Image: Self-affine set

Description: A self-affine fractal set. Build iteratively from a p×q{\displaystyle \scriptstyle {p\times q}} array with p≤q{\displaystyle \scriptstyle {p\leq q}}. Its Hausdorff dimension equals log⁡(∑k=1pnka)/log⁡p{\displaystyle \log {\left(\sum _{k=1}^{p}n_{k}^{a}\right)}/\log {p}} with a=log⁡p/logq{\displaystyle a=\log {p}/log{q}} and nk{\displaystyle n_{k}} is the number of elements in the kth{\displaystyle k^{th}} column. Here it is 1.8272. The box-countig dimension yields a different formula, therefore, a different value. Unlike self-similar sets, the Hausdorff dimension of self-affine sets depends on the position of the iterated elements and there is no formula, so far, for the general case
Title: Self-affine set
Credit: Own work
Author: Prokofiev
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License: CC BY-SA 3.0
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