Fourteen years seems like a good time to dust off one of the strangest and potentially most interesting things I ever dug up--the Fourier series for the gasket-filling curve. The coefficients are easily computed by a rapidly convergent 2x2 matrix product which is not termwise equivalent to a sum nor anything else familiar. It seems an absolute poster child for the superiority of matrix products. In fact, it overperforms, computing in its four elements the desired A(x) four different ways: A(x) -A(x) ( ). A(-x) -A(-x) (Unfortunately, you don't gain a factor of 2 from this because you typically want A(k+1/m) and A(-k+1/m), rather than A(-k-1/m), for some integer m>1. But it makes sense to cache both anyway.) I was able to express A(x) as a *limit* of a sum of 3^n/2 terms, but divided by 2^n instead of 3^n/2--a sort of cockeyed Stieltjes integral. But not really, because the summand involved a stutter-free integer sequence d(n) (A108964) with a weird rule of formation individually, and a different weird rule of formation for the set {d(n)}. So there was no taking the limit for the integral anyway. The final form is oo gasket(t,m) = Sum(A(k+1/m) exp(i pi t (k + 1/m))) k=-oo which traces |m| gaskets on the sides of a regular |m|-gon as -n < t < n. (Inside for m<0.) We usually think of (and draw) this fractal as a bunch of line segments representing the boundaries of the deleted triangles. But this is a set of measure zero in the true fractal, and seems to be the values of gasket(t,3) for which the ternary expansion of t has only finitely many 0s, 1s, or 2s. Formulae and pix: http://gosper.org/gasky.pdf . --rwg Some new weirdness has crept in. The case m=2 used to draw two gaskets base-to-base as expected, but Mathematica now draws crazy squiggles. Also, does anybody know how to prevent Mma from idiotically spending *minutes* trying to find a closed form or a fancy (but disastrous) numerical method when all you want is a Sum of a few dozen terms? I wind up having to Total a Table! PS: Are there uncountably many "squarefree" (stutter-free) strings base 3? PPS, There are no straight lines in Julian's gasket since no segments in the uninverted gasket extend through the origin.