Abstract
We consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere. After establishing the existence of solutions for any given thickness between 0 and 1, we explicitly construct for each given thickness \({\Theta_n:= {\rm sin}\, \pi/(2n),}\) \({n\in\mathbb{N}}\), exactly \({\varphi(n)}\) solutions, where \({\varphi}\) is Euler’s totient function from number theory. Then we prove that these solutions are unique, and also provide a complete characterisation of sphere filling curves on the unit sphere; that is of those curves whose spherical tubular neighbourhood completely covers the surface area of the unit sphere exactly once. All of these results carry over to open curves as well, as indicated in the last section.
Similar content being viewed by others
References
Bach E., Shallit J.: Algorithmic Number Theory. MIT Press Cambridge, Massachusetts (1996)
Cantarella J., Fu J.H.G., Kusner R.B., Sullivan J.M., Wrinkle N.C.: Criticality for the Gehring link problem. Geom. Topol. 10, 2055–2116 (2006)
Cantarella J., Kusner R.B., Sullivan J.M.: On the minimum ropelength of knots and links. Invest. Math. 150, 257–286 (2002)
Cantarella, J., Piatek, M., Rawdon, E.: Visualizing the tightening of knots. VIS’05: Proceedings of the 16th IEEE Visualization 2005, 575–582. IEEE Computer Society, Washington, DC, 2005
Carlen, M., Laurie, B., Maddocks, J.H., Smutny, J.: Biarcs, global radius of curvature, and the computation of ideal knot shapes. Physical and Numerical Models in Knot Theory, Ser. on Knots and Everything 36 (Eds. Calvo M. and Rawdon S.) World Scientific, Singapore, 75–108, 2005
Do Carmo M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, New Jersey (1976)
Durumeric O.C.: Local structure of ideal shapes of knots. Top. Appl. 154, 3070–3089 (2007)
Gerlach, H.: Der Globale Krümmungsradius für offene und geschlossene Kurven im \({\mathbb{R}^N}\). Diploma thesis at Bonn University, 2004. http://www.littleimpact.de/permanent/math/2009/dipl/
Gerlach, H.: Ideal Knots and other packing problems of tubes. PhD thesis No. 4601, EPFL Lausanne (2010). http://library.epfl.ch/theses/?display=detail&nr=4601
Gerlach, H.: Construction of sphere-filling ropes. Website: http://www.littleimpact.de/permanent/math/sphere_filling/
Gonzalez O., de la Llave R.: Existence of ideal knots. J. Knot Theory Ramif. 12, 123–133 (2003)
Gonzalez O., Maddocks J.H.: Global curvature, thickness and the ideal shapes of knots. Proc. Natl. Acad. Sci. USA 96, 4769–4773 (1999)
Gonzalez O., Maddocks J.H., Schuricht F., von der Mosel H.: Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. 14, 29–68 (2002)
Gray, A.: Tubes., 2nd edn. Progress in Mathematics, 221. Birkhäuser Verlag, Basel, 2004
Hotelling H.: Tubes and spheres in n-spaces. Am. J. Math. 61, 440–460 (1939)
Katzav E., Adda-Bedia M., Boudaoud A.: A statistical approach to close packing of elastic rods and to DNA packaging in viral capsids. Proc. Natl. Acad. Sci. USA 103, 18900–18904 (2006)
Kusner, R.B.: On thickness and packing density for knots and links. Physical Knots: Knotting, Linking, and Folding Geometric Objects in \({\mathbb{R}^3}\) (Las Vegas, NV, 2001) (Eds. Calvo M. and Rawdon S.) AMS Providence, Contemp. Math. 304, 175–180, (2002)
Pireranski, P.: In search of ideal knots. Ideal Knots, Ser. on Knots and Everything 19 (Eds. Stasiak, Katritch and Kauffman) World Scientific, Singapore, 20–41, 1998
Smutny, J.: Global radii of curvature and the biarc approximation of spaces curves: in pursuit of ideal knot shapes. PhD thesis No. 2981, EPFL Lausanne (2004). http://library.epfl.ch/theses/?display=detail&nr=2981
Schuricht F., von der Mosel H.: Global curvature for rectifiable loops. Math. Z. 243, 37–77 (2003)
Schuricht F., von der Mosel H.: Characterization of ideal knots. Calc. Var. Partial Differ. Equ. 19, 281–305 (2004)
Strzelecki P., von der Mosel H.: On rectifiable curves with L p-bounds on global curvature: self-avoidance, regularity, and minimizing knots. Math. Z. 257, 107–130 (2007)
Varea C., Aragon J.L., Barrio R.A.: Turing patterns on a sphere. Phys. Rev. E 60, 4588–4592 (1999)
Weyl H.: On the volume of tubes. Am. J. Math. 61, 461–472 (1939)
Wiggs, C.C., Taylor, C.J.C.: Bead puzzle. US Patent D269629 (issued 1983)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Otto
Rights and permissions
About this article
Cite this article
Gerlach, H., von der Mosel, H. What are the Longest Ropes on the Unit Sphere?. Arch Rational Mech Anal 201, 303–342 (2011). https://doi.org/10.1007/s00205-010-0390-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-010-0390-y