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>Date: Fri, 16 Oct 1998 19:23:24 +0200
>To: struck at xs4all.nl
>From: Gerald de Jong <gerald at beautifulcode.nl>
>Subject: struck: Fundamentals of Elastic Interval Geometry
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>(continued...)
>
>when entertaining ideas involving substantial elastic intervals connected
>at their ends to other elastic intervals, you might feel compelled to look
>more closely at the joining points and find out what they might look like.
>i've already described them as singularities, or foci, or even portals from
>one interval to the next, and these are fine images if you can accept the
>notion of a singularity.  however some people might want to work with a
>fractal definition of joints, where zooming in on a joint reveals ever more
>complexity, rather than ending up at a black-hole singularity.  indeed, as
>current theory goes, black holes themselves are more than slightly restless
>at and around their event horizons.  basically, when we choose to call the
>elastic interval connections by the name singularity, we're saying that
>we're willing to forget about looking deeper inside.  but suppose that we
>choose to have something appear as we zoom in.  what might we see?
>
>i would suggest that it's a fairly believable image if we were to encounter
>a very very tiny elastic interval network when we zoom in on one
>connection.  it would also seem natural to attribute some sort of
>regularity to it, because it is apparently capable of pulling elastic
>interval 'ends' so close together that we see it as a simple connection
>from a distance.  choosing to see it as some sort of regular geometrical
>form would limit us very much with respect to how many connections could
>legitimately be involved (ie. tetrahedron=4, octahedron=6, icosahedron=12,
>but where are the other numbers?).  the only seemingly sensible solution is
>to consider the connection to be what is called a 'complete graph', where
>each of its joints are connected to all the others.  that would certainly
>keep things tight inside there.  (incidentally, i hesitate to call them
>elastic intervals of span zero, but hopefully i can explain that a little
>later on).  naturally, if we were to zoom in on one of these inner joints,
>we would encounter another complete network, and so on, ad infinitum.  it's
>an interesting view of things, but it would not be my first choice.  the
>reason being that it seems to not resonate with physics at all.
>
>the physics that we observe has some very fascinating general properties,
>and one of them is that 'size matters'.  our movements must look quite
>dinosaur-like to an ant, and their erratic scavenging looks to us like
>unreal fast-motion.  small insects fly without even the use of wings
>because to them the air is positively viscous like syrup.  a jet airplane
>encountering water at high velosity will crumple as if it encountered
>concrete.  now once again, the goal of EIG is not to faithfully represent
>the laws of physics by any means, but it does seem extremely interesting to
>at least represent such a general principle as 'size matters'.  as it turns
>out, this can very easily be done, and it has the much more important
>consequence of actually simplifying the model of structural form.  we all
>know that fractals are infinitely complex, but we also know that a rock is
>not as elusive as an electron.  avoiding the fractal aspect (in which size
>does not at all matter, at least in the same sense), is already a
>simplification, so if we can say that smaller than this-and-that you cannot
>go without profoundly different behavior, we've simplified.
>
>this brings us to the domain of elastic interval definitions.  given two
>connections between which an interval is mediating, there must be a
>definition available to describe how much an interval pulls when it finds
>itself to be too long, and how much it pushes when too short.  remember
>that an interval knows of nothing other than it's two connections and
>itself.  one essential piece of information that the interval must contain
>is its rest span, the length at which it is 'satisfied', thus neither
>pushing nor pulling.  what remains to be defined is how intensely the
>spring wishes to return to its rest span given a particular perturbation.
>Hooke determined that for all practical purposes, springs obey a linear law
>involving a constant, so we could opt for Hooke's law, but that would
>clearly require that we not only  have the interval know about its rest
>span, but a constant would also have to be maintained.  that is more
>information than we strictly need, because this defines a whole family of
>behaviors and not just one.  believe it or not, Hooke's simple law is too
>complex for us.
