I'll see what I can do. On a long bicycle ride yesterday, I began to have doubts about 5; perhaps the number needs to be divisible by 2 or 4. But the construction below doesn't require r1=r2, r3=r4 (the mechanical drawing approach), so this gives me some hope. At 01:25 AM 3/14/2012, Bill Gosper wrote:
Henry, can you exhibit a smooth loop with five radius changes? A special case of four is the "four point ellipse" from mechanical drawings of yore. These can co-rotate in continuous tangential contact<http://gosper.org/pump1.gif> .
Similarly, "six point Reuleaux triangles <http://gosper.org/reuleaux.gif>". (Rich's observation.) --rwg
hgb> I now think that it is impossible to create a simple closed C1 curve from only 3 circular arc segments. The following construction for 4 segments shows why this is. 1. Draw a circle of radius r1. 2. Draw a circle of radius r2 that intersects circle #1. 3. Draw a circle of radius r3 inside the intersection that is tangent to the first 2 two circles. 4. Draw another circle of radius r4 inside the intersection of #1 & #2 that is tangent to #1 and #2. A circular arc segment is taken from each of the 4 circles to produce a closed C1 curve. Basically, it is the boundary of the intersection region, with both sharp ends cut off by circular arcs from circles #1 & #2. The construction shows that r3<r1, r3<r2, r4<r1, r4<r2. There are probably interesting relationships between the centers of these circles, considered as complex numbers, and the various radii. There is a paper by someone at Bell Labs that showed some similar relationships of tangent circles & complex coordinates.