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Minimal Surface evolution from Torus - reversible?

  1. Sep 2, 2006 #1
    This is speculation about the possible relationship of QM to GR to biophysical evolution - apparently shared to some extent with Leonardo da Vinci.

    1 - Evolution [in the sense of continuous transformations] of both biophysiology and physics [QM and GR] may involve transformations of minimal surfaces into tori and perhaps back again.

    Luboš Motl has several threads on his blog ‘the reference frame’ that may relate to this:

    a - K3 - simplest after torus Tuesday, 24 May 2005

    b - ADE classification, McKay correspondence, and string theory, 7 May 2006

    c - Alain Connes: a theory of everything, 31 Aug 2006
    “... correct the myth that rabbit is topologically a sphere. Of course, its surface is a torus because of the tunnel between the mouth and the "hole below its tail", to put it politely. I would say that genus one is probably an underestimate because there are two more holes in the nose. At least, this would be true for humans ...’

    2 - I found this dynamic graphic of a torus transformed into a Genus One, Three-Ended Minimal Surface of the Costa-Hoffman-Meeks family.


    It is from Stewart Dickson [“... a pioneer in the field of visualizing mathematical objects in three physical dimensions using computer-aided rapid mechanical prototyping technologies“ ...] in ‘Project Proposal: A Three-Dimensional Zoetrope’, formerly MathArt.org of CalArts Computer Animation Lab but apparently now part of Silicon Graphics, Inc.

    The original text includes:
    “Costa's Minimal Surface was the first example of an infinitely large class of minimal surfaces discovered by David Hoffman, WIlliam Meeks III and James Hoffman at the University of Massachusetts, Amherst between 1983 and 1985. The object was initially proposed by a graduate student named Celsoe Costa in Rio de Janeiro, but the mathematics were so complicated, they obscured the underlying geometry.
    “When David Hoffman began to make pictures (graphs) of aspects of Costa's mathematics, it became clear that the surface contained symmetry no-one had expected when looking at the mathematics alone. Hoffman and his team then modified the mathematics to generalize the surface based upon its periodic symmetry. This breakthrough in the mathematics of minimal surfaces (the previous major discovery in the field occurred in the late 1800's) was brought about by visual computing.
    “Costa's surface is also known as the Genus One, Three-Ended Minimal Surface of the Costa-Hoffman-Meeks family. The surface is obtained topologically by pulling a simple torus inside-out, opening it at three points. Looking at Costa's minimal surface by itself, it is not obvious how this is done. But, by looking at a computer-generated metamorphosis, it is immediately clear.”

    3 - If one were to adopt pitch, yaw and roll with a calculated eigen axis per NASA,

    rather than the conventional spatial dimensions, then one may have an easier correlation with both momenta and neurosensory organs such as semicircular canals [apparently identical size] in most vertebrates

    and Statocysts of cnidaria, mollusks, and crustacea.

    4 - This may allow for dimensions to be interpreted as ENERGY degrees of freedom rather than SPATIAL degrees of freedom? Biological evolution may have been an experiment of mathematical objects?

    5 - Can K3 be analagously transformed [still and movie from Harvard]?

    This looks remarkably like ‘Scientists: Cosmic Blob Biggest Thing in Universe’ [credit National Astronomical Observatory of Japan at Space.com].

    6 - Newsweek reports in ‘The Drawings of a Master’ [RE: “new exhibition at London's Victoria and Albert Museum“] that Leonardo da Vinci suspected that “... his exploration of the ancient idea of microcosm and macrocosm—the belief that the human body contains a miniature model of the world and universe ...”
  2. jcsd
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