>
>so what is needed is a definition in which small things are as restless and
>elusive as subatomic particles and large things are as plodding and syrupy
>as slowly-rotating galaxies.  with such a definition, size affects
>behavior.  possibly the easiest way to achieve this is by using a
>logarithmic impulse definition for the elastic intervals.  a core
>consequence of this is that doubling of size is equivalent-but-opposite to
>halving of size, or more generally, if an elastic interval that is X times
>as long as it wishes pulls just as intensely as that same interval when
>compressed to 1/X its rest span.  let ln(x) symbolize the natural logarithm
>of x.  an impulse formula might look like this:
>
>	PushOutwards = ln(RestSpan/CurrentSpan) = ln(RestSpan) -
>ln(CurrentSpan)
>
>this rule is simpler than Hooke's law in that there is no explicit
>constant.  the natural logarithm is a function that has its origins in
>things like biological systems.  its rate of increase is equal to the rate
>of increase of its rate of increase, and so on.  there is an implicit
>constant involved, but it's not an arbitrary one by any means.
>
>examining the behavior of this function, we see that when the rest span is
>small the variation in push will be comparatively large.  for example, an
>interval with a rest span of 0.10 and a current span of 0.09 produces a
>push of 0.105, which is huge since it's larger than the interval itself.  a
>10% perturbation pruduces such an explosion!  the same amount of push is
>produced when an interval with rest span 1000 is compressed to 900, of
>course, but then 0.105 represents one ten thousandth of the interval's rest
>length.  the push is barely noticeable, while the interval has been made
>100 shorter.  it should be clear that an interval as small as 0.10 will go
>completely awry during the first few iterations, while the 1000 interval
>will be adjusting ever so slowly.
>
>it may be strange to realize that no use is made of units in the above
>discussion.  there is indeed no applicable unit, since it's purely in terms
>of numbers.  needless to say this will prompt the physicists to say that
>there is little merit in this exploration.  the mathematicians are back to
>their regular activities, having seen nothing really complex to sink their
>teeth into.  the artists are confused by all the logarithms and numbers and
>ask where the feelings are.  what dicipline are we in with our elastic
>intervals?  sometimes i wonder.  not to worry, however, since people from
>all disciplines seem more than a bit surprised and pleased when they view
>the results that the Struck project has produced, in the form of Quicktime
>movies and other animations and images.  is it art, or is it science?  i'd
>prefer to let the children of the 21st century decide.  i guess that was an
>aside.  now we can return to more aspects of the geometry.
>
>for the purposes of simulated-life animation, it turns out that the results
>of using the logarithmic impulse formula were quite appropriate.  small
>intervals making up the 'body' of a creature were relatively rigid and
>responsive, and the longer intervals corresponding to the 'legs' or 'wings'
>of the creature move with grace and flexibility.  the resulting animations
>are somehow
>intuitively correct, despite the lack of impulse formulas that are
>specifically informed by physics principles.  this was a happy coincidence,
>or the consequence of a successful simplification.
>
>when a logarithmic or similar impulse formula is used, there is always a
>potential for resonances and chaotic behavior.  on the one hand this is an
>interesting area of study.  the process of searching for 'the right size'
>involves progressively scaling down until things start to wobble about.
>scaling up slightly will bring things back to rest.  each structure has its
>own characteristic natural minimum size, under which it wobbles out of
>control.  this is the famous border between order and chaos, which has been
>some of the most fruitful areas of research in recent years.  elastic
>interval geometry provides yet another context in which to study it.
>
>another indispensible characteristic of the logarithmic impulse formula is
>that it flatly denies zero-rest-span intervals.  the outward push becomes
>outrageously large when an interval is compressed to a relatively small
>span.  furthermore, it is impossible to have a span of zero with such
>intervals because the natural logarithm of zero is undefined.  this brings
>me back to one of the points i made at the outset: that elastic intervals
>are twonesses.  a zero-span interval is arguably not a twoness, so this
>kind of impulse formula is a good enforcer of that rule.  furthermore, the
>fractal view in which interval connections actually represent minescule
>interval networks themselves is out of the question, because clearly these
>would exhibit wild chaotic behavior.
>
>(to be continued..)
>
>
>---
>Gerald de Jong, Beautiful Code B.V.
>Rotterdam, The Netherlands
>http://www.beautifulcode.nl
>