This seminar will start with a reformulation of Trigonometry attributed to Ptolemy, followed by the ingenious methods his geocentric model of the universe employed. Although this model was scientifically unsound, the mathematics were excellent and form the foundation of Fourier's methods 1500 years later. Today this technique is used for decomposing arbitrary signals (light, sound, economic, etc.) into a superposition of component frequencies.
This course reinvents and extends the power of trigonometric functions. Review this document contains examples of questions you should be able to understand to ensure you are prepared for the explorations in this seminar.
This document is an introduction to our first expedition, and you are encouraged to think about the questions before our first meeting. It also contains some additional expeditions that we may undertake throughout the year.
Trigonometry is such a widely applicable collection of tools, the modern presentation of the material relies heavily on identities and memorization. Ptolemy’s formulation is more geometric and builds an intuition that is hard to develop otherwise. Not surprisingly, his insights into circular motion helped create a remarkably accurate model of celestial movement. We will connect this to modern methods of spectroscopy and advanced imaging to show how this ancient insight is still bearing fruit today.
Ptolemy did his trigonometry inside a circle with a unit diameter, and by the extended Law of Sines he identified the sine of an angle with the length of the opposite side. Clever constructions lead naturally to angle addition formulas and deepen the connection between circles and inscribed triangles. His geocentric model can be described in modern terms as a Complex Fourier Series, where each successive term corresponds to another epicycle of the orbit.