I've been working on modeling an orchid using the union of three 
spherical products, and making some decent progress, and if it gets good 
enough, I'll post the result, but I keep getting sidetracked by other 
interesting thoughts.

A rather famous multifractal Fourier series is the function W(x), 
referred to variously as the Riemann multifractal function or the 
Weierstrass multifractal function, defined and described here, along 
with its interesting pathologies:

http://mathworld.wolfram.com/WeierstrassFunction.html

I took the first four terms, not scaled by pi (as in the above 
reference) and over a full period (0 to 2pi), i.e. considered as a 
classical Fourier series, and crossed W(u)W(v) with Sin(3v) -- just 
because tracing one petal basically gives the shape of the R1 function 
in the xz plane, while rendering the whole object gives a nice 
presentation of mirror images, as it were, of the R1 function (along 
with a "stem," not visible from above.)

So, for those keeping score, we have the parametric surface:

X(u, v) = W(u)*W(v)*Cos(u)*Sin(3*v)*Cos(v)
Z(u, v) = W(u)*W(v)*Sin(u)*Sin(3*v)*Cos(v)
Y(u, v) = Sin(3*v)*Sin(v)

In the first image, 0 <= u <= 2pi, and 0 <= v <= pi/3, while in the 
second,  0 <= u <= 2pi, 0 <= v <= pi.

I think the first (one petal) looks like a moth (wings sloped back, 
fuzzy antennae) and the second (both upper petals) is more of a butterfly.

The camera is looking down from positive y, and in the first image, the 
positive z-axis is down, while in the second, the positive z-axis is to 
the right.

I apologize for the gaudy coloring, but while I was working on it, one 
of my colleagues wandered by and informed me that my original and (in my 
opinion) tasteful color scheme was "boring,"  so I changed it.  It also 
serves to highlight the onset of the complexities.  (Yes, I'm POVing at 
the office, but that's what summers are for, if you're a teacher in 
America.)

Dave Matthews

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