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Penrose Tiling on the Hydra — Parallax Forums

Penrose Tiling on the Hydra

LoopyBytelooseLoopyByteloose Posts: 12,537
edited 2010-10-13 00:11 in Propeller 1
http://en.wikipedia.org/wiki/Penrose_tiling

The above is a link to Penrose tiling. I am wondering if gamers have given any thought to using it. I do realize that it is quite problematic as rectangular tiling is optimal for code and storage.

Nonetheless, I am quite interested in exploring the 5 fold symmetry of Penrose tiles and even the more recent discovery of quasi-crystals - which are 3-D Penrose tiles.

I'd love to find some code to learn from.

Comments

  • trodosstrodoss Posts: 577
    edited 2010-09-29 07:51
    Loopy,
    Penrose tiling could be implemented using 3D rendering techniques in 2D.

    Phil Pilgrim had a 3D Graphics demo, which you might be able to use the transformation matrix math (translation, scaling, rotation) from to 'place' your shapes.

    http://forums.parallaxinc.com/forums/default.aspx?f=25&m=358882

    Baggers also had a demo of polygon rendering for his PropGFX code, however I do not know if the source was released.

    Hopefully that helps,
    --trodoss
  • LeonLeon Posts: 7,620
    edited 2010-09-29 07:52
    Kleenex used the original Penrose tiling for their lavatory paper a few years ago; Penrose sued Kimberly Clark and there was an out-of-court settlement. I remember buying it at the time, and thinking that the pattern looked familiar. They didn't use it for long, which must have been part of the settlement; Penrose presumably didn't like the idea of people wiping their posteriors on his discovery.
  • LoopyBytelooseLoopyByteloose Posts: 12,537
    edited 2010-09-29 10:42
    Hmmm.... I'd forgotten that the Penrose family had sued Kleenex for using 'their tiles'. It is a sad day when advances in mathematics are subject to ownership. In the past 15 years, the discovery 3-D Penrose structures have explained structures in metallurgy that were being ignored by traditional crystallography. So their existence in nature and in 3-D make them more profound than a mere 2-D tiling oddity.

    I suppose this 'intellectual property' issue makes it nearly impossible to write a game or even a video demo without some sort of royalty to the Penroses. They seem to want a royalty for any decorative use and buying ceramic tiles of their 'shape' are quite expensive. Nonetheless, there are examples of Islamic artwork that demonstrate that tiles very near to the Penrose tiles were understood long before their property claims. Those were mostly on a 10-fold symmetry rather than the 5-fold symmetry of the Penrose tile. But the results in both cases were a 'non-periodic' pattern.

    I do realize that a big chuck of brute force matrix math may get them to work on the Propeller, but I was wondering if something akin to a tile engine for games could find a way to implement them. It really comes down to fitting a 72 degree module in a 90 degree presentation. But of course, there is an awful lot of rotation of the tiles to consider. I guess I should try to go through cases and make some tables.
  • LeonLeon Posts: 7,620
    edited 2010-09-29 10:54
    Penrose just copyrighted the pattern in the same way as, say, a font can be copyrighted. Kimberley Clark copied the design and patented it, which was rather naughty of them.
  • trodosstrodoss Posts: 577
    edited 2010-09-29 11:03
    By 'tiling' are you meaning texture mapping on the polygon? A lot of raycasting engines use that method.

    Somewhere in the Hydra forum, Baggers posted his Wolf3D clone code that used a form of texture mapping.
  • Phil Pilgrim (PhiPi)Phil Pilgrim (PhiPi) Posts: 23,514
    edited 2010-09-29 21:50
    The thing is, though, that there is not one, single Penrose tiling of the plane. Since the tiling is quasi-periodic, there have to be scads of distinct Penrose-style tilings. There can't possibly be copyrights on all of them! That would be like saying the copyright holders of Happy Birthday to You could sue anyone who used the same notes but in a different order.

    -Phil
  • LoopyBytelooseLoopyByteloose Posts: 12,537
    edited 2010-09-30 06:47
    I thought that the Penrose tiles were all based on two polygonal shapes - a vee and a diamond (aka, a kite and dart).

    http://en.wikipedia.org/wiki/Penrose_tiling

    It is an interesting issue as to what exactly Penrose did and could claim ownership to. A patent would be more appropriate than a copyright. I'd have to do some research to find out.

    The 'other problem' is that maths seems to really prefer a rectangular coordinate system and that scheme leaks over into computer as rows and columns in display and memory.

    Tiles in mathematics are generally 2-D mappings. We all know that squares, regular triangles, and regular hexagons will create nice well behaved tiles. For ages it was thought that was all their was. They kinds of tiles make for memory saving algorithms in video display.
  • LeonLeon Posts: 7,620
    edited 2010-09-30 06:59
    I don't think a mathematical concept can be patented, it isn't really an invention. That will get us into deep philosophical waters, however. Patterns can be copyright, of course.
  • ericballericball Posts: 774
    edited 2010-10-01 10:34
    Kite and Dart is only one set of basic shapes which can be used for Penrose Tiling.

    The challenge with any tiling is more the coordinate system. Square grids are easy - X & Y (and Z for 3D) locate each square and can also be used to determine the distance between squares. The coordinates for hexagonal tiling can also use X & Y - column & diagonal row. But Penrose tiling is, by definition, non-periodic. Therefore each tile can only be described in relation to the neighboring tiles - not as an absolute location.
  • LoopyBytelooseLoopyByteloose Posts: 12,537
    edited 2010-10-02 09:27
    I guess it all comes down to periodic cannot be easily combined with non-periodic tilings.

    The nature of graphic video display is row and column, which in the Propeller is often organized in a modular periodic tile. That saves a lot of ram and games exploit this economy.

    But the Penrose tile is definitely non-periodic and I just don't see how to use anything other than brute force of pixel-by-pixel rendering in order to get them to display (okay, maybe vector graphics will work - but I have to really thing about it).

    Still, I find the whole topics of Penrose tiles and Quasi-crystals quite exciting. Before 1997, chemistry generally ignored that there were other organizations of crystalline structure than the periodic. It opens a whole new frontier of material science and challenges basic math with all the X, Y, and Z co-ordinates.

    It is nice to see that there are new things to explore. I just wish I could figure out how to get it into a game.
  • UnknownUnknown
    edited 2010-10-11 07:31
    How about Riemannian Geometry?

    http://en.wikipedia.org/wiki/Riemannian_geometry
  • LoopyBytelooseLoopyByteloose Posts: 12,537
    edited 2010-10-13 00:11
    Hopefully you are kidding. I wanted to tile a background, not become the next Alfred Einstein.

    I believe that Penrose tiles do exist in Euclidean geometry, even if they don't conform to X-Y-Z axises and are not periodic.
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