Seven mutually touching infinite cylinders -...
Transcript of Seven mutually touching infinite cylinders -...
Seven mutually touching
infinite cylinders
Sándor BOZÓKI 1,2, Tsung-Lin LEE 3, Lajos RÓNYAI 1,4
1 Institute for Computer Science and Control,Hungarian Academy of Sciences (MTA SZTAKI)
2 Corvinus University of Budapest
3 National Sun Yat-sen University, Taiwan
4 Institute of Mathematics,Budapest University of Technology and Economics
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Mutually touching cigarettesMartin Gardner popularized the puzzle in ScientificAmerican. Surprisingly, a few solvers found 7 mutuallytouching cigarettes:
source: Gardner (1959), p. 115(condition: length/diameter ≥ 7
√3/2 ≈ 6.06.)
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Littlewood’s problem
“Is it possible in 3-space for seven infinite circularcylinders of unit radius each to touch all the others?Seven is the number suggested by constants.”
(Littlewood, 1968, p. 20)
In Ogilvy (1962),„How many lines can be drawn in 3-space, each aunit distant from every one of the others? It is con-jectured that seven is the maximum number, but noproof is available. Seven might be too high or toolow.” (p. 61)„The question on skew lines in 3-space was sug-gested by Littlewood.” (p. 153)
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Kuperberg’s arrangement with 8 infinite cylinders
source: Ambrus, Bezdek,2008, p. 1804
Ambrus and Bezdek disproved Kuperberg’s proposal, theyshowed that at least one pair is not touching in theconfiguration above.
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Lower and upper bounds
Theorem (Bezdek, 2005): The maximal number of mutuallytouching identical infinite cylinders is at most 24.
Best known lower bound: 6.
It is shown in the rest of the talk that the lower bound canbe improved to 7. Results of our paper
Bozóki, S., Lee, T.L., Rónyai, L. (2015):Seven mutually touching infinite cylinders,Computational Geometry: Theory and Applications,48(2):87–93.
are to be summarized.
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The model
Let the radius of cylinders be 1, i.e., the distance of lines is2.
ℓi(s) = Pi + swi
is a parametric representation of line ℓi for i = 1, . . . , 7. HerePi ∈ R
3 is a point of ℓi, wi ∈ R3 is a direction vector and s is
a real parameter. If lines ℓi and ℓj are skew, then theirdistance can be obtained as
d(ℓi, ℓj) =|(−−−→PiPj) · (wi × wj)|
||wi × wj||.
With d = 2,
|(−−−→PiPj) · (wi × wj)|2
− 4||wi × wj||2 = 0.
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The model
|(−−−→PiPj) · (wi × wj)|2
− 4||wi × wj||2 = 0.
Let Pi = (xi, yi, zi), wi = (ti, ui, vi).Apply the determinantal form of the triple product:
det
xj − xi yj − yi zj − zi
ti ui vi
tj uj vj
2
− 4[
(uivj − viuj)2+
+ (vitj − tivj)2 + (tiuj − uitj)
2]
= 0.
This is a polynomial equation of degree 6 in 12 variables.The polynomial on the left is a linear combination of 84monomials.
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Reduction of the number of variables
We call a line horizontal if it is parallel to the plane z = 0.Assume without of loss of generality that ℓ1 goes throughthe point P1(0, 0,−1) and it is horizontal with direction vectorw1 = (1, 0, 0). Let the touching point of cylinders C1 and C2
be (0, 0, 0), that is, ℓ2 goes through the point P2(0, 0, 1), andit is horizontal, too.
The direction of ℓ2 is the only degree of freedom when thefirst two lines are considered.
We may assume without loss of generality that ℓi
(i = 3, . . . , 7) is not horizontal (otherwise it would be parallelto ℓ1 or ℓ2), consequently it intersects properly the planez = 0, i.e., zi = 0 (i = 3, . . . , 7).
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Reduction of the number of variables
Finally, let the direction vectors be normalized byti + ui + vi = 1 (i = 3, . . . , 7). Note that this excludesdirection vectors with ti + ui + vi = 0, however, it will turn outthat we do not lose all solutions.
We are left with:1 + 5 × 4 = 21 variables and5 + 5 +
(
5
2
)
= 20 equations.
Let ℓ2 be orthogonal to ℓ1, that is, the first two cylinders arefixed. Now we have 20 variables and 20 equations.
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The system of equations
The distance of lines ℓ1 and ℓj (3 ≤ j ≤ 7):
y2
j t2
j + 2y2
j tjuj − 2y2
j tj + y2
ju2
j − 2y2
juj + y2
j + 2yjtjuj + 2yju2
j −−2yjuj − 4t2j − 8tjuj + 8tj − 7u2
j + 8uj − 4 = 0, j = 3, . . . , 7.
