Which direction is Mecca?

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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.

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5 Responses to “Which direction is Mecca?”

  1. Aleya Says:

    That’s great! Thanks! I’m subscribed to your blog and I am a Muslim, so it’s nice to open up my google reader and see an article about the direction to Mecca. Thanks a lot!

  2. Ξ Says:

    You’re welcome — I’m glad you enjoyed the post! In college I learned that very little math was done in the Middle Ages, and it was only once I began teaching that I learned that that wasn’t true worldwide, and in fact a great deal of math was being done in the Islamic world (e.g. spherical trigonometry and finding direction and distance to Mecca).

  3. Hakim Says:

    I’m a Muslim, and I personally think someone has pulled a fast one on us in the NorthEast of the Americas…..According to this NE calc. I face the same direction towards Mecca as would a person in central Yemen. NE…but we are NOT in the same location. But Mecca is only in 1 place, while we are in 2 different places. Therefore, I am NOT facing the qiblah when satisfying some math. problem. Its so simple. Mecca is SE. NOT NE. I dont want to start a debate. So, when I am in mosques that want to be sophisticated, I go along..Allah knows whats in my heart. At home, I pray SE. Where I have control, I pray SE. I suggest to Muslims to do the same.
    Salam/peace.

  4. TwoPi Says:

    Hakim: You might find this Globe Applet interesting and illuminating:

    http://www.joelduffin.com/opensource/globe/

    If you click on two of the cities indicated on the globe, it illustrates the Great Circle route between them in green on the globe, as well as the Mercator straight-line route in red,

    Mecca isn’t one of the handful of cities they’ve included, but the path from NYC to Bombay is pretty similar in its geometric features.

  5. sultan hakim Says:

    The distance between Detroit and Mecca is the same whether you look away or face 12 degrees south of East. You cannot face N.E.
    and be facing Mecca,S.A.

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