Symmetry in the Basic Sciences
Final rept. Aug 1988-Apr 1989
AIR FORCE ACADEMY COLORADO SPRINGS CO
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This report summarizes talks given by USAF Academy faculty members at an interdisciplinary colloquium held 7-10 Nov 88. Chapter 1 presents the basic mathematical theory behind plane symmetry groups. This theory is then applied in classifying the symmetry of bounded figures, frieze patterns and wallpaper patterns. Recently developed algorithms are included to help analyze complex designs. Chap. 2 discusses symmetry operations relevant to three-dimensional crystallography. In particular, the seven crystal systems that classify the thirty-two crystallographic point groups are described. These are then used to construct the Bravais lattices. Chap. 3 investigates the role of symmetry in biological forms. Specifically, DArcy Thompsons work on growth and form of molluscan shells is reviewed with an attempt to explain the consequences of that growth and form to the natural history of the Chambered Nautilus and its ancestors. Chap. 4 looks at the central role symmetry has increasingly played in physics by examining the Principle of Least Action and the invariance of the Lagrangian under a transformation. Noethers Theorem guarantees that a conservation law is associated with each of these symmetries. Examples include the conservation of energy, linear momentum, and angular momentum, as well as the purely quantum mechanical symmetry of invariance under an exchange operation. A brief look at gauge theories is the final example of how symmetry has become a guiding principle in the formulation of new theories.
- Theoretical Mathematics
- Quantum Theory and Relativity