This chapter describes, analyzes and attempts to explain the highly original and visionary approach to the fundamental nature of living organisms presented by D'Arcy Wentworth Thompson (1860-1948) in his classic, idiosyncratic book, On Growth and Form (1917). In opposition to the then common explanatory concept of vitalism, he argued that the adaptedness and integrity of structure seen in the anatomical shapes of organisms could be explained in terms of physical forces and precisely described in mathematical terms. He can be seen as part of the modernizing of biology. The book has continued to inspire distinguished biologists to this day and has been a significant influence on artists, architects and engineers. It is argued that D'Arcy Thompson's influence owes much to the imposing language and rich pictorial content of his book, but that book represents a uniquely powerful expression of the challenges still facing us in fully explaining the structural complexity of biological forms.
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