# Dense Sphere Packings: A Blueprint for Formal Proofs by Thomas Hales

By Thomas Hales

The 400-year-old Kepler conjecture asserts that no packing of congruent balls in 3 dimensions may have a density exceeding the popular pyramid-shaped cannonball association. during this e-book, a brand new facts of the conjecture is gifted that makes it available for the 1st time to a large mathematical viewers. The publication additionally offers strategies to different formerly unresolved conjectures in discrete geometry, together with the powerful dodecahedral conjecture at the smallest floor quarter of a Voronoi cellphone in a sphere packing. This ebook can also be at the moment getting used as a blueprint for a large-scale formal evidence undertaking, which goals to envision each logical inference of the facts of the Kepler conjecture via desktop. this is often an critical source should you are looking to be cited thus far with study at the Kepler conjecture.

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It follows that sin is nonnegative on [0, π/2]. It is enough to check that sin2 (π/2) equals 1. Then sin2 (π/2) = 1 − cos2 (π/2) = 1. 11 sin(π/2 − x) = cos(x), cos(π/2 − x) = sin(x). 6), sin(π/2 − x) = sin(π/2) cos(−x) + cos(π/2) sin(−x) and use sin(π/2) = 1 and cos(π/2) = 0. Then use that cos is an even function. The second identity is similar. 30 Trigonometry Similarly, cos(π/2 + x) = − sin(x), sin(π/2 + x) = cos(x). 13 = sin(x), cos(2π + x) = − cos(π + x) = cos(x). 12) The sine function is nonnegative on [0, π].

By contrast, a graph, endowed with a fixed embedding into the plane, is a plane graph. A planar graph has too little structure for our purposes because it does not single out a particular embedding and the plane graph has too much structure because it gives a topological object where combinatorics alone should suﬃce. A hypermap gives just the right amount of structure. It is a purely combinatorial notion, yet encodes the relations among nodes, edges, and faces determined by the embedding. An entire chapter of this book is about hypermaps.

This definition gives a compact notation that encompasses many of the standard polyhedra (points, lines, planes, rays, half-planes, half-spaces, convex hulls, aﬃne hulls) that appear throughout the book. From there, the reader can consult the definition of two important polynomials ∆ and υ, make a note of the unorthodox notation arc(a, b, c) for the angle opposite c of a triangle with sides of lengths a, b, c, stop a moment to admire Euler’s formula for the solid angle of a spherical triangle; and then jump directly to the final section, which introduces polar cycle.