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This is an illustration of the method of constructing different sizes of Kakeya sets using the "sprouting" method, in which the final shapes are what are called Perron trees. One starts with a triangle, and then partitions it into ${\displaystyle 2^{n}}$ subtriangles (for some integer ${\displaystyle n}$) and then shifts each consecutive pair of triangles over each other so they overlap, then shifts the consecutive remaining pieces, and so on until a single "tree" shape is obtained. For large enough ${\displaystyle n}$, one can shift these triangles in such a way to get a tree as small as desired. The image is based on the construction described in Falconer's book "Geometry of Sets and Fractals". I made this image myself on Flash MX.

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Derivative works of this file:  Perron tree.svg