Math 495 Projects
Potential Projects in Mathematics Research (MAT 495)
The Mathematics of 19th Century Navigation
Navigation in the 19th Century was a matter of life and death. An error of less than a mile on a chart could mean the difference between clear sailing and sinking on hidden shoals. Determining latitude was easy; determining longitude was not. Finding latitude simply required clear skies and a sextant to fix the altitude of the sun at its zenith. Finding longitude required either a highly accurate timepiece that would function in the extreme environments encountered on a square-rigged sailing ship or a mathematical means of finding longitude from astronomical observations. An American, Nathaniel Bowditch, first described a method of fixing longitude from observations of known stars and spherical trigonometry.
The Mathematics of Calendars
We take our Gregorian calendar for granted, but there is nothing simple about it. It is a complex blend of astronomy, religion and mathematics. The Gregorian calendar is motivated by the religious desire to keep Easter near Passover and prevent it from moving through the seasons. It is a purely solar calendar. In contrast, the Islamic calendar is a purely lunar calendar whose religious observations, Ramadan and Hajj, move through the seasons. The Jewish calendar is a blend, a lunisolar calendar, whose months are lunar but with leap months inserted to present seasonal drift of the holidays. Its current version, delineated by the 12th Century Jewish scholar Maimonides, is based on Metonic cycles, a curious pattern of the moon discovered by the Greek scholar Meton. The solar and lunisolar calendars are approximations and over time will drift from their desired behavior. The mathematics of calendars have wide-ranging implications many fields of study.
Mathematics of Musical Scales
Musical instruments and musical scales are based on the possibilities of standing waves on strings and in pipes. The major and minor scales of Western music are only two of many possible musical scales. The mathematics of scales are obscure but straightforward; they are the basis for interesting questions like, “What other scales are possible?” and “What (mathematically) makes a chord pleasing or dissonant?”
The production of pleasing sounds by stringed instruments relies on the coupling of vibrating strings to the instrument body, particularly the soundboard. It is possible to model stringed instruments as coupled oscillators. A combination of analytic and numeric work is possible in this approach.
Mapping – Projecting Spheres onto Planes
A map is a two-dimensional representation of a shape on a spherical surface. There is more than one way to skin a cat – and there is more than one way to project a portion of a sphere onto a plane.