Caption: The Tusi couple (see the animation to the right) is a mathematical device in which a smaller circle (of radius r) rotates around its center while its center rotates (with radius r) around the center of a larger circle twice the diameter of the smaller circle. The compounded motions of the smaller circle cause a point on the circumference of the smaller circle to undergo a sinusoidal oscillation along a diameter of the larger circle. The Tusi couple is a 2-cusped hypocycloid.

Note that along a diameter of any orientation, there will be a sinusoidal oscillation of a some point on the smaller circle. This result follows from symmetry. The animation shows explicitly the sinusoidal oscillations along the x-axis and the y-axis.

Features:

1. The Tusi couple was first introduced in the 13th century by Nasir al-Din al-Tusi (1201--1274) (a Persian polymath: architect, astronomer, mathematician, philosopher, physician, scientist, theologian) in his Tahrir al-Majisti = Commentary on the Almagest (1247) as a model for the latitudinal motion of the inferior planets. It was later often used as a replacement for the equant introduced by Ptolemy (c.100--c.170 CE) in the Almagest. The equant was the crime of Ptolemy since it violated the principle of uniform circular motion which was usually posited as essential by Greek astronomy and by Ptolemy himself.

2. The term Tusi couple is modern and was introduced by science historian Edward Stewart Kennedy (1912--2009) in 1966.

3. Nicolaus Copernicus (1473--1543) made use of the Tusi couple in his planetary models in his De Revolutionibus Orbium Coelestium = On the Revolutions of the Heavenly Spheres (1543).

   It is NOT known how Copernicus knew of the Tusi couple. There are 3 possibilities:

   1. Some more or less complicated transmission path from Nasir al-Din al-Tusi.
   2. Some more or less complicated transmission path from some unknown independent discoverer.
   3. Copernicus may have discovered the Tusi couple for himself---it's not rocket science.

4. The transformation of rotation (AKA rotary motion) to sinusoidal oscillation (in technology jargon, reciprocating motion) and vice versa is an important ingredient in technology. For example, the internal combustion engine (ICE) works by converting the reciprocating motion of a piston in an engine cylinder into the rotation of car wheels. See the figure below (local link / general link: internal_combustion_engine.html).

   Yours truly CANNOT think of any exact realization of the Tusi couple in technology.

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