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 More Numberphile videos with Dr Krieger: http://bit.ly/HollyKrieger Her Twitter: https://twitter.com/hollykrieger 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://www.youtube.com/watch?v=0hlvhQZIOQw Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Subscribe: http://bit.ly/Numberphile_Sub Videos by Brady Haran Patreon: http://www.patreon.com/numberphile Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/ Brady's latest videos across all channels: http://www.bradyharanblog.com/ Sign up for (occasional) emails: http://eepurl.com/YdjL9