The Verification of the Proof of the Kepler Conjecture (slides)
Symposium on Evidence in the Natural Sciences
May 30, 2014
Thomas Hales Pittsburgh University
The Verification of the Proof of the Kepler Conjecture
1.2 Face-Centered Cubic
7
Figure 1.4 [PTFTWZM] Newton’s claim – twelve is the maximum number of congruent balls that can be tangent to a given congruent ball– was confirmed in the 1953. Musin and Tarasov only recently proved that the arrangement shown here is the unique arrangement of thirteen congruent balls that shrinks the thirteen by the least possible amount to permit tangency [32]. Each node of the graph represents one of the thirteen balls and each edge represents a pair of touching balls. The node at the center of the graph corresponds to the uppermost ball in the second frame. The other twelve balls are perturbations of the FCC tangent arrangement.
Figure 1.5 [NTNKMGO] The pyramid on a square base is the same lattice packing as the pyramid on a triangular base. The only differences are the orientation of the lattice in space and the exposed facets of the lattice. Their orientation and exposed facets are matched as shown.
cking with its set V of centers. For our purposes, a packing is just a set of nts in R3 in which the elements are separated by distances of at least 2. he density of a packing is the ratio of the volume occupied by the balls to volume of a large container. The purpose of a finite container is to prevent volumes from becoming infinite. To eliminate the distortion of the packing sed by the shape of the its boundary, we take the limit of the densities within ncreasing sequence of spherically shaped containers, as the diameter tends nfinity. he FCC packing is obtained from a cubic lattice, by inserting a ball at each he eight extreme points of each cube and then inserting a another ball at center of each of the six facets of each cube (Figure 1.6). The name face-
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!
1.3 Hexagonal-Close Packing C
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B
A
A
B
B FCC
HCP
Figure 1.11 [SGIWBEN] The patterns of twelve neighboring points in the FCC and HCP packings. In both cases, the convex hull of the twelve points is a polyhedron with six squares and eight triangles, but the top layer of the HCP pattern is rotated 60 degrees with respect to the FCC pattern. The FCC pattern is a cuboctahedron. In the HCP pattern, there is a uniquely determined plane of reflectional symmetry, containing six of the twelve points.
Thursday, May 29, 2014
√ There are, in fact, uncountably many packings of density π/ 18 in which the tangent arrangement around each ball is either the FCC pattern or the HCP pattern. A hexagonal layer (Figure 1.12) is a translate of the two-dimensional hexagonal lattice (also known as the triangular lattice). That is, it is a translate of the planar lattice generated by two vectors of length 2 and angle 2π/3. The FCC packing is an example of a packing built from hexagonal layers. If L is a hexagonal layer, then a second hexagonal layer L" can be placed parallel to the first so that each lattice point of L" has distance 2 from three different points of L, which is the smallest possible distance from first layer. A choice of a unit normal vector e to the plane of L determines an upward direction. There are two different positions in which L" can be closely placed above L (Figure 1.12). Each successive layer (L, L" , L"" , and so forth) offers two further choices for the placement of that layer. Running through different sequences of choices gives uncountably many packings. In each of these packings the tangent arrangement around each ball is the FCC or HCP arrangement. As a packing is constructed, each layer may be labeled A, B, or C depending
The face-centered cubic packing is “the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container.” (Kepler, 1611)
3
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“Many mathematicians believe and all physicists know” [that the pyramid arrangement is best]. (Rogers, 1958) The “problem in 3-dimensions remains unsolved. This is a scandalous situation since the (presumably) correct answer has been known since the time of Gauss . . . All this is missing is a proof.” (Milnor, 1976)
Thursday, May 29, 2014
In 1975, Buckminster Fuller claimed to have a proof, but his arguments were faulty. In the early 1990s, a Berkeley professor claimed to have a proof. His arguments turned out to be incorrect as well. As recently as last month, there were new unconfirmed announcements on the Kepler conjecture Wiki of a new proof.
Thursday, May 29, 2014
After working on the conjecture for several years, with the help of a graduate student Sam Ferguson, I announced a proof of the Kepler conjecture in 1998.
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THE KEPLER CONJECTURE
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Section A.2.7†. If the circumradius of a quasi-regular tetrahedron is ≥ 1.41, then by [I.9.17], τ > 1.8 pt, and many of the inequalities hold. In Sections A.2.7 and A.2.8, let S1 , . . . , S5 be 5 simplices arranged around a common edge (0, v), with |v| ∈ [2, 2.51]. Let yi (Sj ) be the edges, with y1 (Sj ) = |v| for all j, y3 (Sj ) = y2 (Sj+1 ), and y5 (Sj ) = y6 (Sj+1 !). where the subscripts j are extended modulo 5. In Sections A.2.7 and A.2.8, dih(Sj ) ≤ 2π. Set π0 = 2ξV + ξΓ if σ ˆ = vor0 in the cases (y4 ≥ 2.6, y1 ≥ 2.2) and (y4 ≥ 2.7). Set π0 = 0, otherwise. √ (551665569) τ (S1 ) + τ (S2 ) + τ (S4 ) > 1.4 pt, if y4 (S3 ), y4 (S5 ) ≥ 2 2. √ (824762926) τ (S1 ) + τ (S2 ) + τ (S3 ) > 1.4 pt, if y4 (S4 ), y4 (S5 ) ≥ 2 2. √ τ (S1 ) + τ (S2 ) + (ˆ τ (S3 ) − π0 ) + τ (S4 ) > 1.4 pt + D(3, 1), if y4 (S3 ) ∈ [2.51, 2 2], y4 (S5 ) ≥ 2.51, dih(S5 ) > 1.32, (675785884) √ τ (S1 ) + τ (S2 ) + τ (S3 ) + (ˆ τ (S4 ) − π0 ) > 1.4 pt + D(3, 1), if y4 (S4 ) ∈ [2.51, 2 2], y4 (S5 ) ≥ 2.51, dih(S5 ) > 1.32. (193592217) Section A.2.8†. As in A.2.7, the quasi-regular tetrahedra are ! generally compression scored. Define π0 as in Section A.2.7. The constraint (5) dih(Sj ) = 2π is assumed. √ (325738864) τ (S1 ) + τ (S2 ) + τ (S3 ) + τ (S4 ) > 1.5 pt, if y4 (S5 ) ≥ 2 2. √ τ (S1 )+τ (S2 )+τ (S3 )+τ (S4 )+(ˆ τ (S5 )−π0 ) > 1.5 pt+D(3, 1), if y4 (S5 ) ∈ [2.51, 2 2]. (314974315)
The proof was nearly 300 pages long and relied on long computer calculations that had been made over a period of years. The there were about 3GB of data backing up the calculations.
