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A generic geometric transformation that unifies a wide range of natural and abstract shapes
Author(s) -
Gielis Johan
Publication year - 2003
Publication title -
american journal of botany
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.218
H-Index - 151
eISSN - 1537-2197
pISSN - 0002-9122
DOI - 10.3732/ajb.90.3.333
Subject(s) - euclidean geometry , logarithm , transformation (genetics) , trigonometry , pythagorean theorem , simple (philosophy) , range (aeronautics) , natural (archaeology) , variety (cybernetics) , computer science , trigonometric functions , mathematics , artificial intelligence , geometry , mathematical analysis , biology , paleontology , biochemistry , philosophy , materials science , epistemology , composite material , gene
To study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man‐made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non‐Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.