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 coins

source: Dudeney: Amusements in mathematics (1917),p. 143

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Mutually touching coins

source: Dudeney: Amusements in mathematics (1917),p. 248

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Mutually touching cigarettes

source: Grätzer: Rébusz (1935), p. 115

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Mutually touching cigarettes

source: Grätzer: Rébusz (1935), p. 233

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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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6 infinite cylinders

source: Brass, Moser, Pach, 2005, p. 98

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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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Homotopy continuation method

H(x, t) = (1 − t)Q(x) + tP (x) = 0

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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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Solution 1

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Solution 2

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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.

bozoki.sandor@sztaki.mta.hu

http://www.sztaki.mta.hu/∼bozoki

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