Abstract
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.
A cautious man should above all be on his guard against resemblances; they are a very slippery sort of thing.
Plato, The Sophist, translated on p. 180 of F. M. Cornford, Plato’s Theory of Knowledge, 1935
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References
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.
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© 2001 Springer Science+Business Media New York
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Winfree, A.T. (2001). The Varieties of Phaseless Experience: In which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways. In: The Geometry of Biological Time. Interdisciplinary Applied Mathematics, vol 12. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3484-3_10
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DOI: https://doi.org/10.1007/978-1-4757-3484-3_10
Publisher Name: Springer, New York, NY
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