Abstract
Squig intervals are a class of hierarchically constructed fractals introduced by the author. They can be visualized as the final outcome upon a straight interval of a suitable cascade of local perturbative “eddies” ruled by two processes called decimation and separation. Their theory is summarized and their scope is extended in several new directions, especially by introducing new forms of separation. Squig intervals are generalized in two dimensions, with fractal dimensions ranging from 1.2886 to 1.589. Squig sheets are constructed in three dimensional space with fractal dimensions ranging from 8/3 up. They should prove useful in modeling the fractal surfaces associated with turbulence and related phenomena. Squig intervals are constructed in three dimensions. Nonsymmetric “eddies” and the resulting squigs are tackled. Squig trees and intervals are drawn on unconventional lattices, either in the plane or in a prescribed fractal surface. Peyriére'sM systems are mentioned: their study includes the proof that the informal “renormalization” argument (involving a transfer matrix) is exact for squigs.
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The reader's attention should be drawn to the fact that the second and later printings of this book include an update chapter and additional references. Though it should not have been necessary, it may be useful also to mention here that most of the material in this book that concerns physics, e.g., polymers and percolation clusters, wasnot found in either of my two earlier Essays on fractals,Les objects fractals: forme, hasard et dimension (Flammarion,
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Mandelbrot, B.B. Squig sheets and some other squig fractal constructions. J Stat Phys 36, 519–539 (1984). https://doi.org/10.1007/BF01012918
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DOI: https://doi.org/10.1007/BF01012918