http://gosper.org/fst.pdf (pp130-131, 137) shows how to construct the Fourier series for m copies of an arc z(t), -1<=t<=1, arranged around a regular m-gon. Replacing m by -m flips the arcs across the corresponding sides of the m-gon. Or more likely, rotates the arcs by pi about the centers of the sides of the m-gon, and then sweeps them in the opposite order. Using my peculiar 2x2 matrix product for the Sierpinski gasket Fourier series, m=6 makes a spongy star of David, and -6 makes a lace doily. But what I really wanted was six gaskets pointing alternately in and out, which makes the boundary of the region of the complex plane represented by the base 2 pure fractions, using the digits 0, 1, (-1)^(2/3), and (-1)^(4/3) ! (http://www.tweedledum.com/rwg/rad2.htm, Mandelbrot calls (part of) this the Arrowhead Fractal.) Unable to salvage my spastic attempts, Julian finally found the general "inny-outy" transformation, after a 12 hr struggle. ( http://gosper.org/sierpclock.png http://gosper.org/base2sierp.png) Since we were unable to rederive my 2x2 product, Julian also derived for good measure a 3x3 (with 9^-n convergence vs my 3^-n). We have yet to demonstrate their equivalence. Presumably, these (infinite) series (exactly) represent a "space"fill of the closed version of the D=lg 3 gasket fractal. (Which, if accurately drawn, is completely invisible, since it has zero area. Unless it is implicit as the boundary of a 2D region.) The six-gasket boundary arc appears to have uncountably many double points, and one triple point. Spacefilling any set makes it an arc. Unlike the gasket, many such sets, with 1<=dimension<2, can be spacefilled 1-1. The "open" gasket, constructed by deleting closed triangles from a open one, is a countable intersection of open sets. Is it actually open? --rwg