Editing Hypocycloid (section)

==Relationship to group theory==

[[File:Rolling Hypocycloids.gif|thumb|Hypocycloids "rolling" inside one another.  The cusps of each of the smaller curves maintain continuous contact with the next-larger hypocycloid.]]

Any hypocycloid with an integral value of ''k'', and thus ''k'' cusps, can move snugly inside another hypocycloid with ''k''+1 cusps, such that the points of the smaller hypocycloid will always be in contact with the larger.  This motion looks like 'rolling', though it is not technically rolling in the sense of classical mechanics, since it involves slipping.

Hypocycloid shapes can be related to [[special unitary group]]s, denoted SU(''k''), which consist of ''k'' × ''k'' unitary matrices with determinant 1.  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.<ref>{{cite web|last=Baez|first=John|title=Deltoid Rolling Inside Astroid|url=http://blogs.ams.org/visualinsight/2013/12/01/deltoid-rolling-inside-astroid/|work=AMS Blogs|publisher=American Mathematical Society|access-date=22 December 2013}}</ref><ref>{{cite web|last=Baez|first=John|title=Rolling hypocycloids|url=http://johncarlosbaez.wordpress.com/2013/12/03/rolling-hypocycloids/|work=Azimuth blog|date=3 December 2013|access-date=22 December 2013}}</ref>