The distance of lines ℓ2 and ℓj (3 ≤ j ≤ 7):
x2
j t2
j + 2x2
j tjuj − 2x2
j tj + x2
ju2
j − 2x2
juj + x2
j − 2xjtjuj − 2xjt2
j +
+2xjtj − 4u2
j − 8tjuj + 8tj − 7t2j + 8uj − 4 = 0, j = 3, . . . , 7.
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The system of equationsThe distance of lines ℓi and ℓj (3 ≤ i < j ≤ 7):−4xiyitiuitjuj +4xixjtiuitjuj +4xiyjtiuitjuj +4yixjtiuitjuj +4yiyjtiuitjuj
−4xjyjtiuitjuj −2x2
itiuitjuj −2y2
itiuitjuj −2x2
jtiuitjuj −2y2
jtiuitjuj −4xixjtiuiuj
+4xixjtiu2
j+4xixju2
itj −4xixjuitjuj +4yiyjt2
iuj −4yiyjtiuitj −4yiyjtitjuj
+4yiyjuit2
j+4xixjuiuj +4yiyjtitj +x2
it2iu2
j+x2
iu2
it2j
+y2
it2iu2
j+y2
iu2
it2j
+x2
jt2iu2
j
+x2
ju2
it2j
+y2
jt2iu2
j+y2
ju2
it2j
+2xiyit2
iu2
j+2xiyiu
2
it2j−2xixjt2
iu2
j−2xixju2
it2j−2xiyjt2
iu2
j
−2xiyju2
it2j−2yixjt2
iu2
j−2yixju2
it2j−2yiyjt2
iu2
j−2yiyju2
it2j
+2xjyjt2iu2
j+2xjyju2
it2j
−2xiyit2
iuj −2xiyitiu
2
j+2xiyjt2
iuj +2xiyjtiu
2
j+2xiyju2
itj +2xiyjuit
2
j−2xiyiu
2
itj
−2xiyiuit2
j+2yixjt2
iuj +2yixjtiu
2
j+2yixju2
itj +2yixjuit
2
j−2xjyjt2
iuj −2xjyjtiu
2
j
−2xjyju2
itj −2xjyjuit
2
j−2x2
itiu
2
j−2x2
iu2
itj −2y2
it2iuj −2y2
iuit
2
j−2x2
jtiu
2
j−2x2
ju2
itj
−2y2
jt2iuj −2y2
juit
2
j+2x2
itiuiuj +2x2
iuitjuj +2y2
itiuitj +2y2
ititjuj +2x2
jtiuiuj
+2x2
juitjuj +2y2
jtiuitj +2y2
jtitjuj +2xiyitiuitj +2xiyitiuiuj +2xiyititjuj
+2xiyiuitjuj −2xiyjtiuitj −2xiyjtiuiuj −2xiyjtitjuj −2xiyjuitjuj −2yixjtiuitj
−2yixjtiuiuj −2yixjtitjuj −2yixjuitjuj +2xjyjtiuitj +2xjyjtiuiuj +2xjyjtitjuj
+2xjyjuitjuj −2x2
iuiuj −2y2
ititj −2x2
juiuj −2y2
jtitj −2xiyitiui +2xiyitiuj
+2xiyiuitj −2xiyitjuj +2xiyjtiui −2xiyjtiuj −2xiyjuitj +2xiyjtjuj +2yixjtiui
−2yixjtiuj −2yixjuitj +2yixjtjuj −2xjyjtiui +2xjyjtiuj +2xjyjuitj −2xjyjtjuj
−2xixju2
i−2xixju2
j−2yiyjt2
j−2yiyjt2
i+24tiuitjuj +x2
iu2
i+x2
iu2
j+y2
it2i
+y2
it2j
+x2
ju2
i
+x2
ju2
j+y2
jt2i
+y2
jt2j−12t2
iu2
j−12u2
it2j−4t2
i−4u2
i−4t2
j−4u2
j−8tiuitj −8tiuiuj
−8titjuj +8tiu2
j+8t2
iuj +8u2
itj +8uit
2
j−8uitjuj +8titj
+8uiuj = 0, i = 3, . . . , 6, j = i + 1, . . . , 7.– p. 16/40
Homotopy continuation method
Drexler (1977)
Garcia, Zangwill (1979)
Morgan, Sommese (1987)
Huber, Sturmfels (1995)
Li (1997)
Lee, Li, Tsai (2008)
Chen, Lee, Li (2013)
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Two solutions by polyhedral homotopy continuation method
We obtain 180, 734 mixed cells of the system by softwareMixedVol-2.0 (Chen, Lee, Li, 2013), which provide121, 098, 993, 664 homotopy curves to be tracked.
In order to track so many curves efficiently, we use thesubroutines in the TBB library (Thread Building Blocks) todistribute data over multiple cores for parallel computation.Employing total 12 cores in 2 Intel Xeon X5650 2.66 GHzCPUs, 20 million curves are completed in a month.
