[math-fun] fractal optimizer?
A few months back on this list (I'm looking back thru the archives) somebody wanted to know if fractals were optimal solutions to something [of real-world interest]. Other than maximizing the amount of bullshit. :) Let me suggest two interesting candidates. 1. One problem I've long been interested in is this: the strongest beam with a given length and given mass of material (or: given the strength & length, minimize the mass). "Strong" means against buckling instability, as first examined by Euler. The optimum convex shape is known. However, much stronger beams are nonconvex and I believe probably fractal. See, you get more stiffness by making the beam wider, make a truss out of lots of triangles for example. This is way stiffer (=more stable against buckling) than any long thin beam of the same mass which is convex. But then, each strut in the truss, presumably also wants to be an optimum beam, I.e. also a truss viewed magnified. And so on recursively. Fractal. --- Another problem I'm interested in is this. You've all heard of the Penrose tiling (set of two tiles which only tile aperiodically). It is a famous open problem ("Einstein" problem) whether a single 2D tile exists which only tiles aperiodically. I would like to pose the same problem in 1D where now I do not demand that the tile be a connected set. I mean: 1D tiling? Now THAT'S simple. But... I do not know the answer, even in 1D. I feel quite embarrassed about it. I mean, you have to be an idiot not to be able to solve 1D tiling. If there is such a tile, it probably has a fractal-esque nature. For a (not quite) example Cantor's "exclude middle third" set is tiled by two copies of itself and is fractal in nature, so you could say it is a solution of this optimization problem: "minimize area subject to requirement two copies tile itself and it contains >=2 points."
If your single 1D tile can be reduced to a finite set of nonnegative integers, including 0 for definiteness, then any tiling can be converted to a periodic one. Suppose the integers are covered by some tiling. For each K, consider the set of integers >K which are covered by some tile that begins at a value <=K. This set is of bounded width, not exceeding the width of the tile. So there are at most 2^width possible shapes for the set. Somewhere in the first 2^width+1 integers, there will be a K' and K'' with congruent sets (i.e., shifted by K''-K'). The supertile defined by the shift, i.e. tiles covering up to K'' but not entirely <=K', has a set of holes on the left that matches the covered squares on the right. So the supertile makes a periodic covering of the integers. There are easy tiles which can't be 'integrated' (rationalized?), but do cover the real line, so maybe there's an irrational aperiodic disconnected 1D tile. Rich -------- Quoting Warren Smith <warren.wds@gmail.com>: <clip>
Another problem I'm interested in is this.
You've all heard of the Penrose tiling (set of two tiles which only tile aperiodically). It is a famous open problem ("Einstein" problem) whether a single 2D tile exists which only tiles aperiodically. I would like to pose the same problem in 1D where now I do not demand that the tile be a connected set. I mean: 1D tiling? Now THAT'S simple. But... I do not know the answer, even in 1D. I feel quite embarrassed about it. I mean, you have to be an idiot not to be able to solve 1D tiling.
If there is such a tile, it probably has a fractal-esque nature. For a (not quite) example Cantor's "exclude middle third" set is tiled by two copies of itself and is fractal in nature, so you could say it is a solution of this optimization problem: "minimize area subject to requirement two copies tile itself and it contains >=2 points."
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