Jan Kåhre at A New Kind of Science NKS 2003
The content of my poster presented at the NKS2003 in Boston. Typed comments are added later and build upon my notes taken at the conference. As Steve Wolfram said in the beginning: "It is not necessary to wait for actual work to be done on an application before publishing the ideas". Here are some ideas:
NKS in its current form relies on visual substantiation. As Steven Wolfram said (from lecture notes): "You run the program and you see what happens". Examples: It looks like snowflakes (NKS, page 371), it looks like turbulence (NKS, page 380), and it looks like fractures (NKS, page 375). It is not meaningful to say "how an atom looks". A graphical representation of a Schrödinger solution is the best we got for visual substantiation of an atomic model.
Steven Wolfram raised an objection to a similar kind of calculation: Finding the densest packing of balls by pushing them together. It will not work because the balls get stuck (See also NKS, page 349). There is a thermodynamic analogy: trying to reach absolute zero temperature. This is impossible (Nernst's law). In contrast, the molecules in the Bernoulli ideal gas model don't get stuck; they don't even interact.
The NKS snowflake model sets no limits to size except by the number of steps taken. How should other size constraints be introduced?
One solution could be to induce stability by applying feedback to the isoperimetric problem. During the conference, Robert Lurie asked about implementing feedback within NKS, and Steven Wolfram answered that "feedback is hard to emulate". A bit unexpectedly, because cellular automata are themselves pure feedback systems: the output of one step is the input to the next.
Another solution contributed during the poster session was to use two sets of cellular automata working in parallel: One set of ideal gas molecules bouncing around, one set of sticks connected end to end forming a flexible ring.
The isoperimetric problem belongs to a large class of relevant problems where size is limited by constraints: In the balloon case volume is the restriction. Biologic (Darwinistic) evolution is driven by limited resources forcing competition.
Now we are ready to apply the calculus of variations:
P.S. From the horse's mouth, Steve Wolfram when his Mathematica demonstration went astray:
Contact: jankahre (at) hotmail.com