We have been working to develop a generalization of the trigonometry called the fractional trigonometry. This work is closely related to the fractional calculus and is best understood in that context. Thus, the talk begins with an overview of the ideas of fractional order integration and differentiation, conceptualizations, and examples of where this mathematics can be applied. We will see some new kinds of differintegrals made possible by the fractional calculus. Because this mathematics generalizes the integer order calculus it is expected to enhance scientific and engineering modeling.
Our stroll will take us to the fractional trigonometry, made possible by the generalization of the exponential function. The fractional trigonometry creates new fractional spiral functions that contain the classical circular functions. The fractional spirals appear to mimic the behaviors of the galaxies and other natural phenomena such as hurricanes and tornados. Current efforts to determine if galactic evolution may be a fractional order process and other potential applications will be discussed.
Sorry, but there are no derivations on this short tour!

Carl Lorenzo is a Distinguished Research Associate (an emeritus position) at the NASA Glenn Research Center in Cleveland. Previous positions there include Chief Engineer of the Instrumentation and Controls Div. and Chief of the Advanced Controls Technology and Systems Dynamics branches. His Awards include the American Rocket Society (now AIAA) Chrysler Award, The NASA Lewis Distinguished Publication award, the NASA Exceptional Service Medal and the NASA Medal for Exceptional Achievement. He received a B.S. in mechanical engineering from Fenn College (Now Cleveland State University) in 1958 and a M.S. in mathematics from John Carroll University in 1965.