Section A.3.1. τ − 0.2529 dih > −0.3442, if y1 ∈ [2.3, 2.51], and dih ≥ 1.51. (572068135) √ τ0 − 0.2529 dih > −0.1787, if y1 ∈ [2.3, 2.51], y6 ∈ [2 2, 3.02], 1.26 ≤ dih ≤ 1.63. (723700608) √ (560470084) τˆ − 0.2529 dih2 > −0.2137, if y2 ∈ [2.3, 2.51], y4 ∈ [2.51, 2 2], τ0 − 0.2529 dih > −0.1371, if y1 ∈ [2.3, 2.51], y5 , y6 ∈ [2.51, 3.02], 1.14 ≤ dih ≤ 1.51. (535502975) Section A.3.8. A. dih < 1.63, if y6 ≥ 2.51, y2 , y3 ∈ [2, 2.168]. (821707685) B. dih < 1.51, if y5 = 2.51, y6 ≥ 2.51, y2 , y3 ∈ [2, 2.168]. (115383627) √ C. dih < 1.93, if y6 ≥ 2.51, y4 = 2 2, y2 , y3 ∈ [2, 2.168]. (576221766) √ D. dih < 1.77, if y5 = 2.51, y6 ≥ 2.51, y4 = 2 2, y2 , y3 ∈ [2, 2.168]. (122081309) τ0 − 0.2529 dih > −0.2391, if y6 ≥ 2.51, dih ≥ 1.2, y2 , y3 ∈ [2, 2.168]. (644534985) τ0 − 0.2529 dih > −0.1376, if y5 = 2.51, y6 ≥ 2.51, dih ≥ 1.2, and y2 , y3 ∈ [2, 2.168]. (467530297) √ τ0 −0.2529 dih > −0.266, if y6 ≥ 2.51, y4 ∈ [2.51, 2 2], dih ≥ 1.2, y2 , y3 ∈ [2, 2.168]. (603910880) √ τ0 − 0.2529 dih > −0.12, if y5 = 2.51, y6 ≥ 2.51, y4 ∈ [2.51, 2 2], dih ≥ 1.2, y2 , y3 ∈ [2, 2.168]. (135427691) dih < 1.16, if y5 = 2.51, y6 ≥ 2.51, y4 = 2, y2 , y3 ∈ [2, 2.168]. (60314528) τ0 − 0.2529 dih > −0.1453, if y2 , y3 ∈ [2, 2.168], y5 ∈ [2.51, 3.488], y6 = 2.51. (312132053)
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46
THOMAS C. HALES
Section A.3.8, Case 2-b. dih2 > 0.74, if y1 ∈ [2.51, 2.696], y2 , y3 ∈ [2, 2.168]. (751442360) τ0 − 0.2529 dih > −0.2391, if ∆(y52 , 4, 4, 8, 2.512, y62 ) ≥ 0, y2 , y3 ∈ [2, 2.168], y5 ∈ [2.51, 3.488]. (893059266) dih +0.5(2.402 − y4 ) < π/2, if y5 ≥ 2.51, y2 , y3 ∈ [2, 2.168]. (690646028) Section A.3.9. √ dih > 1.78, if y4 = 3.2, y1 ∈ [2.51, 2 2], y2 + y3 ≤ 4.6.
(161665083)
Section A.4.4. The following inequalities hold for flat quarters. In these inequalities the fourth edge is the diagonal.
− dih2 + 0.35y2 − 0.15y1 − 0.15y3 + 0.7022y5 − 0.17y4 > −0.0123,
dih2 − 0.13y2 + 0.631y1 + 0.31y3 − 0.58y5 + 0.413y4 + 0.025y6 > 2.63363, − dih1 + 0.714y1 − 0.221y2 − 0.221y3 + 0.92y4 − 0.221y5 − 0.221y6 > 0.3482, dih1 − 0.315y1 + 0.3972y2 + 0.3972y3−
0.715y4 + 0.3972y5 + 0.3972y6 > 2.37095, − sol − 0.187y1 − 0.187y2 − 0.187y3 + 0.1185y4 + 0.479y5 + 0.479y6 > 0.437235†, sol + 0.488y1 + 0.488y2 + 0.488y3 − 0.334y5 − 0.334y6 > 2.244†, −ˆ σ − 0.145y1 − 0.081y2 − 0.081y3 − 0.133y5 − 0.133y6 > −1.17401,
−ˆ σ − 0.12y1 − 0.081y2 − 0.081y3 − 0.113y5 − 0.113y6 + 0.029y4 > −0.94903, σ ˆ + 0.153y4 + 0.153y5 + 0.153y6 < 1.05382, σ ˆ + 0.419351 sol +0.19y1 + 0.19y2 + 0.19y3 < 1.449, σ ˆ + 0.419351 sol < −0.01465 + 0.0436y5 + 0.0436y6 + 0.079431 dih, σ ˆ < 0.0114, τˆ > 1.019 pt, (4.4.1)
Inequalities (4.4.1) (867513567) If there are four quasi-regular tetrahedra {S1 , . . . , S4 } at the central vertex of the flat quarter Q, then (4.4.2†)
σ ˆ (Q) +
!
