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Strange Crystal
Darsh Ranjan '05
Department of Mathematics
This crystal grows on its black substrate from a pentagonal seed by reflecting it across its 5 vertices and rescaling the new pentagons by a factor of 0.61803..., the “golden mean,” back towards the point of reflection, and repeating this for all the new pentagons, ad infinitum. Each seed can have its own rule to determine its color and the colors of its descendants. The growth of a single seed has finite area but infinite detail (possessing a fractal dimension of 2). This crystal is strange because crystals in nature do not possess 5-fold symmetry on any large scale, while this one can fill the entire plane very nicely with appropriately placed seeds. In understanding this shape, the arithmetic of the integers extended by the fifth roots of unity proves very helpful.