The Varieties of Phaseless Experience: In which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways

  • Arthur T. Winfree
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 12)


In Chapter 1 we dwelt on the notion of smooth maps from one space to another. In Chapter 2 it emerged that certain kinds of mapping involving circles cannot be contrived smoothly. As an application we saw that certain kinds of experimentally observed continuity and smoothness involving measures that are periodic in space or time inescapably imply an unobserved (but observable) discontinuity. A phase singularity is one way to resolve this crisis of continuity implicit in the observation of nonzero winding number.


Physical Nature Spiral Wave Cholesteric Liquid Crystal Ambiguous Phase Diverse State 
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  1. 1.
    If you love geometry, look at Bouligand’s papers on dislocations in liquid crystals (Journal de Physique, 1972–1974). They are full of astounding singularities, including cusps, screws, and even Möbius bands. At least some of these have been identified in biological materials and may play essential roles in the mechanics of plant cell elongation and of chromosome condensation. The literature of liquid crystal spherulites (of polypeptides; DNA, etc.) likewise is a study in singularities, since there is no way to smoothly map a sphere’s surface to the ring of possible compass orientations for molecules on that surface. See also Slonczewski and Malozemoff (1978) on orientational singularities in three dimensional magnetic bubbles.Google Scholar

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© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Arthur T. Winfree
    • 1
  1. 1.Department of Ecology and Evolutionary BiologyUniversity of ArizonaTucsonUSA

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