σ(Si ) < 0.114.
(4)
Inequalities 4.4.2
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(867359387)
THE KEPLER CONJECTURE
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Section A.4.5. In a quadrilateral cluster, with a given edge y4 as the diagonal, the other diagonal will be denoted y4! . The following relations for upright quarters (scored by ν) hold. (We use the inequalities of III.A for upright quarters in quad clusters, which are scored by a different function.) In these inequalities the upright diagonal is the first edge. We include in this group, the inequalities IV.A2 , IV.A3 for upright quarters.
y1 > 2.51, √ y1 < 2 2, dih1 − 0.636y1 + 0.462y2 + 0.462y3 − 0.82y4 + 0.462y5 + 0.462y6 > 1.82419,
− dih1 + 0.55y1 − 0.214y2 − 0.214y3 + 1.24y4 − 0.214y5 − 0.214y6 > 0.75281, dih2 + 0.4y1 − 0.15y2 + 0.09y3 + 0.631y4 − 0.57y5 + 0.23y6 > 2.5481, − dih2 − 0.454y1 + 0.34y2 + 0.154y3 − 0.346y4 + 0.805y5 > −0.3429, dih3 + 0.4y1 − 0.15y3 + 0.09y2 + 0.631y4 − 0.57y6 + 0.23y5 > 2.5481, − dih3 − 0.454y1 + 0.34y3 + 0.154y2 − 0.346y4 + 0.805y6 > −0.3429, sol + 0.065y2 + 0.065y3 + 0.061y4 − 0.115y5 − 0.115y6 > 0.2618,
− sol − 0.293y1 − 0.03y2 − 0.03y3 + 0.12y4 + 0.325y5 + 0.325y6 > 0.2514, −ν − 0.0538y2 − 0.0538y3 − 0.083y4 − 0.0538y5 − 0.0538y6 > −0.5995, ν ≤ 0,
(Calculations F.3.13.3, F.3.13.4)
τν − 0.5945 pt > 0.
(A.4.5.1)
(498839271)
ν − 4.10113 dih1 < −4.3223,
ν − 0.80449 dih1 < −0.9871, ν − 0.70186 dih1 < −0.8756,
ν − 0.24573 dih1 < −0.3404,
ν − 0.00154 dih1 < −0.0024, ν + 0.07611 dih1 < 0.1196.
(A.4.5.2, IV.A2 )
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THOMAS C. HALES
τν + 4.16523 dih1 > 4.42873, τν + 0.78701 dih1 > 1.01104, τν + 0.77627 dih1 > 0.99937, τν + 0.21916 dih1 > 0.34877, τν + 0.05107 dih1 > 0.11434, (A.4.5.3, IV.A3 )
τν − 0.07106 dih1 > −0.07749.
The following additional inequalities are known to hold if the upright diagonal has height at most 2.696. νΓ denotes the restriction of ν to a simplex of compression type. y1 < 2.696, dih1 − 0.49y1 + 0.44y2 + 0.44y3 − 0.82y4 + 0.44y5 + 0.44y6 > 2.0421, − dih1 + 0.495y1 − 0.214y2 − 0.214y3 + 1.05y4 − 0.214y5 − 0.214y6 > 0.2282, dih2 + 0.38y1 − 0.15y2 + 0.09y3 + 0.54y4 − 0.57y5 + 0.24y6 > 2.3398,
− dih2 − 0.375y1 + 0.33y2 + 0.11y3 − 0.36y4 + 0.72y5 + 0.034y6 > −0.36135, sol + 0.42y1 + 0.165y2 + 0.165y3 − 0.06y4 − 0.135y5 − 0.135y6 > 1.479,
− sol − 0.265y1 − 0.06y2 − 0.06y3 + 0.124y4 + 0.296y5 + 0.296y6 > 0.0997, −ν + 0.112y1 − 0.142y2 − 0.142y3 − 0.16y4 − 0.074y5 − 0.074y6 > −0.9029, ν + 0.07611 dih1 < 0.11,
νΓ − 0.015y1 − 0.16(y2 + y3 + y4 ) − 0.0738(y5 + y6 ) > −1.29285, τν − 0.07106 dih1 > −0.06429, τν > 0.0414.
(A.4.5.4) (319046543) In connection with Inequalities 4.5.4, we occasionally use the stronger constant 0.2345 instead of 0.2282. To justify this constant, we have checked using interval arithmetic that the the bound 0.2345 holds if y1 ≤ 2.68 or y4 ≤ 2.475. Further interval calculations show that the anchored simplices can be erased if they share an upright diagonal with such a quarter. Inequalities (4.5.5)
(365179082)
vor0 < −0.043/2, if y6 = 2.51, y1 ∈ [2.51, 2.696]. (368244553†) √ vor0 (S) + vor0 (S(2, y2 , y3 , y4 , 2, 2)) < −0.043, if y1 ∈ [2.51, 2.696], y4 ∈ [2 2, 3.2], y4! ≥ 2.51. (820900672†) √ vor0 (S)+vor0 (S(2.51, y2, y3 , y4 , 2, 2)) < −0.043, if y1 ∈ [2.51, 2.696], y4 ∈ [2 2, 3.2], y4! ≥ 2.51. (961078136†) Inequalities (4.5.7), (The last of these was verified by S. Ferguson.) (424186517)
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The proof was solicited for publication in the Annals of Math, a leading math journal. A dozen referees were assigned to check the proof. In January 1999, a conference was organized at the Institute for Advanced Study in Princeton. All the referees were invited to attend, and I gave a long series of lectures going through the details of the proof.
