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With Dr Holly Krieger from Murray Edwards College, University of Cambridge.
Have a look at Brilliant (and get 20% off) here: https://brilliant.org/Numberphile
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Extra detail via Holly: Here's a link to an article (meant for a somewhat general audience) by Bob Devaney explaining the numbers of components, which are called periods: https://plus.maths.org/content/os/issue40/features/devaney/index
And more from Bob: http://math.bu.edu/DYSYS/FRACGEOM2/FRACGEOM2.html
If checking out https://brilliant.org/Numberphile ---
Try the complex algebra course at http://bit.ly/Brilliant_Complex
Editing and animation by Pete McPartlan
Farey Sums: https://youtu.be/0hlvhQZIOQw
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Videos by Brady Haran
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