Thanks for the reply Bill -- I looked at those PDFs when you sent them last week. They are part of the reason I asked my question. My simplistic pentagon-triangle tiling came out of this discussion I had with a reader back in 2004: http://www.mrob.com/users/ivan-freyman/20040506.html I see how "my" tiling is related to your spacefilling rules, and if I am tracing the curve correctly, I would guess that your "hemis.gif" example is using the trapezoid (hemi-hexagon) rep-tile rule (as seen in the image http://www.mrob.com/users/ivan-freyman/www.armcorp.us/200405-trapezoid.gif). As you can see from those images, my purpose for tilings is related to computer art. It does not matter to me whether there is "a fixed number of different tiles" (as Tom Karzes points out), or that "density variations [...] become unbounded in the limit" (Bill Gosper, referring to his pentagonfill.pdf) because the computer can be programmed to subdivide a tile only when that tile is larger than a desired minimum size. Of course, I knew about the various Penrose tilings but I wanted the artwork to contain actual regular pentagons, and for aesthetic reasons only one other shape which would be an isosceles triangle. I was expecting someone would reply and say "oh, that's a Jenkins quasi-foo tiling, read his 1962 paper" or "that's a class B3 recursive quux tessellation", but apparently there is as yet no such nomenclature. I suppose it is because these tilings have little application beyond art. Marjorie Rice's page (on archive.org here: http://web.archive.org/web/20071011074636/home.comcast.net/~tessellations/te... ) looks familiar, at any rate seems to cover well-known territory (see e.g. http://www.mathpuzzle.com/tilepent.html) - Robert On Mon, Sep 6, 2010 at 03:53, Bill Gosper <billgosper@gmail.com> wrote:
This is precisely the tiling behind www.tweedledum.com/rwg/pentagonfill.pdf and http://gosper.org/pentfill.pdf . (And that stick-length illusion.)
Martin Gardner once fowarded me a set of ingenious but cryptic frac-tiles by Ammann. I don't know where they are, so I hope someone else has a copy. They all lead to spacefills, probably of the pentfill type, since they were based on algebraic numbers. Martin was intrigued by Ammann's constantly shifting return address, and repeatedly requested some biographical data. Ammann finally replied that he was a parking lot attendant. Uh-huh.
Here's an easy one with two equal tiles. Your hemispheres are not quite alike: www.tweedledum.com/rwg/hemis.gif .
I think I've finally figured out that pentagram snowflake S. Witham found (and my tutors rediscovered) in Minsky rug plots. Has anybody else worked on it? --rwg
On Thu, 19 Aug 2010, Bill Gosper wrote:
Yesterday I told tutor Julian an idea for spacefilling a pentagon, and he said I could just conformally map Peano's filled square onto whatever, but then he got into it and helped make www.tweedledum.com/rwg/pentagonfill.pdf . It self-contacts, but the median curve doesn't. Note the density variations, which become unbounded in the limit. I don't have a fix, but I can turn a slight variant of this one into one of those exact continuous maps of the rational unit interval onto some dense set of algebraic points. This would permit sampling at various frequencies and phases, which has proven interesting in previous cases (www.tweedledum.com/rwg/sampeano.htm, www.tweedledum.com/rwg/sam306090.htm ). --rwg
-- Robert Munafo -- mrob.com -- twitter.com/mrob_27