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“It is very unusual to have such a large set of reviewers. The main portion of the reviewing took place in a siminar at E¨otvos University, Budapest, over a three year period. Some reviewers made computer experiments, in a detailed check of specific parts of the proof.. . . In this process detailed checking of many specific assertions found them to be essentially correct in every case. This result of the reviewing process produced in these reviewers a strong degree of conviction of the essential correctness of this proof approach,. . . ” (J. Lagarias, editor)
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Robert MacPherson, editor of the Annals, wrote a report that states “The news from the referees is bad, from my perspective. They have not been able to certify the correctness of the proof, and will not be able to certify it in the future, because they have run out of energy to devote to the problem. This is not what I had hoped for.” “Fejes Toth thinks that this situation will occur more and more often in mathematics. He says it is similar to the situation in experimental science - other scientists acting as referees can’t certify the correctness of an experiment, they can only subject the paper to consistency checks. He thinks that the mathematical community will have to get used to this state of affairs.”
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The Kepler paper prompted the editors to issue a new policy on computer-assisted proofs. Statement by the Annals Editors on Computer-Assisted Proofs “The computer part may not be checked line-by-line, but will be examined for the methods by which the authors have eliminated or minimized possible sources of error” http://annals.math.princeton.edu/board
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In a broader context • Historically, no branch of human knowledge makes stronger claims to certainty than mathematics. • Mathematics has reached a level of unprecedented complexity. • Mathematics is making the transition to computer based systems. • Computers are very mathematical in some respects and very unmathematical in other respects.
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"#$!%$&'$()!*$+,-!.'!/+)#$-+)0(1 Mathematical Certainty
Historically, no branch of human knowledge makes Myth and Reality stronger claims to certainty than mathematics.
Ma
Mathematics has reached a level of unprecedented complexity.
!!!"##$%&'(&%)%*%)%+%,
Mathematics is making the transition to computer based systems.
)%*%)%+%)%*%-./%0 Computers %%+%-./%1)%*%02 are very mathematical in some respects and very unmathematical %%+%-./%) in other respects. %%+%, Thursday, May 29, 2014
Mathematics has reached a level of unprecedented complexity. Mathematics is making the transition to computer based systems. Computers are very mathematical in some respects and very unmathematical in other respects.
+ Four-color theorem, Grothendieck, Weil conjectures, Harish-Chandra, Lafforgue, Langlands, Arthur, Almgren, Kepler conjecture, geometrization theorem, etc. Thursday, May 29, 2014
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complexity. Mathematics is making the transition to computer based systems. Birch-Swinnerton Dyer conjecture, Sato-Tate Conjecture, Computers are very mathematical in some respects Lyons-Sims group of order 28 37 56 71 111 311 371 671 , original proof of the Calalan conjecture xm − xn = 1, qWZ proof of and very unmathematical in other respects. the Rogers-Ramunujan identities, conjectural optimal packings of tetrahedra (Chen-Engel-Glotzer), 4-color theorem, finite projective plane of order 10, Smale’s 14th problem on strange attractors in the Lorenz oscillator, Mandelbrot’s conjectures in fractal geometry, visualization of sphere eversions and Costa surface embeddings, the double bubble conjecture, construction of counterexamples to the Kelvin conjecture, calculation of kissing numbers (Sloane-Odlyzko), the character table for E 8 (Atlas project), Cohn-Kumar proof of the packing optimality of the Leech and E 8 packings among lattices, classification of fake projective planes, weak Goldbach, twin prime problem (2013).
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systems. Computers mathematical in some respects Historically,are novery branch of human knowledge makes stronger claims to certainty than mathematics. and very unmathematical in other respects. Mathematics has reached a level of unprecedented complexity. Mathematics is making the transition to computer based systems. Computers are very mathematical in some respects and very unmathematical in other respects. 16
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Will the transition to computer-based mathematics mark the end of mathematical conviction?
99% convinced. . . check of specific parts. . . strong degree of conviction of the essential correctness of this proof approach. . . similar to the situation in experimental science . . . the mathematical community will have to get used to this . . . not checked line-by-line
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IJCAR 2001 - Siena
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Computers were once human Referees were once human
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HOL Light - a formal proof assistant
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real numbers is denoted (&). We identify a packing of balls in R3 with the set V of centers of the balls. The balls in the packing are normalized to have unit length, so that the distance between distinct elements of V is at least 2. More formally, ‘(packing V (!u v. u IN V /\ v IN V /\ dist(u,v) < &2 ==> u = v))‘ We define the constant the kepler conjecture to be the term ‘the_kepler_conjecture (!V. packing V ==> (?c. !r. &1 &(CARD(V INTER ball(vec 0,r))) the_kepler_conjecture Figure 8.1 [HBMVLMY] Here are some of the tens of thousands of planar graphs whose hypermaps are tame. The ones depicted here are the ones that are the most difficult to eliminate through linear programming.
Lemma 8.12 (standard fan) [CKQOWSA] [formal proof by Alexey Solovyev]. Let V ⊂ B be a packing. Then (V, E std ) is a fan.
The project to check the Kepler conjecture by computer is over 95% complete. (Stayed tuned!) It automates the work that was once done by referees. The process is orders of magnitude more reliable than the traditional refereeing process.
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Credits: The project to formalize the Kepler conjecture is a large collaboration. Special thanks to Alexey Solovyev and Hoang Le Truong. Many images from this presentation do not originate with me. The images have been included for nonprofit educational purposes.
