal radius balls in three dimensions cannot exceed $pi/sqrt{18}$. The original proof of the Kepler Conjecture, from 1998, relies extensively on computer calculations. Because the proof relies on relatively few external results, it is a natural choice for a formalization effort. 

A Revision of the Proof of the Kepler Conjecture

The current status of a long-term initiative to reorganize the original proof of the Kepler conjecture into a more transparent form and to provide a greater level of certification of the correctness of the computer code and other details of the proof is summarized.


This paper constitutes the official published account of the now completed Flyspeck project and describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants.

hosted at the Radboud Repository of the

This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published

The HOL Light Theory of Euclidean Space

  • J. Harrison
  • Mathematics
    Journal of Automated Reasoning
  • 2012
This formalization was started in 2005 and has been extensively developed since then, partly in direct support of the Flyspeck project, partly out of a general desire to develop a well-rounded and comprehensive theory of basic analytical, geometrical and topological machinery.

Ordered geometry in Hilbert's Grundlagen der Geometrie

The Grundlagen der Geometrie brought Euclid’s ancient axioms up to the standards of modern logic, anticipating a completely mechanical verification of their theorems. There are five groups of axioms,

Formal computations and methods

This work is an integral part of the Flyspeck project (a formal proof of the Kepler conjecture) and it is shown how developed formal procedures solve formal computational problems in this project.

Formal Proofs for Nonlinear Optimization

The implementation tool interleaves  semialgebraic approximations with sums of squares witnesses to form certificates and produces both valid underestimators and lower bounds for each approximated constituent.

Formal Proofs for Global Optimization - Templates and Sums of Squares. (Preuves formelles pour l'optimisation globale - Méthodes de gabarits et sommes de carrés)

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions and to prove their correctness by checking the certificates in the Coq proof system, using a software package named NLCertify.

Mechanising Hilbert's Foundations of Geometry in Isabelle

This project continues and revises Meikle’s mechanisation of Hilbert’s Foundations of Geometry in Isabelle/HOL, focusing on declarative-style proofs to create readable and maintainable proof

Initial Experiments on Deriving a Complete HOL Simplification Set

This paper describes the initial experiments with the aim to close the gap and use rewriting to compute a complete rst-order simplication set for a HOL-based proof assistant fully automatically.



Some Algorithms Arising in the Proof of the Kepler Conjecture

  • T. Hales
  • Computer Science, Mathematics
  • 2002
A series of nonlinear optimization algorithms are given and it is shown how a systematic application of these algorithms would bring substantial simplifications to the original proof of the Kepler conjecture.

An overview of the Kepler conjecture

This is the first in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than

Sphere packings, I

  • T. Hales
  • Physics, Mathematics
    Discret. Comput. Geom.
  • 1997
A program to prove the Kepler conjecture on sphere packings is described and it is shown that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.

Bounds for Local Density of Sphere Packings and the Kepler Conjecture

The particular local density inequality underlying the Hales and Ferguson approach to prove Kepler’s conjecture is described and some features of their proof are sketched.

The Jordan-Scho¨nflies theorem and the classification of surfaces

INTRODUCTION. The Jordan curve theorem says that a simple closed curve in the Euclidean plane partitions the plane into precisely two parts: the interior and the exterior of the curve. Although this

Theory on plane curves in non-metrical analysis situs

JORDAN'St explicit formulation of the fundamental theorem that a simple closed curve lying wholly in a plane decomposes the plane into an inside and an outside region is justly regarded as a most

The Kissing Problem in Three Dimensions

  • O. Musin
  • Mathematics
    Discret. 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.

A Proof of the

We present a new proof of the version of the Shauder-Tychonov theorem provided by Coppel in Stability and Asymptotic Behavior of Difierential Equations, Heath Mathematical Monographs, Boston (1965).

A Proof-Producing Decision Procedure for Real Arithmetic

This is the first generally useful proof-producing implementation of a quantifier elimination procedure for real closed fields, and convincing examples of its value in interactive theorem proving are demonstrated.

Euclidean and Non-Euclidean Geometry: An Analytic Approach

Preface Notation and special symbols Historical introduction 1. Plane Euclidean geometry 2. Affine transformations in the Euclidean plane 3. Finite groups of isometries of E2 4. Geometry on the