The first real solution is found after tracking 80 millionpaths, and the second one is found after tracking another25 million paths.
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x3 11.675771704477 2.075088491891
y3 −4.124414157636 −2.036516392124
t3 0.704116159640 −0.030209763440
u3 0.235129952793 0.599691085438
x4 3.802878122730 −2.688893665930
y4 −2.910611127075 4.070505903499
t4 0.895623427074 0.184499043058
u4 −0.149726023342 0.426965115851
x5 8.311818491659 −4.033142850644
y5 −1.732276613733 −2.655943449984
t5 2.515897624878 0.251380280590
u5 −0.566129665502 0.516678258430
x6 −6.487945444917 6.311134419772
y6 −8.537495065091 −5.229892181735
t6 0.785632006191 −0.474742889365
u6 0.338461562103 1.230302197822
x7 −3.168475045360 3.914613907006
y7 −2.459640638529 −7.881492743224
t7 0.192767499267 1.698198197367
u7 0.536724141124 −1.164062857743– p. 20/40
Verification of the rootsAren’t they just good approximations? Do there exist exactsolutions around them?
alphaCertified, based on Smale α-theory
interval Krawczyk method
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Verification of the roots: alphaCertified
Smale’s α-theory (1986) provides a positive, effectivelycomputable constant α(F,x) for a polynomial systemF : C
n → Cn and a point x ∈ C
n with the property that if
α(F,x) ≤ 13−3√
17
4≈ 0.1576, then Newton’s iteration starting
from x converges quadratically to a solution ξ close to x ofthe system F = 0.
Based on Smale’s theory Hauenstein and Sottile (2012)developed algorithms which, for given F and x compute anupper bound on α(F,x) and on some related quantities. Onthat basis they have built a multipurpose verificationsoftware called alphaCertified. It certifies that
(i) x is an approximate solution of F = 0 in the above sense;(ii) an approximate solution corresponds to an isolatedsolution;(iii) the solution ξ corresponding to x is real (for real F ).
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Verification of the roots: alphaCertified
We have used alphaCertified v1.2.0 (August 15, 2011,GMP v4.3.1 & MPFR v2.4.1-p5) with Maple 13 interface.The input of alphaCertified is our polynomial system of 20variables/equations and the approximate solutions.
We need to write the first solution up to at least 12 digits,otherwise algorithm alphaCertified does not certify it. Theoutput of alphaCertified with the first solution consists ofα = 4.4333 · 10−2, β = 3.1668 · 10−12, γ = 1.3999 · 1010.
The second solution has to be written up to at least 11digits in order to be certified. The output of alphaCertifiedwith the second solution (truncated at 11 digits) consists ofα = 6.578 · 10−2, β = 2.2387 · 10−11, γ = 2.9392 · 109.
Both solutions have been certified to be real and isolatedsolutions.
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Verification of the roots: interval Krawczyk method
The interval Krawczyk method (1969) is based on thefollowing fact: for a smooth function F : R
n → Rn and a
point x ∈ Rn, let [x]r ⊂ R
n be the ball centered at x withradius r > 0. Namely,
[x]r = {y ∈ Rn : ‖y − x‖∞ ≤ r} ,
where ‖ ‖∞ is the infinity norm. Assuming that thederivative of F at x, denoted by DF (x), is nonsingular, theKrawczyk set of F associated with [x]r is defined as
K(F, [x]r) = x−DF (x)−1F (x)+[
I − DF (x)−1DF ([x]r)]
([x]r−x).
If the Krawczyk set is contained in the interior of [x]r, thenthere exists a unique zero of F in [x]r.
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Verification of the roots: interval Krawczyk method
The task of verification is implemented by using the intervalarithmetic in INTLAB (INTerval LABoratory) by Rump.
In this implementation each numerical solution x is taken asthe center of the ball [x]r with radius r = 10−8.
Again, both solutions have been certified to be real andisolated solutions.
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Main result
We have shown that
Theorem: The maximal number of mutually touchingidentical infinite cylinders is at least 7.
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Open problems: mutually touching finite cylinders
Can we find more than 7?
Intermediate length/diameter ratios
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Open problems: mutually touching infinite cylinders
Are there more than 7? The system of equations writtenfor 8 cylinders has 25 variables and 27 equations.
Other angles for ℓ1 and ℓ2?
Direction vectors with ti + ui + vi = 0?
What is the maximal number of lines in Rn (n > 3)
having the same nonzero pairwise distance?
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Main references to mutually touching coins
Dudeney, H.E. (1917): Amusements in mathematics,Thomas Nelson and Sons, London, Edingburgh, New York,p. 143., p. 248.https://archive.org/details/amusementsinmath00dude
https://openlibrary.org/books/OL178183M/Amusements_in_mathematics
Gardner, M. (1959): The Scientific American book ofmathematical puzzles and diversions, Simon and Schuster,New York, pp. 110–115.
