Summary. This part of this book, which is the first of the four foundational chapters, presents a systematic development of trigonometry, volume, hypermap, and fan. There is a separate chapter on each of these topics. The purpose of the this material is to build a bridge between the foundations of mathematics, as presented in formal theorem proving systems such as HOL Light, and the solution to the packing problem.

In this chapter, trigonometry is developed analytically. Basic trigonometric functions are defined by their power series representations, and calculus of a single real variable is used to develop the basic properties of these functions. Basic vector geometry is presented.

Background Knowledge

formal proof

We repeat that our purpose is to give a blueprint of the formal proof of Kepler's conjecture that no packing of congruent balls in three-dimensional Euclidean space has density greater than the familiar cannonball packing. The blueprint of a formal proof is not the same as a formal proof, which is a fleeting pattern of bits in a computer. The book describes to the reader how to construct the computer code that produces and then reliably reproduces that pattern of bits.

A more traditional book might take as its starting point the imagined mathematical background of a typical reader. The blueprint of a formal proof starts instead with the current mathematical background of a formal proof assistant. I surveyed the knowledge base of my formal proof assistant and compared it with what is needed in the construction of our formal proof. It turns out that the proof assistant already has an adequate background in real analysis, basic topology, and plane trigonometry, including the trigonometric addition laws, and formulas for derivatives.

This book is just a blueprint, which gives instructions about how to construct the formal proof. Flyspeck is the name of an ongoing project to construct a formal proof of the Kepler conjecture in the HOL Light proof assistant, along the lines described in this book. The eventual aim of the project is to give a formal verification of the computer portions of the proof as well as the standard text portions of the proof. The project is about 80% complete as of May 2012. The source code for the project and information about the current project status are available at [21].

Here is the fine print about the current project status. (I hope that this status report is out of date by the time this book is printed.) There are four components to the formalization project (text, hypermaps, linear programs, and nonlinear inequalities), at various stages of completion.

History

This section gives a brief history of the study of dense sphere packings. Further details appear at [43] and [20]. The early history of sphere packings is concerned with the face-centered cubic (FCC) packing, a familiar pyramid arrangement of congruent balls used to stack cannonballs at war memorials and oranges at fruit stands (Figure 1.1).

Sanskrit sources

The study of the mathematical properties of the FCC packing can be traced to a Sanskrit work (the Āryabhaṭīya of Āryabhaṭa) composed around 499 CE. The following passage gives the formula for the number of balls in a pyramid pile with triangular base as a function of the number of balls along an edge of the pyramid [40].

In modern notation, the passage gives two formulas for the number of balls in a pyramid with n balls along an edge (Figure 1.2):

Harriot and Kepler

The modern mathematical study of spheres and their close packings can be traced to Harriot. His work – unpublished, unedited, and largely undated – shows a preoccupation with sphere packings. He seems to have first taken an interest in packings at the prompting of Sir Walter Raleigh.

Summary. A planar graph, which is a graph that admits a planar embedding, has too little structure for our purposes because it does not specify a particular embedding. A plane graph carries a fixed embedding, which gives it a topological structure where combinatorics alone should suffice. A hypermap gives just the right amount of structure. It is a purely combinatorial object, but carries information that the planar graph lacks by encoding the relations among nodes, edges, and faces. This chapter is about hypermaps.

In the original proof of the Kepler conjecture, the basic combinatorial structure was that of a planar map, as defined by Tutte [47]. Although planar maps appear throughout that proof, they are lightweight objects, in the sense that no significant structural results are needed about them.

Gonthier makes hypermaps the fundamental combinatorial structure in his formal proof of the four-color theorem [16]. His formal proof eliminates topological arguments such as the Jordan curve theorem in favor of purely combinatorial arguments. When I learned of Gonthier's work, I significantly reorganized the proof by replacing planar maps with hypermaps, making them heavyweight objects, in the sense that significant structural results about them are needed.

As a result of these changes, many parts of the proof that were originally done topologically can now be done combinatorially, a change that significantly reduces the effort required to formalize the proof. These changes also make it possible to treat rigorously what was earlier done by geometric intuition.