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May 30, 2014
Thomas Hales Pittsburgh University
The Verification of the Proof of the Kepler Conjecture
1.2 Face-Centered Cubic
7
Figure 1.4 [PTFTWZM] Newton’s claim – twelve is the maximum number of congruent balls that can be tangent to a given congruent ball– was confirmed in the 1953. Musin and Tarasov only recently proved that the arrangement shown here is the unique arrangement of thirteen congruent balls that shrinks the thirteen by the least possible amount to permit tangency [32]. Each node of the graph represents one of the thirteen balls and each edge represents a pair of touching balls. The node at the center of the graph corresponds to the uppermost ball in the second frame. The other twelve balls are perturbations of the FCC tangent arrangement.
Figure 1.5 [NTNKMGO] The pyramid on a square base is the same lattice packing as the pyramid on a triangular base. The only differences are the orientation of the lattice in space and the exposed facets of the lattice. Their orientation and exposed facets are matched as shown.
cking with its set V of centers. For our purposes, a packing is just a set of nts in R3 in which the elements are separated by distances of at least 2. he density of a packing is the ratio of the volume occupied by the balls to volume of a large container. The purpose of a finite container is to prevent volumes from becoming infinite. To eliminate the distortion of the packing sed by the shape of the its boundary, we take the limit of the densities within ncreasing sequence of spherically shaped containers, as the diameter tends nfinity. he FCC packing is obtained from a cubic lattice, by inserting a ball at each he eight extreme points of each cube and then inserting a another ball at center of each of the six facets of each cube (Figure 1.6). The name face-
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!
1.3 Hexagonal-Close Packing C
11
B
A
A
B
B FCC
HCP
Figure 1.11 [SGIWBEN] The patterns of twelve neighboring points in the FCC and HCP packings. In both cases, the convex hull of the twelve points is a polyhedron with six squares and eight triangles, but the top layer of the HCP pattern is rotated 60 degrees with respect to the FCC pattern. The FCC pattern is a cuboctahedron. In the HCP pattern, there is a uniquely determined plane of reflectional symmetry, containing six of the twelve points.
Thursday, May 29, 2014
√ There are, in fact, uncountably many packings of density π/ 18 in which the tangent arrangement around each ball is either the FCC pattern or the HCP pattern. A hexagonal layer (Figure 1.12) is a translate of the two-dimensional hexagonal lattice (also known as the triangular lattice). That is, it is a translate of the planar lattice generated by two vectors of length 2 and angle 2π/3. The FCC packing is an example of a packing built from hexagonal layers. If L is a hexagonal layer, then a second hexagonal layer L" can be placed parallel to the first so that each lattice point of L" has distance 2 from three different points of L, which is the smallest possible distance from first layer. A choice of a unit normal vector e to the plane of L determines an upward direction. There are two different positions in which L" can be closely placed above L (Figure 1.12). Each successive layer (L, L" , L"" , and so forth) offers two further choices for the placement of that layer. Running through different sequences of choices gives uncountably many packings. In each of these packings the tangent arrangement around each ball is the FCC or HCP arrangement. As a packing is constructed, each layer may be labeled A, B, or C depending
The face-centered cubic packing is “the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container.” (Kepler, 1611)
3
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“Many mathematicians believe and all physicists know” [that the pyramid arrangement is best]. (Rogers, 1958) The “problem in 3-dimensions remains unsolved. This is a scandalous situation since the (presumably) correct answer has been known since the time of Gauss . . . All this is missing is a proof.” (Milnor, 1976)
Thursday, May 29, 2014
In 1975, Buckminster Fuller claimed to have a proof, but his arguments were faulty. In the early 1990s, a Berkeley professor claimed to have a proof. His arguments turned out to be incorrect as well. As recently as last month, there were new unconfirmed announcements on the Kepler conjecture Wiki of a new proof.
Thursday, May 29, 2014
After working on the conjecture for several years, with the help of a graduate student Sam Ferguson, I announced a proof of the Kepler conjecture in 1998.
9
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THE KEPLER CONJECTURE
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Section A.2.7†. If the circumradius of a quasi-regular tetrahedron is ≥ 1.41, then by [I.9.17], τ > 1.8 pt, and many of the inequalities hold. In Sections A.2.7 and A.2.8, let S1 , . . . , S5 be 5 simplices arranged around a common edge (0, v), with |v| ∈ [2, 2.51]. Let yi (Sj ) be the edges, with y1 (Sj ) = |v| for all j, y3 (Sj ) = y2 (Sj+1 ), and y5 (Sj ) = y6 (Sj+1 !). where the subscripts j are extended modulo 5. In Sections A.2.7 and A.2.8, dih(Sj ) ≤ 2π. Set π0 = 2ξV + ξΓ if σ ˆ = vor0 in the cases (y4 ≥ 2.6, y1 ≥ 2.2) and (y4 ≥ 2.7). Set π0 = 0, otherwise. √ (551665569) τ (S1 ) + τ (S2 ) + τ (S4 ) > 1.4 pt, if y4 (S3 ), y4 (S5 ) ≥ 2 2. √ (824762926) τ (S1 ) + τ (S2 ) + τ (S3 ) > 1.4 pt, if y4 (S4 ), y4 (S5 ) ≥ 2 2. √ τ (S1 ) + τ (S2 ) + (ˆ τ (S3 ) − π0 ) + τ (S4 ) > 1.4 pt + D(3, 1), if y4 (S3 ) ∈ [2.51, 2 2], y4 (S5 ) ≥ 2.51, dih(S5 ) > 1.32, (675785884) √ τ (S1 ) + τ (S2 ) + τ (S3 ) + (ˆ τ (S4 ) − π0 ) > 1.4 pt + D(3, 1), if y4 (S4 ) ∈ [2.51, 2 2], y4 (S5 ) ≥ 2.51, dih(S5 ) > 1.32. (193592217) Section A.2.8†. As in A.2.7, the quasi-regular tetrahedra are ! generally compression scored. Define π0 as in Section A.2.7. The constraint (5) dih(Sj ) = 2π is assumed. √ (325738864) τ (S1 ) + τ (S2 ) + τ (S3 ) + τ (S4 ) > 1.5 pt, if y4 (S5 ) ≥ 2 2. √ τ (S1 )+τ (S2 )+τ (S3 )+τ (S4 )+(ˆ τ (S5 )−π0 ) > 1.5 pt+D(3, 1), if y4 (S5 ) ∈ [2.51, 2 2]. (314974315)
The proof was nearly 300 pages long and relied on long computer calculations that had been made over a period of years. The there were about 3GB of data backing up the calculations.
