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QuasiG V1.4 is a freeware Penrose tiling program that will show and print full-colour Penrose tiling patterns, and more general quasi-crystal patterns, on any Windows 95/98 or NT/2000/XP PC. At first sight, these tilings may seem esoteric, but they have found practical application in coating non-stick cookware, and making more attractive toilet paper rolls. A subset of quasicrystals (Penrose tiles) have even funded the retirement dreams of the legions of lawyers that prosecuted their use on the toilet rolls...

Background

Quasi-crystal tilings are assembled from two rhomb shaped tiles (squashed squares with equal length sides). The smaller angle in one rhomb is half that of the smaller angle in the other rhomb. The angle divides into p (PI) an odd number of times (n). p (PI) is 180 degrees, or half a circle. The number (n) gives the degrees of symmetry that can be observed in the pattern (you can find parts of the pattern which can be rotated n times through the smaller angle and still look the same).

tile marksPenrose tilings are a subset of these in which there are 5 degrees of symmetry (n = 5 ), and in which tile edges are matched to satisfy the patterns in Figure 1 at left (see examples in Penrose Marking section below )

Tilings are constructed by finding ways of combining the 2 angles possible with each of two tile orientations so as to add up to 2p (2 x PI) - and thus span a full circle around a vertex. For example, with n = 5, there are 7 different ways to arrange the tiles at a vertex (or 8 ways if you count two star patterns that look the same except for the markings). vertex stars

Eric Weeks's site provides an explanation for the methods used in an dual grid algorithm based on N.G.deBruijn's dual grid. It can generate penrose tilings ( see quasi.c http://www.physics.emory.edu/~weeks/software/quasic.html. http://www.physics.emory.edu/~weeks/software/quasic.html ) ..

For other sites explaining more about non-periodic tilings, see the links section below.

Eric's quasi.c, on which this is based, does not enforce the strict tile-matching rules that the classic Penrose tilings have. Nor could it draw them. But it's source code can be adapted to produce them. And, the non-penrose patterns can be just as interesting.

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Using QuasiG

Using QuasiG is meant to be simple. It has default input values that will produce a simple image of penrose tiling. Details QuasiG features and its options are described in here.

Example Screenshots images below demonstrate some of the patterns possible with QuasiG.

Click here to download QuasiG.

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QuasiG Examples

The image below is an example of QuasiG's screen output for 5 degrees of symmetry, auto-scaling on, central pattern only, fill on, black edge on, color-doubling on, color gradient off, all 4 quadrants displayed, and 30 generating lines (I used PaintShop Pro's screen capture to get this). The title bar summarises the options selected - in this version Offset Multiplier was 1.

Offset Multiplier Image

And here's an example of the full pattern display (or at least the viewport that fits on one screen, on the real thing you can scroll to view the rest) (also Offset Multiplier 1):

penrose tiles q530all.gif

And here's an extract of a pattern not available in Eric's version (offset multiplier set to 10, black edges off):

ten thin pattern

The Border image

The border image for this page was produced by using QuasiG in non-colour mode, and then capturing the image with Paint Shop Pro. I then did lots of fiddling, starting with embossing the image (hoping to make a watermark), and adjusting the colors, until this image emerged (i.e. I can't remember exactly what I did, but eventually I liked it enough and saved it as a JPG).

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Animation Examplenew1 gif

The image below is an animated GIF comprised of 3 frames showing several subsets of the 5-symmetry patterns. The first frame starts 20% into the generating sequence (Start Tile % as 20%), and ends at 30%; the next two frames extend this by 10% each frame.

Selection of tile subsets such as these was a new feature in QuasiG V1.3.

animated penrose construction

Click here to download a longer animation sequence (184K bytes) sym5_30_100_ani.gif. This longer sequences finishes with a frame showing 100% of the tiles.

Caution: Animated gif's drawn by your browser should have 1.5 seconds between frames, but a heavily loaded PC or an otherwise slow drawer won't get the right effect here. These animated GIF's were tested on an NEC Versa LX laptop with 300MHz Pentium II processor. Using Internet Explorer 5.5 as the browser, each frame tended to get rendered in 3 steps (from top to bottom of image, not how QuasiG< works !).The longer sequence taxed it's capabilities heavily.

