Hypocycloid

The 2-cusped hypocycloid called Tusi couple was first described by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in Tahrir al-Majisti (Commentary on the Almagest).

[1][2] German painter and German Renaissance theorist Albrecht Dürer described epitrochoids in 1525, and later Roemer and Bernoulli concentrated on some specific hypocycloids, like the astroid, in 1674 and 1691, respectively.

Specially for k = 2 the curve is a straight line and the circles are called Tusi Couple.

Nasir al-Din al-Tusi was the first to describe these hypocycloids and their applications to high-speed printing.

[4][5] If k is a rational number, say k = p/q expressed in simplest terms, then the curve has p cusps.

The hypocycloid with two "cusps" is a degenerate but still very interesting case, known as the Tusi couple.

This motion looks like 'rolling', though it is not technically rolling in the sense of classical mechanics, since it involves slipping.

For example, the allowed values of the sum of diagonal entries for a matrix in SU(3), are precisely the points in the complex plane lying inside a hypocycloid of three cusps (a deltoid).

Likewise, summing the diagonal entries of SU(4) matrices gives points inside an astroid, and so on.

Thanks to this result, one can use the fact that SU(k) fits inside SU(k+1) as a subgroup to prove that an epicycloid with k cusps moves snugly inside one with k+1 cusps.

The Pittsburgh Steelers' logo, which is based on the Steelmark, includes three astroids (hypocycloids of four cusps).

In his weekly NFL.com column "Tuesday Morning Quarterback," Gregg Easterbrook often refers to the Steelers as the Hypocycloids.

Chilean soccer team CD Huachipato based their crest on the Steelers' logo, and as such features hypocycloids.

The astroids on the doors and turntable were removed when the show switched to high definition broadcasts starting in 2008, and only the giant price tag prop still features them today.