Section A.3.1. τ − 0.2529 dih > −0.3442, if y1 ∈ [2.3, 2.51], and dih ≥ 1.51. (572068135) √ τ0 − 0.2529 dih > −0.1787, if y1 ∈ [2.3, 2.51], y6 ∈ [2 2, 3.02], 1.26 ≤ dih ≤ 1.63. (723700608) √ (560470084) τˆ − 0.2529 dih2 > −0.2137, if y2 ∈ [2.3, 2.51], y4 ∈ [2.51, 2 2], τ0 − 0.2529 dih > −0.1371, if y1 ∈ [2.3, 2.51], y5 , y6 ∈ [2.51, 3.02], 1.14 ≤ dih ≤ 1.51. (535502975) Section A.3.8. A. dih < 1.63, if y6 ≥ 2.51, y2 , y3 ∈ [2, 2.168]. (821707685) B. dih < 1.51, if y5 = 2.51, y6 ≥ 2.51, y2 , y3 ∈ [2, 2.168]. (115383627) √ C. dih < 1.93, if y6 ≥ 2.51, y4 = 2 2, y2 , y3 ∈ [2, 2.168]. (576221766) √ D. dih < 1.77, if y5 = 2.51, y6 ≥ 2.51, y4 = 2 2, y2 , y3 ∈ [2, 2.168]. (122081309) τ0 − 0.2529 dih > −0.2391, if y6 ≥ 2.51, dih ≥ 1.2, y2 , y3 ∈ [2, 2.168]. (644534985) τ0 − 0.2529 dih > −0.1376, if y5 = 2.51, y6 ≥ 2.51, dih ≥ 1.2, and y2 , y3 ∈ [2, 2.168]. (467530297) √ τ0 −0.2529 dih > −0.266, if y6 ≥ 2.51, y4 ∈ [2.51, 2 2], dih ≥ 1.2, y2 , y3 ∈ [2, 2.168]. (603910880) √ τ0 − 0.2529 dih > −0.12, if y5 = 2.51, y6 ≥ 2.51, y4 ∈ [2.51, 2 2], dih ≥ 1.2, y2 , y3 ∈ [2, 2.168]. (135427691) dih < 1.16, if y5 = 2.51, y6 ≥ 2.51, y4 = 2, y2 , y3 ∈ [2, 2.168]. (60314528) τ0 − 0.2529 dih > −0.1453, if y2 , y3 ∈ [2, 2.168], y5 ∈ [2.51, 3.488], y6 = 2.51. (312132053)
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THOMAS C. HALES
Section A.3.8, Case 2-b. dih2 > 0.74, if y1 ∈ [2.51, 2.696], y2 , y3 ∈ [2, 2.168]. (751442360) τ0 − 0.2529 dih > −0.2391, if ∆(y52 , 4, 4, 8, 2.512, y62 ) ≥ 0, y2 , y3 ∈ [2, 2.168], y5 ∈ [2.51, 3.488]. (893059266) dih +0.5(2.402 − y4 ) < π/2, if y5 ≥ 2.51, y2 , y3 ∈ [2, 2.168]. (690646028) Section A.3.9. √ dih > 1.78, if y4 = 3.2, y1 ∈ [2.51, 2 2], y2 + y3 ≤ 4.6.
(161665083)
Section A.4.4. The following inequalities hold for flat quarters. In these inequalities the fourth edge is the diagonal.
− dih2 + 0.35y2 − 0.15y1 − 0.15y3 + 0.7022y5 − 0.17y4 > −0.0123,
dih2 − 0.13y2 + 0.631y1 + 0.31y3 − 0.58y5 + 0.413y4 + 0.025y6 > 2.63363, − dih1 + 0.714y1 − 0.221y2 − 0.221y3 + 0.92y4 − 0.221y5 − 0.221y6 > 0.3482, dih1 − 0.315y1 + 0.3972y2 + 0.3972y3−
0.715y4 + 0.3972y5 + 0.3972y6 > 2.37095, − sol − 0.187y1 − 0.187y2 − 0.187y3 + 0.1185y4 + 0.479y5 + 0.479y6 > 0.437235†, sol + 0.488y1 + 0.488y2 + 0.488y3 − 0.334y5 − 0.334y6 > 2.244†, −ˆ σ − 0.145y1 − 0.081y2 − 0.081y3 − 0.133y5 − 0.133y6 > −1.17401,
−ˆ σ − 0.12y1 − 0.081y2 − 0.081y3 − 0.113y5 − 0.113y6 + 0.029y4 > −0.94903, σ ˆ + 0.153y4 + 0.153y5 + 0.153y6 < 1.05382, σ ˆ + 0.419351 sol +0.19y1 + 0.19y2 + 0.19y3 < 1.449, σ ˆ + 0.419351 sol < −0.01465 + 0.0436y5 + 0.0436y6 + 0.079431 dih, σ ˆ < 0.0114, τˆ > 1.019 pt, (4.4.1)
Inequalities (4.4.1) (867513567) If there are four quasi-regular tetrahedra {S1 , . . . , S4 } at the central vertex of the flat quarter Q, then (4.4.2†)
σ ˆ (Q) +
!