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Are these Penrose Tiles ?new1 gif

Many of the patterns generated by QuasiG allow clusters of thin tiles where more than 2 thin tiles share a vertex, and are all adjoining. These don't follow the Penrose tile edge matching rules. The next image almost seems to satisfy the matching rules - it's difficult to spot any "lines" running through it. But the Penrose matching rules can not be applied to it.

If you want to investigate this further, one interesting line might be to note that Durand's QuasiTiler needs irrational numbers to select tilings that will be Penrose tilings. Maybe the initial positions of Eric Weeks' generating lines needs to obey some constraint that's equivalent to Durands.

The next image used an Offset Multiplier of 0.2137.

0.2317 tiles

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Penrose Tile Marking

Now for an image that is a penrose tiling. This one has OM=0.10, and has penrose marking applied to the tiles:

penrose marking

The corresponding diagram for OM=0.2137 shows the mismatches:

not penrose tiles

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Penrose Tiling at Storey Hall

Storey Hall Interior, Click for full size image, 134 KbFor a remarkable application of Penrose Tiling, see the RMIT University's Storey Hall Auditorium in Swanson Street, Melbourne, Australia (click on image at right to enlarge).

The facade and interior of this building were decorated with patterns derived from penrose tiling It is an eye-catching statement which advertises the University's technical history. Can you imagine how much work and money must have gone into this ?

Located in the heart of the Melbourne CBD, the second largest city in Australia, RMIT is a University which morphed from the Royal Melbourne Institute of Technology. It seems fitting that a building housing an institution with a strong history and interest in technology should be morphed into a statement by clever mathematics and architectural sculpture.

The work, completed in October 1995, won numerous Architectural awards. I became aware of it only after posting this web-page, thanks to a architecture book review in a Sydney newspaper magazine. You can find out lots more about the Storey Hall at the RMIT web site (http://www.rmit.edu.au) (but at writing time, they'd messed up their picture links).

The Storey Hall patterns uses Penrose matching marks which inspired me to add them to QuasiG. In Storey Hall's patterns, both circle markers have been given the same colour and the fat tile arcs have been drawn as straight line segments. That's easier to construct (even in MS Windows, more so in the Hall) - and the difference is only aesthetic. My first attempts used straight lines, and looked pretty much like the Storey Hall images.

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Printing this Page

Printing with many browsers (including IE5.5 and earlier) often doesn't work real well because most browser print functions don't handle the horizontal scrolling view you get on screen. This page has been put together to facilitate printing the text on an A4 page - essentially you will see everything within the white area of the background.

However, many of the images extending outside this area will get clipped out. You could try changing your print setup options to select landscape printing layout (but they'll mostly be split across a page).

If you really want to print the images, make them with QuasiG and print directly from its File/Print menu item. This will print the screen image across 4 A4 size pages (A4 tiles !) - the screen plot area is 32 cm by 32 cm, and prints isometrically (1 cm of screen = 1 cm of page).

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Links

The following links all worked as of May 8th 2004, when I last ran Xenu Link Sleuth. If you find them broken, Click Here To Email QuasiG

In July 2002, Dutchman Frans C Mijlhoff advised me of his freeware QCTilingsC Windows tiling program which is like QuasiG and can be downloaded from Simtel. QCTilingsC (v 1.3) can show the generating lines as well as the pattern, and does even-symmetry pattterns. There other differences with QuasiG, so you might want to have both !

Frans' gave me a tip for a great site for links related to tiling : the Geometry Junkyard - Tiling page. The Geometry Junkyard - Penrose page has numerous links related to penrose and non-periodic tiling.

