James Steiner wrote:
Its an application of basic geometry.
If the struts of the triangle are made of materials that do not
stretch, compress, or flex (outside of acceptable parameters for the
construction in question), then the triangle is *stable*--even if the
joints are frictionless pivots. This is essentially because the struts
hold their opposing joints at fixed angles--something no other 2d
arrangement does.
So, I guess you could say that the stability of a triangle is an
emergent property of the geometry.
Then again, I"m not a wise man.
James -
Nicely succinct. I sent a much more elaborate version of this to Nick
privately.
But there is a point I want to make publicly:
Aggregate or Composite properties is not equivalent to Emergence.
Sadly, defining Emergence (or Complexity or ...) is a bit like defining
Art.
I have a wonderful collection of "ingenious mechanisms for inventors"
from the Industrial Age
with wonderful collections of levers and wheels and gears and cams and
rivet patterns. Within
those mechanism are hidden a number of triangles and quadrilaterals
whose properties
of rigidity and limited degrees of freedom are exploited to practical ends.
All of these have wonderful collective properties, but none of which I
would call
"Emergent".
The key (upon reflection) to what I insist on to call something
"Emergent" is
nonlinearity. These rigid-body structures all have linear
properties. If you
draw a phase-space diagram, there will be nothing but a series of
well-defined
lines (or areas) describing the paths of the components through space-time.
I suspect (but can't muster the intellectual will to prove or even
illustrate it)
that if we add the art and science of tolerances to the discussion, that
some
emergent properties might come in to play. That a well-toleranced
mechanism,
under repeated use, friction, and wear, will settle into an attractor...
that this
is what tolerancing is about.
I postulate that a mechanism designed without tolerancing is poised on a
ridge between basins of attraction and that such mechanisms "wear out"
by falling off that ridge and undergoing accelerated wear and eventually
catastrophic failure. In a properly toleranced mechanism, we are already,
by design starting out in the middle of a (specifically and well-designed)
basin of attraction and the natural wear of the mechanism will simply cause
it to wander around within that basin.
Perhaps, in the words of Nick, there are "wise men" here who can speak
to this more eloquently? Does anyone know of such a study on tolerancing
and nonlinear dynamics?
- Steve
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