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The Mathematical Intelligencer

, Volume 40, Issue 1, pp 26–35 | Cite as

The Mathematics of Taffy Pullers

  • Jean-Luc Thiffeault
Article
Taffy is a type of candy made by first heating sugar to a critical temperature, letting the mixture cool on a slab, then repeatedly “pulling”—stretching and folding—the resulting mass. The purpose of pulling is to get air bubbles into the taffy, which gives it a nicer texture. Many devices have been built to assist pulling, and they all consist of a collection of fixed and moving rods, or pins. Figure  1 shows the action of such a taffy puller from an old patent. Observe that the taffy (pictured as a dark mass) is stretched and folded on itself repeatedly. As the rods move, the taffy is caught on the rods and its length is forced to grow exponentially. The effectiveness of a taffy puller is directly proportional to this growth, since more growth implies a more rapid trapping of the air bubbles. Given a pattern of periodic rod motion, regarded as orbits of points in the plane, the mathematical challenge is to compute the growth.

Notes

Acknowledgments

Supported by NSF grant CMMI-1233935. The author thanks Alex Flanagan for helping to design and build the 6-rod taffy puller, and Phil Boyland and Eiko Kin for their comments on the manuscript.

References

  1. Binder, B. J. (2010). “Ghost rods adopting the role of withdrawn baffles in batch mixer designs.” Phys. Lett. A 374, 3483–3486.Google Scholar
  2. Binder, B. J. and S. M. Cox (2008). “A mixer design for the pigtail braid.” Fluid Dyn. Res. 40, 34–44.Google Scholar
  3. Boyland, P. L., H. Aref, and M. A. Stremler (2000). “Topological fluid mechanics of stirring.” J. Fluid Mech. 403, 277–304.Google Scholar
  4. Boyland, P. L. and J. Harrington (2011). “The entropy efficiency of point-push mapping classes on the punctured disk.” Algeb. Geom. Topology 11(4), 2265–2296.Google Scholar
  5. Connelly, R. K. and J. Valenti-Jordan (2008). “Mixing analysis of a Newtonian fluid in a 3D planetary pin mixer.” 86(12), 1434–1440.Google Scholar
  6. Dickinson, H. M. (Sept. 1906). “Candy-pulling machine.” Pat. US831501 A.Google Scholar
  7. Farb, B. and D. Margalit (2011). A Primer on Mapping Class Groups. Princeton, NJ: Princeton University Press.Google Scholar
  8. Fathi, A., F. Laundenbach, and V. Poénaru (1979). “Travaux de Thurston sur les surfaces.” Astérisque 66-67, 1–284.Google Scholar
  9. Finn, M. D. and J.-L. Thiffeault (Dec. 2011). “Topological optimization of rod-stirring devices.” SIAM Rev. 53(4), 723–743.Google Scholar
  10. Firchau, P. J. G. (Dec. 1893). “Machine for working candy.” Pat. US511011 A.Google Scholar
  11. Franks, J. and E. Rykken (1999). “Pseudo-Anosov homeomorphisms with quadratic expansion.” Proc. Amer. Math. Soc. 127, 2183–2192.Google Scholar
  12. Halbert, J. T. and J. A. Yorke (2014). “Modeling a chaotic machine’s dynamics as a linear map on a “square sphere”.” Topology Proceedings 44, 257–284.Google Scholar
  13. Hall, T. (2012). Train: A C++ program for computing train tracks of surface homeomorphisms. http://www.liv.ac.uk/~tobyhall/T_Hall.html
  14. Hironaka, E. and E. Kin (2006). “A family of pseudo-Anosov braids with small dilatation.” Algebraic & Geometric Topology 6, 699–738.Google Scholar
  15. Hudson, W. T. (Feb. 1904). “Candy-working machine.” Pat. US752226 A.Google Scholar
  16. Kirsch, E. (Jan. 1928). “Candy-pulling machine.” Pat. US1656005 A.Google Scholar
  17. Kobayashi, T. and S. Umeda (2007). “Realizing pseudo-Anosov egg beaters with simple mecanisms.” In: Proceedings of the International Workshop on Knot Theory for Scientific Objects, Osaka, Japan. Osaka, Japan: Osaka Municipal Universities Press, pp. 97–109.Google Scholar
  18. – (2010). “A design for pseudo-Anosov braids using hypotrochoid curves.” Topology Appl. 157, 280–289.Google Scholar
  19. Lanneau, E. and J.-L. Thiffeault (June 2011). “On the minimum dilatation of braids on the punctured disc.” Geometriae Dedicata 152(1), 165–182.Google Scholar
  20. MacKay, R. S. (2001). “Complicated dynamics from simple topological hypotheses.” Phil. Trans. R. Soc. Lond. A 359, 1479–1496.Google Scholar
  21. Nitz, C. G. W. (Sept. 1918). “Candy-puller.” Pat. US1278197 A.Google Scholar
  22. Richards, F. H. (May 1905). “Process of making candy.” Pat. US790920 A.Google Scholar
  23. Robinson, E. M. and J. H. Deiter (Mar. 1908). “Candy-pulling machine.” Pat. US881442 A.Google Scholar
  24. Thibodeau, C. (Aug. 1903). “Method of pulling candy.” Pat. US736313 A.Google Scholar
  25. Thiffeault, J.-L. and M. D. Finn (Dec. 2006). “Topology, braids, and mixing in fluids.” Phil. Trans. R. Soc. Lond. A 364, 3251–3266.Google Scholar
  26. Thiffeault, J.-L. and M. Budišić (2013–2017). Braidlab: A Software Package for Braids and Loops. http://arXiv.org/abs/1410.0849, Version 3.2.2.Google Scholar
  27. Thurston, W. P. (1988). “On the geometry and dynamics of diffeomorphisms of surfaces.” Bull. Am. Math. Soc. 19, 417–431.Google Scholar
  28. Venzke, R. W. (2008). Braid Forcing, Hyperbolic Geometry, and Pseudo-Anosov Sequences of Low Entropy. PhD thesis. California Institute of Technology.Google Scholar
  29. Zorich, A. (2006). “Flat surfaces.” In: Frontiers in Number Theory, Physics, and Geometry. Ed. by P. Cartier et al. Vol. 1. Berlin: Springer, pp. 439–586.Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA

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