D'Arcy Thompson's Affine Fish Transformations

D'Arcy Wentworth Thompson (1860–1948) was a pioneer in mathematical biology, best known for his classic On Growth and Form (1917). He showed that in several classes of organisms, notably fish, the morphology of related species could be generated by simple geometric transformations. This Demonstration considers a subset of these, namely affine transformations—rotations, scaling, and shearing—that are readily carried out by Mathematica. These can be represented by a matrix equation , in which you control the rows and by dragging two locators.



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Snapshot 1: Argyropelecus olfersi
Snapshot 2: Sternoptyx diaphana, obtained by a 70° shearing transformation
Snapshot 3: a seemingly reasonably possible fish, but, according to a colleague in ichthyology, "there ain't no such animal"
Reference: D. W. Thompson, On Growth and Form, abridged ed., New York: Cambridge University Press, 1984 pp. 298–301.
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Related Curriculum Standards

US Common Core State Standards, Mathematics