σ(Si ) < 0.114.
(4)
Inequalities 4.4.2
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(867359387)
THE KEPLER CONJECTURE
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Section A.4.5. In a quadrilateral cluster, with a given edge y4 as the diagonal, the other diagonal will be denoted y4! . The following relations for upright quarters (scored by ν) hold. (We use the inequalities of III.A for upright quarters in quad clusters, which are scored by a different function.) In these inequalities the upright diagonal is the first edge. We include in this group, the inequalities IV.A2 , IV.A3 for upright quarters.
y1 > 2.51, √ y1 < 2 2, dih1 − 0.636y1 + 0.462y2 + 0.462y3 − 0.82y4 + 0.462y5 + 0.462y6 > 1.82419,
− dih1 + 0.55y1 − 0.214y2 − 0.214y3 + 1.24y4 − 0.214y5 − 0.214y6 > 0.75281, dih2 + 0.4y1 − 0.15y2 + 0.09y3 + 0.631y4 − 0.57y5 + 0.23y6 > 2.5481, − dih2 − 0.454y1 + 0.34y2 + 0.154y3 − 0.346y4 + 0.805y5 > −0.3429, dih3 + 0.4y1 − 0.15y3 + 0.09y2 + 0.631y4 − 0.57y6 + 0.23y5 > 2.5481, − dih3 − 0.454y1 + 0.34y3 + 0.154y2 − 0.346y4 + 0.805y6 > −0.3429, sol + 0.065y2 + 0.065y3 + 0.061y4 − 0.115y5 − 0.115y6 > 0.2618,
− sol − 0.293y1 − 0.03y2 − 0.03y3 + 0.12y4 + 0.325y5 + 0.325y6 > 0.2514, −ν − 0.0538y2 − 0.0538y3 − 0.083y4 − 0.0538y5 − 0.0538y6 > −0.5995, ν ≤ 0,
(Calculations F.3.13.3, F.3.13.4)
τν − 0.5945 pt > 0.
(A.4.5.1)
(498839271)
ν − 4.10113 dih1 < −4.3223,
ν − 0.80449 dih1 < −0.9871, ν − 0.70186 dih1 < −0.8756,
ν − 0.24573 dih1 < −0.3404,
ν − 0.00154 dih1 < −0.0024, ν + 0.07611 dih1 < 0.1196.
(A.4.5.2, IV.A2 )
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THOMAS C. HALES
τν + 4.16523 dih1 > 4.42873, τν + 0.78701 dih1 > 1.01104, τν + 0.77627 dih1 > 0.99937, τν + 0.21916 dih1 > 0.34877, τν + 0.05107 dih1 > 0.11434, (A.4.5.3, IV.A3 )
τν − 0.07106 dih1 > −0.07749.
The following additional inequalities are known to hold if the upright diagonal has height at most 2.696. νΓ denotes the restriction of ν to a simplex of compression type. y1 < 2.696, dih1 − 0.49y1 + 0.44y2 + 0.44y3 − 0.82y4 + 0.44y5 + 0.44y6 > 2.0421, − dih1 + 0.495y1 − 0.214y2 − 0.214y3 + 1.05y4 − 0.214y5 − 0.214y6 > 0.2282, dih2 + 0.38y1 − 0.15y2 + 0.09y3 + 0.54y4 − 0.57y5 + 0.24y6 > 2.3398,
− dih2 − 0.375y1 + 0.33y2 + 0.11y3 − 0.36y4 + 0.72y5 + 0.034y6 > −0.36135, sol + 0.42y1 + 0.165y2 + 0.165y3 − 0.06y4 − 0.135y5 − 0.135y6 > 1.479,
− sol − 0.265y1 − 0.06y2 − 0.06y3 + 0.124y4 + 0.296y5 + 0.296y6 > 0.0997, −ν + 0.112y1 − 0.142y2 − 0.142y3 − 0.16y4 − 0.074y5 − 0.074y6 > −0.9029, ν + 0.07611 dih1 < 0.11,
νΓ − 0.015y1 − 0.16(y2 + y3 + y4 ) − 0.0738(y5 + y6 ) > −1.29285, τν − 0.07106 dih1 > −0.06429, τν > 0.0414.
(A.4.5.4) (319046543) In connection with Inequalities 4.5.4, we occasionally use the stronger constant 0.2345 instead of 0.2282. To justify this constant, we have checked using interval arithmetic that the the bound 0.2345 holds if y1 ≤ 2.68 or y4 ≤ 2.475. Further interval calculations show that the anchored simplices can be erased if they share an upright diagonal with such a quarter. Inequalities (4.5.5)
(365179082)
vor0 < −0.043/2, if y6 = 2.51, y1 ∈ [2.51, 2.696]. (368244553†) √ vor0 (S) + vor0 (S(2, y2 , y3 , y4 , 2, 2)) < −0.043, if y1 ∈ [2.51, 2.696], y4 ∈ [2 2, 3.2], y4! ≥ 2.51. (820900672†) √ vor0 (S)+vor0 (S(2.51, y2, y3 , y4 , 2, 2)) < −0.043, if y1 ∈ [2.51, 2.696], y4 ∈ [2 2, 3.2], y4! ≥ 2.51. (961078136†) Inequalities (4.5.7), (The last of these was verified by S. Ferguson.) (424186517)
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The proof was solicited for publication in the Annals of Math, a leading math journal. A dozen referees were assigned to check the proof. In January 1999, a conference was organized at the Institute for Advanced Study in Princeton. All the referees were invited to attend, and I gave a long series of lectures going through the details of the proof.
