The Tusi Couple

The Tusi Couple was a brief diversion on the way from Bernoulli Numbers and the Euler Maclaurin Sum Function to some new destination, as yet undecided.

I came across reference to Tusi while reading Lost Enlightenment.  Central Asia’s Golden Age from the Arab Conquest to Tamerlane by S Frederic Starr.  Nasir al-Din al-Tusi (1201 – 1274) was a polymath who founded what was then the world’s largest astronomic observatory at Maragha or Maragheh in what is now Azerbaijan.

What is now referred to as the Tusi Couple, and naturally was studied by Tusi, is a planar geometric construct consisting of two circles, the smaller of which is half the diameter of the larger and inside it, touching at a single point.  The smaller circle rolls around the circumference of the larger without sliding in a fixed direction, anticlockwise say.

The singular feature of the Tusi Couple is that each point on the circumference of the smaller circle exhibits a periodic motion along a fixed diameter of the larger.



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