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Main references to mutually touching cigarettes
Gardner, M. (1959): The Scientific American Book ofMathematical Puzzles and Diversions, Simon and Schuster,New York, pp. 110–115.
Grätzer, J. (1935): Rébusz (in Hungarian), Singer ésWolfner Irodalmi Intézet, Budapest, p. 115., p. 233.
Méro, L. (1997): Észjárások - A racionális gondolkodáskorlátai és a mesterséges intelligencia (in Hungarian),Tericum Kiadó, puzzle R8, pp. 17–18, pp. 183–184
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Main references to mutually touching infinite cylinders
Ambrus, G., Bezdek, A. (2008): On the number of mutuallytouching cylinders. Is it 8?, European Journal ofCombinatorics 29(8):1803–1807.
Bezdek, A. (2005): On the number of mutually touchingcylinders, Combinatorial and Computational Geometry,MSRI Publication, 52:121–127.
Brass, P., Moser, W., Pach, J. (2005): Research problems indiscrete geometry, Springer.
Littlewood, J.E. (1968): Some problems in real and complexanalysis, Heath Mathematical Monographs, RaytheonEducation, Lexington, Massachusetts.
Ogilvy, C.S. (1962): Tomorrow’s math: unsolved problemsfor the amateur, Oxford Univesity Press, New York.
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Main references to mutually touching infinite cylinders
Bozóki, S., Lee, T.L., Rónyai, L. (2015): Seven mutuallytouching infinite cylinders, Computational Geometry:Theory and Applications, 48(2):87–93.
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Main references to homotopy continuation method
Chen, T., Lee, T.L., Li, T.Y. (2013): Mixed VolumeComputation in Parallel, Taiwanese Journal ofMathematics, accepted, DOI 10.11650/tjm.17.2013.3276
Drexler, F.J. (1977): Eine Methode zur Berechnungsämtlicher Lösungen von Polynomgleichungssystemen,Numerische Mathematik 29(1):45–58.
Garcia, C.B., Zangwill, W.I. (1979): Finding all solutions topolynomial systems and other systems of equations,Mathematical Programming 16(1):159–176.
Huber, B., Sturmfels, B. (1995): A polyhedral method forsolving sparse polynomial systems, Mathematics ofComputation 64(212): 1541–1555.
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Main references to homotopy continuation method
Lee, T.L., Li, T.Y., Tsai, C.H. (2008): HOM4PS-2.0, Asoftware package for solving polynomial systems by thepolyhedral homotopy continuation method, Computing83(2-3):109–133.
Li, T.Y. (1997): Numerical solution of multivariatepolynomial systems by homotopy continuation methods,Acta Numerica 6:399–436.
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Main references to root verification
Blum, L., Cucker, F., Shub, M., Smale, S. (1997):Complexity and real computation, Springer-Verlag, NewYork.
Hauenstein, J.D., Sottile, F. (2012): Algorithm 921:alphaCertified: certifying solutions to polynomial systems,ACM Transactions on Mathematical Software 38(4): Article28. DOI 10.1145/2331130.2331136
Krawczyk, R. (1969): Newton-Algorithmen zur Bestimmungvon Nullstellen mit Fehlerschranken, Computing4(3):187–201.
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Main references to root verification
Rump, S.M. (1999): INTLAB – INTerval LABoratory, in:Csendes, T., editor, Developments in reliable computing,Kluwer Academic Publishers, Dordrecht, pp. 77–104.
Rump, S.M.: INTLAB – INTerval LABoratory.http://www.ti3.tu-harburg.de/rump/intlab/
Smale, S. (1986): Newton’s method estimates from data atone point, in Ewing, R.E., Gross, K.I., Martin, C.F. (editors):The merging of disciplines: new directions in pure, applied,and computational mathematics, Springer, New York,pp. 185–196.
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Main references to materials science and auxetic lattices
Pikhitsa, P.V. (2004): Regular network of contactingcylinders with implications for materials with negativePoisson ratios, Physical Review Letters 93(1) Article015505.
Pikhitsa, P.V. (2007): Architecture of cylinders withimplications for materials with negative Poisson ratio,Physica Status Solidi B 244(3):1004–1007.
Pikhitsa, P.V., Choi, M., Kim, H-J., Ahn, S-H. (2009): Auxeticlattice of multipods, Physica Status Solidi B246(9):2098–2101.
Pikhitsa, P.V., Choi, M. (2014): Seven, eight, and ninemutually touching infinitely long straight round cylinders:Entanglement in Euclidean space, manuscript,arXiv:1312.6207
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Thank you for your attention.
http://www.sztaki.mta.hu/∼bozoki
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