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Mathematics and Optimal Form (Scientific American Library)
Mathematics and Optimal Form - Scientific American Library Author:Stefan Hildebrandt, Anthony Tromba Combining striking photographs with a compelling text, authors Stefan Hildebrandt and Anthony Tromba give us a thoughtful account of the symmetry and regularity of nature's forms and patterns. Although often easily seen, these forms and patterns are not always easily explained. Are there simple universal laws that allow us to comprehend them? Wh... more »y are celestial bodies spherical rather than square or pyramidal? Nature does not abhor sharp polyhedral structures; we see them in crystalline formations. Can these jagged structures be explained by the same underlying principle that explains the shape of a perfectly round and symmetric soap bubble?
This book examines attempts by scientists and mathematicians throughout history to find answers to these riddles. It is the story of the development of the branch of mathematics called the calculus of variations, which concerns questions of optimization-finding forms or patterns that maximize or minimize a particular quantity. Is the igloo the optimal housing form for minimal heat loss to the outside? Do bees really use the least possible amount of wax in constructing their hexagonal cells? Moreover, is there an underlying principle that describes the infinite variety of forms in our world?
These questions have no final answers, but scientists continue to explore the notion that nature is governed by the principle of the economy of means-that nature proceeds in the simplest, most efficient way. The authors look at familiar examples of nature's tendency to economize, from soap films, soap bubbles, and the nuclei of atoms, to the planets of our solar system. They demonstrate how the laws of physics can be applied to this phenomenon and trace the development of the principle of economy of means from antiquity to the nuclear age.« less