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 usual pyramid-shaped cannonball association. during this booklet, a brand new evidence of the conjecture is gifted that makes it available for the 1st time to a wide 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 quarter of a Voronoi phone in a sphere packing. This publication is usually at the moment getting used as a blueprint for a large-scale formal facts undertaking, which goals to envision each logical inference of the facts of the Kepler conjecture via machine. this is often an essential source if you happen to are looking to be stated to this point with study at the Kepler conjecture.
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Additional resources for Dense Sphere Packings: A Blueprint for Formal Proofs
Let a be the angle between v1 and v3 , and let b be the angle between v1 and v2 at v0 . Let A = cos a, B = cos b, C = cos c, A = sin a, B = sin b, C = sin c. The spherical law of cosines gives B − AC 2 p sin2 β = 1 − = 2 2, AC A C where p = 1 − A2 − B2 − C 2 + 2ABC. In particular, p ≥ 0. A computation of sin2 α and the remaining terms in the same way gives p sin α sin β = A BC2 C − AB A − BC B − AC pC + = . AB BC AC A BC2 The result follows by real arithmetic. cos γ + cos α cos β = The following lemma gives a formula for the dihedral angle of a tetrahedron along an edge in terms of its edge lengths.
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. Since the proof assistant already has a significant library of theorems in real analysis and point-set topology, we use background facts in these areas wherever they help.
On this interval, sin is nonnegative. By real analysis, it is enough to check that the squares of the two nonnegative numbers are equal. 23. The following lemma shows how to rewrite arccos in terms of arctan2 . 25 33 If y ∈ [−1, 1], then arccos(y) + arctan2 1 − y2 , y = π/2. Proof The brief justification is simply that arccos(y/z) gives one acute angle of a right triangle with hypotenuse z and sides x and y, and arctan2 (x, y) gives the other acute angle. The two acute angles of a right triangle have sum π/2.
Dense Sphere Packings: A Blueprint for Formal Proofs by Thomas Hales