The month of Ramadan begins today. This is an opportunity to refrain from food and drink (during daylight hours) and to focus on God. During this and all other months, Muslims pray five times a day towards Mecca. But this brings about an interesting mathematical question: what direction does Mecca lie in?
Perhaps surprisingly, there are two different answers for calculating the Qibla (the direction for prayer). One method is the “constant compass” distance. To visualize this, imagine a world map that uses the Mercator projection. Mark your location and Mecca, and draw a line between them. From Rochester, NY that direction is about 103° clockwise from North, or roughly Southeast (but more East than South).
Another method is to use the shortest distance between two cities. The easiest way to visualize that is to get a globe, put one finger on your location and one on Mecca, and then to turn the globe so that both fingers are lying on an imaginary “equator”. What you’re really doing is finding a great circle — a circle that cuts the globe in half — that passes through both Mecca and your town. From Nazareth College, this turns out to be about 50°-55° clockwise from North, or roughly Northeast (but more East than North). The north part is somewhat surprising since Mecca is south of Rochester, and I thought I’d simply made a mistake the first few times I calculated the distance using spherical trigonometry while teaching History of Mathematics.
There is not complete agreement about which way is the more appropriate, and some of the online Qibla calculators (such as this one) allow you to use either method. It reminds me a bit of the Millennium, with people celebrating on both January 1, 2000 and January 1, 2001, although the question here isn’t which is technically correct but which of the two mathematical models for direction gives the appropriate direction for religious purposes.