One of these is Stephen Collins free Windows Penrose program called Bob (see http://www.stephencollins.net/web/penrose/), which does some interesting things with walks around the pattern, and illustrates inflation/deflation of penrose tilings.

kite or rhombFor links to other Windows Penrose tiling discussions, see Dr Matrix's Programming Challenge. Dr Matrix's site discusses penrose tiling properties more formally. He uses a kites and darts terminology to describe the tiling - kites and darts are assemblies of rhombs.

Dr Matrix also highlights the ways in which the golden-ratio of (1+SQRT(5))/2 = 1.61803398 . . . can be found in the Penrose patterns.

It is well worth a visit for the mathematically inclined. Here is a quote from Dr Matrix:

Although it is possible to construct Penrose patterns with a high degree of symmetry (an infinity of patterns have bilateral symmetry), most patterns, like the universe, are a mystifying mixture of order and unexpected deviations from order.

As the patterns expand, they seem to be always striving to repeat themselves but never quite managing it. G. K. Chesterton once suggested that an extraterrestrial being, observing how many features of a human body are duplicated on the left and the right, would reasonably deduce that we have a heart on each side. The world, he said,

"looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait." Everywhere there is a "silent swerving from accuracy by an inch that is the uncanny element in everything . . . a sort of secret treason in the universe." The passage is a nice description of Penrose's planar worlds.

There is something even more surprising about Penrose universes. In a curious finite sense, given by the "local isomorphism theorem," all Penrose patterns are alike. Penrose was able to show that every finite region in any pattern is contained somewhere inside every other pattern. Moreover, it appears infinitely many times in every pattern.

Another source expounding on penrose tiling is Andrew Lewis's Some Planar Tilings web site. Or you could try Eugenio Durand's QuasiTiler site - a useful resource for tiling and tesselations facts.

On a lighter side, there are links between M.C.Escher's lithographs and the works of Roger Penrose and his father. Penrose actually patented his tiles, and has a company that distributes games based on them (and can well afford to advertise itself without our help). Strange isn't it: much of what Penrose or other mathemetician's discover is based on the vast body of knowledge accumulated before them and around them in the universities they work in and with. Even Roger Penrose acknowledged this:

In 1995, computer expert Roger Schlafly received a patent on two extremely large prime numbers. Among the chorus of protesters against the idea of someone claiming ownership to a number: the eminent Sir Roger Penrose.

"It's absurd," Penrose said of the Schlafly case. "Mathematics is out there for everybody."

In spite of this, in 1997 Penrose took legal action against Kleenex over use of Penrose patterns on Toilet Paper (see Toilet Paper Plagiarism), which was subsequently settled out of court. And while Penrose's discovery of penrose tilings predated it, in 1982 quasicrystals were discovered by a crystallographer taking images of Aluminium, Lead, Maganese alloys: they were something that occurred naturally, and Penrose appears to be patenting nature ! If he is not paying royalties on the food he eats, air he breaths and at the very least the language he uses , he should hang his head in shame.

More gravely, somewhere between the apes and me today, there was a mathematical ancestor of mine that invented the word rhomb, and if Roger was piqued that Kleenex stole his invention of penrose tiles, I'm doubly piqued that he stole my ancestors words when first describing his tilings in his patents.

Another good introductory site for Penrose tiling is Alison Boyle's From Quasi crystals to Kleenex. This also discusses the Kleenex case, the Escher connection, and connections to non-stick frypans.

Yet another good site, with an excellent applet, is Greg Evan's deBruijn applet. Greg also has other fascinating applets in his gallery.

 

Visual Mathematics

Prof E.Arthur Robinson from the George Washington University has a rich vein of web-resources and bibliography for Visual Mathematics (site visited 8th May 2004). His site includes links to numerous freeware programs demonstrating tiling and other visual mathematics applications.

 

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Using QuasiG | QuasiG Options | Features | Tile Marking |

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Page Created 19th June 2000, Last Revised
Last Revision: vdeck modification, Visual Mathematics link added
You are visitor since 21st Feb 2004.
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This material may be used for educational non-profit purposes with proper acknowledgement of the source. If any images created with QuasiG are posted on the net, send author the link url. All other uses, please Click Here To Email QuasiG


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