By Thomas Hales

The 400-year-old Kepler conjecture asserts that no packing of congruent balls in 3 dimensions could have a density exceeding the normal pyramid-shaped cannonball association. during this ebook, a brand new evidence of the conjecture is gifted that makes it obtainable for the 1st time to a extensive mathematical viewers. The booklet additionally offers options to different formerly unresolved conjectures in discrete geometry, together with the robust dodecahedral conjecture at the smallest floor region of a Voronoi mobile in a sphere packing. This e-book is additionally at present getting used as a blueprint for a large-scale formal facts undertaking, which goals to ascertain each logical inference of the evidence of the Kepler conjecture by means of desktop. this can be an necessary source should you are looking to be mentioned up to now with learn at the Kepler conjecture.

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**Additional resources for Dense Sphere Packings: A Blueprint for Formal Proofs**

**Example text**

6 by the second line of the same lemma. Then use the definition of the tangent. 16 tan(π/4) = 1. Proof tan(π/4) = sin(π/2 − π/4)/ cos(π/4) = cos(π/4)/ cos(π/4) = 1. 17 The function tan is strictly increasing and one-to-one on the domain (−π/2, π/2). Proof By a derivative test, the tangent is strictly increasing on (−π/2, π/2). By real arithmetic, a strictly increasing function is one-to-one. 4 arctangent This section reviews the properties of the arctangent function. 18 (arctangent) By the inverse function theorem of real analysis and properties of tan, there is a unique function arctan : R → R with image (−π/2, π/2) such that tan(arctan x) = x.

This completes the proof. 6 Dense Packings in a Nutshell This section describes the proof of the Kepler conjecture in general, without getting embroiled in detail. The entire book is a blueprint with all the electrical schematics, plumbing, and ventilation systems. This section is the tourist brochure. 74048 of the FCC packing. For a contradiction, we suppose that an explicit counterexample exists to the Kepler conjecture √ in the form of a packing of balls of unit radius with density greater than π/ 18.

To simplify the exposition, this section presents the original definitions. 1 Trigonometric and inverse trigonometric functions. By real analysis, convergence is absolute for every real number x. Each series can be evaluated at 0: cos(0) = 1, sin(0) = 0. 3) These series may be diﬀerentiated term by term to establish the identities: d d cos(x) = − sin(x), sin(x) = cos(x). 4) dx dx Powers (cos(x))n and (sin(x))n are conventionally written cosn (x) and sinn (x). If two functions are the unique solution of an ordinary diﬀerential equation with given initial conditions, then the two functions are equal.