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The Gypjak Mosque just outside of Ashgabat, Turkmenistan is the largest mosque in Central Asia. It is made out of white marble with gold trim, and the inside of its dome is decorated in a beautiful lotus-like pattern. It is a truly stunning building. But looking up at the design inside the dome, I wondered what it would look like if it continued outward indefinitely. Specifically, I was curious about what would happen if we followed one of the curves formed by a leaf edge indefinitely:
What would happen if we extended this curve forever? Would the curve end up being parabolic? Would it rise like an exponential function? Would it be asymptotic like a hyperbola? Would it curve around inward, back to the very center where it started? This is a very interesting question, which happens to have a fairly simple answer.
Notice that each "ring" of diamonds around the center circle is made of 72 diamonds, with the exception of places where four smaller diamonds are combined to make a larger diamond. As we follow the curve outward, each time we get to the next ring, we've moved counterclockwise by half a diamond. So as the radius increases, the angle we're moving increases as well. This looks similar to the polar curve r = θ, as you can see to the right.
(Note that this is not a perfect model, as the size of the diamonds increases as the radius increases. But this will be a close enough approximation for our discussion.)
Notice that as the radius increases indefinitely, the angle will also increase indefinitely. Specifically, if we were to zoom out and continue the pattern indefinitely, we would end up with a spiral that would continue to loop around the center over and over again.
So if we were able to follow the curve on the mosque's ceiling forever, it wouldn't be parabolic or exponential. It wouldn't be asymptotic. It wouldn't curve back to the center. Instead, it would create an infinite spiral.
Sample Problems
Sample Problems
1. Notice that the equation r = θ does not work perfectly, as you can see from my model that this causes the diamond shapes get distorted. Modify the equations in this model to see if you can make a more accurate one.
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