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...
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).
Penrose 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).
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.
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.
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.
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):
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).
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.
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.
Are these Penrose Tiles ?
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.
For 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)>
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.
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).
In July 2002, Dutchman Frans
C Mijlhoff advised me of his freeware QCTilingsC Windows tiling
program which is like QuasiG
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 ! (links to download now broken)
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.
.
Martin Gardner has written a more formal discussion of penrose
tiling... Penrose Tiles to Trapdoor Ciphers, and the return of Dr Matrix Gardner aka Dr Matrix used a
kites and darts terminology to describe the tiling - kites and darts are assemblies
of rhombs.
Dr Matrix also highlighted the ways in which the golden-ratio of
(1+SQRT(5))/2 = 1.61803398 . . . could be found in the Penrose
patterns.
It is well worth a visit for the mathematically inclined. Here
is a quote from Dr Matrix's now defunct Dr Matrix's Programming Challenge website (
https://www.scientium.com/drmatrix/puzzles/progchal.htm and http://www2.spsu.edu/math/tiling/29.html):
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. Chestertononce 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 was Andrew D Lewis's
"Some Planar Tilings" website (http://penelope.mast.queensu.ca/~andrew/qc/
now defunct, internet archive has a part of it captured in this pdf. 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.
Up til Dec 2000, there was a very good introduction
to quasi crystal's and penrose tiling at Chris Hillman'sTiling Dynamical Systems web site at the University of Washington.
(link is pdf file captured from internet archive wayback machine, Jan 2017).
Prof E.Arthur Robinson from the George Washington University had a rich
vein of
web-resources and bibliography for Visual Mathematics (site visited
8th May 2004, link is to pdf capture from internet archive wayback machine Jan 2017). His site
included links to numerous freeware programs demonstrating tiling and other
visual mathematics applications.
Page Created
19th June
2000
Last Revision: Relocated to new web host, 24th Jan 2017
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
Penrose 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 )
.
Martin Gardner has written a more formal discussion of penrose
tiling... 