I like to view the tiling in terms of the underlying tiling polygon (formed by connecting the vertices in the fractal-tiled plane), which in the case of the terdragon is a 60/120 rhombus (i.e., two equilateral triangles joined at an edge). Three of the rhombuses join to form a hexagon which corresponds to the "fudgeflake". I have illustrated them here: http://karzes.best.vwh.net/gif2.out/terdragon.html http://karzes.best.vwh.net/gif2.out/tri1.html It's a simple javacsript app - give it a minute to load, then click on "next" to step through the rep-tile expansions. Here's a similar treatment of Bill's fractal: http://karzes.best.vwh.net/gif2.out/gosper.html Some cases require more than one polygon. For instance, the snowflake curve is self-tiling, but only when using two different sizes of smaller snowflake curves, the larger corresponding to a hexagon and the smaller to a triangle: http://karzes.best.vwh.net/gif2.out/snowflake.html Additional examples are listed here: http://karzes.best.vwh.net/gif2.html This also includes the two little-known Penrose fractals (which I discovered, but which were independently discovered (and published) by others). Tom Bill Gosper writes:
http://gosper.org/islands.png shows four triadic dragons sharing an endpoint, and fanned out at angular intervals π/6, π/3, and π/2. If we suppose each has unit area, it is a minor puzzle to compute the distance between endpoints (12^(1/4)), and the (slightly greater) diameter, which I haven't bothered to rederive. Perhaps the easiest area derivation is to note that the dragons tile the (infinite) plane in a triangular grid, with 3/2 dragons/triangle. This is a case of using infinity to answer a finite question, and is unrelated to the dragon's infinite perimeter, e.g.
Small constant #1: What is the diameter minus endpoint span.
Notice the purple Hawaiianoid chain of apparently similar islands formed by the purple dragon minus the blue one, and the (necessarily) congruent white chain between the green and gold dragons. While trying to characterize these Islands, Corey & Julian noticed something peculiar.
Define A(ø) := the area of intersection of two coterminous unit area tdragons fanned at angle ø. A(π/3)=0 by virtue of the plane tiling. A(π/6) appears to be some simple function of the area of an island chain. But maybe not an island! On very close inspection, A(π/2) is very slightly positive. This is not a positioning artifact--in the fractal limit, coterminous dragons infinitely entwine, and cannot be slid apart without rotation. This is not visually obvious because they are based on a log spiral with a scale gain of 729 per turn. (kth order dragon ~ (rt3 e^(i pi/6))^k. k=12 makes one turn.)
It thus appears that the islands are very slightly dissimilar!
Small constant #2: What is A(π/2)?
Small constant #3: What is the least positive ϵ for which A(π/2+ϵ)=0? --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun