Fw: message layer cake: FFT of Sampling = Repetition, Graphics Needed, Origin of Universe ...

From: Douglas P. Wilson (dp-wilson@shaw.ca)
Date: Thu 14 Nov 2002 - 08:22:31 GMT

  • Next message: Chris Taylor: "Re: Fw: An Individualistic Step"

    forwarded copy of message posted to Virus (CofV) mailing list -- please see e-mail links to possibly interesting lists at the end of this message

            dpw http://www.SocialTechnology.Org/dpwilson.html

    ----- Original Message ----- From: "Douglas P. Wilson" <dp-wilson@shaw.ca> To: <virus@lucifer.com> Sent: Thursday, November 14, 2002 12:16 AM Subject: Fw: message layer cake: FFT of Sampling = Repetition, Graphics Needed, Origin of Universe ...

    What, Joe & Co. want examples of creative thought. Well, I don't suppose this really qualifies, but it might be a refreshing change from To B or Not To B(omb).

            dpw http://www.SocialTechnology.Org/dpwilson.html

    ----- Original Message ----- From: "Douglas P. Wilson" <dp-wilson@shaw.ca> To: <digitalphysics@yahoogroups.com> Cc: "Robert Neville" <rneville@telus.net>; "Douglas Hofstadter" <dughof@cs.indiana.edu> Sent: Wednesday, November 13, 2002 9:28 PM Subject: message layer cake: FFT of Sampling = Repetition, Graphics Needed, Origin of Universe ...

    This is the preface, and table of contents for a short sequence of messages to follow, including also the three listed in in the Subject line, and a conclusion.

    Preface -- this message layer cake, of (which this is just the icing), reminds us that the (not-necessarily Fast) Fourier Transform of the Sampling operator is the repetition operator, then laments my inability to show you this right now with a nice picture, and turns my rough design for software to generate the graphics into a theory of the origin of the universe, then concludes by pointing out that imperfect sampling (leaky cells) makes for a much more interesting OverAll Universe, in which our neighbouring Local Universes are not mere repetitions but actually the slightly different parallel universes of science fiction.


    (F)FT of Sampling = Repetition -- if you sample a blob-like function, one whose Fourier Transform is also a blob-like function,since the Gaussian normal distribution is an identity or invariant under the transform, then the FT of the result is a sequence of blobs. The Fourier Transform of the Sampling operator (or functor, categorically) is the Repetition operator (functor). (In all this I am actually thinking of the Power Spectrum, not the whole FT, but the PS is the magnitude, and phase is irrelevant to this discussion -- it is almost always better to work with Mag/Phase rather than Real/Imag).

    Graphics Needed -- at least a graph showing original Gaussian curve, Fourier Transform (PS, or Mag, only) of it, which is the same shape, then a sampled version of the same curve,and the FT (magnitude, only) of the sampled curve, which is an endless repetition of the original unsampled blob-like FT. A nice two-D graphic diagram of this would help a lot, and so would an example showing the universe repeated over and over.

    Origin of Universe -- while designing software to illustrate the (well-known, often forgotten) point that the FT (actually "Power Spectrum") of the Sampling operator is the Repetition operator, I started with a 2-D cellular automaton, planned to 99% of the points to 0, with just a handful of 1s, then added mass and gravity to affect that mass, so nearby points would gradually merge into one another. Then I planned to add velocity, thus momentum, and conservation of angular momentum to cause the points to swirl around one another as they approach. Yes, should look like the (Local) Universe. Hmmm. Maybe it is the LU -- look I just reinvented Digital Physics! But ... locally velocities would probably not sum to the nice 0.00 that I assumed for graphics-generation purposes, so locally we would expect to see
    (depending on where we were), so we might (about half the time) see what might look (locally), like a big bang ...

    Conclusion -- well, as long as we are talking Celluar Automata anyway, for generating graphics, why not go all the way to Digital Physics and suppose the Universe is a CA? But a cellular model looks alot like a sampling of the smooth ("no more than second degree") functions of Classical Post-Classical-Physics. Smoothed out, our Galaxy, seen from above the principal coordinate plane, would look like a blob, and many physicists of the Big Bang School seem to think that the Smoothed-Out Universe would be a blob too. Then the (2-D or 3-D) Fourier Transform of the universe would look like a blob too, blobs being invariant under rotation from Cosmological Space into Spatial Frequence Space. But the FT of the
    (rather obviously [somewhat irregularly]) sampled universe would look not like a blob but like an endless repetition of blobs. How boring. Unless the cells leak, in which case the Parallel Universes are not identical, more like those of Science Fiction.

            dpw http://www.SocialTechnology.Org/dpwilson.html

    FIRST Message ONLY -- for rest of sequence, visit website of the Digital Physics mailing list, via http://www.DigitalPhysics.Org or http://groups.yahoo.com/group/digitalphysics/

    web site: http://digitalphysics.org email list: http://groups.yahoo.com/group/digitalphysics to post: digitalphysics@yahoogroups.com to unsubscribe: digitalphysics-unsubscribe@yahoogroups.com

    You might also consider:




















            dpw http://www.SocialTechnology.Org/dpwilson.html

    =============================================================== This was distributed via the memetics list associated with the Journal of Memetics - Evolutionary Models of Information Transmission For information about the journal and the list (e.g. unsubscribing) see: http://www.cpm.mmu.ac.uk/jom-emit

    This archive was generated by hypermail 2.1.5 : Thu 14 Nov 2002 - 08:26:01 GMT