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“It is very unusual to have such a large set of reviewers. The main portion of the reviewing took place in a siminar at E¨otvos University, Budapest, over a three year period. Some reviewers made computer experiments, in a detailed check of specific parts of the proof.. . . In this process detailed checking of many specific assertions found them to be essentially correct in every case. This result of the reviewing process produced in these reviewers a strong degree of conviction of the essential correctness of this proof approach,. . . ” (J. Lagarias, editor)
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Robert MacPherson, editor of the Annals, wrote a report that states “The news from the referees is bad, from my perspective. They have not been able to certify the correctness of the proof, and will not be able to certify it in the future, because they have run out of energy to devote to the problem. This is not what I had hoped for.” “Fejes Toth thinks that this situation will occur more and more often in mathematics. He says it is similar to the situation in experimental science - other scientists acting as referees can’t certify the correctness of an experiment, they can only subject the paper to consistency checks. He thinks that the mathematical community will have to get used to this state of affairs.”
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The Kepler paper prompted the editors to issue a new policy on computer-assisted proofs. Statement by the Annals Editors on Computer-Assisted Proofs “The computer part may not be checked line-by-line, but will be examined for the methods by which the authors have eliminated or minimized possible sources of error” http://annals.math.princeton.edu/board
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In a broader context • Historically, no branch of human knowledge makes stronger claims to certainty than mathematics. • Mathematics has reached a level of unprecedented complexity. • Mathematics is making the transition to computer based systems. • Computers are very mathematical in some respects and very unmathematical in other respects.
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"#$!%$&'$()!*$+,-!.'!/+)#$-+)0(1 Mathematical Certainty
Historically, no branch of human knowledge makes Myth and Reality stronger claims to certainty than mathematics.
Ma
Mathematics has reached a level of unprecedented complexity.
!!!"##$%&'(&%)%*%)%+%,
Mathematics is making the transition to computer based systems.
)%*%)%+%)%*%-./%0 Computers %%+%-./%1)%*%02 are very mathematical in some respects and very unmathematical %%+%-./%) in other respects. %%+%, Thursday, May 29, 2014
Mathematics has reached a level of unprecedented complexity. Mathematics is making the transition to computer based systems. Computers are very mathematical in some respects and very unmathematical in other respects.
+ Four-color theorem, Grothendieck, Weil conjectures, Harish-Chandra, Lafforgue, Langlands, Arthur, Almgren, Kepler conjecture, geometrization theorem, etc. Thursday, May 29, 2014
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complexity. Mathematics is making the transition to computer based systems. Birch-Swinnerton Dyer conjecture, Sato-Tate Conjecture, Computers are very mathematical in some respects Lyons-Sims group of order 28 37 56 71 111 311 371 671 , original proof of the Calalan conjecture xm − xn = 1, qWZ proof of and very unmathematical in other respects. the Rogers-Ramunujan identities, conjectural optimal packings of tetrahedra (Chen-Engel-Glotzer), 4-color theorem, finite projective plane of order 10, Smale’s 14th problem on strange attractors in the Lorenz oscillator, Mandelbrot’s conjectures in fractal geometry, visualization of sphere eversions and Costa surface embeddings, the double bubble conjecture, construction of counterexamples to the Kelvin conjecture, calculation of kissing numbers (Sloane-Odlyzko), the character table for E 8 (Atlas project), Cohn-Kumar proof of the packing optimality of the Leech and E 8 packings among lattices, classification of fake projective planes, weak Goldbach, twin prime problem (2013).
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systems. Computers mathematical in some respects Historically,are novery branch of human knowledge makes stronger claims to certainty than mathematics. and very unmathematical in other respects. Mathematics has reached a level of unprecedented complexity. Mathematics is making the transition to computer based systems. Computers are very mathematical in some respects and very unmathematical in other respects. 16
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Will the transition to computer-based mathematics mark the end of mathematical conviction?
99% convinced. . . check of specific parts. . . strong degree of conviction of the essential correctness of this proof approach. . . similar to the situation in experimental science . . . the mathematical community will have to get used to this . . . not checked line-by-line
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IJCAR 2001 - Siena
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Computers were once human Referees were once human
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HOL Light - a formal proof assistant
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real numbers is denoted (&). We identify a packing of balls in R3 with the set V of centers of the balls. The balls in the packing are normalized to have unit length, so that the distance between distinct elements of V is at least 2. More formally, ‘(packing V (!u v. u IN V /\ v IN V /\ dist(u,v) < &2 ==> u = v))‘ We define the constant the kepler conjecture to be the term ‘the_kepler_conjecture (!V. packing V ==> (?c. !r. &1 &(CARD(V INTER ball(vec 0,r))) the_kepler_conjecture Figure 8.1 [HBMVLMY] Here are some of the tens of thousands of planar graphs whose hypermaps are tame. The ones depicted here are the ones that are the most difficult to eliminate through linear programming.
Lemma 8.12 (standard fan) [CKQOWSA] [formal proof by Alexey Solovyev]. Let V ⊂ B be a packing. Then (V, E std ) is a fan.
The project to check the Kepler conjecture by computer is over 95% complete. (Stayed tuned!) It automates the work that was once done by referees. The process is orders of magnitude more reliable than the traditional refereeing process.
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Credits: The project to formalize the Kepler conjecture is a large collaboration. Special thanks to Alexey Solovyev and Hoang Le Truong. Many images from this presentation do not originate with me. The images have been included for nonprofit educational purposes.
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