I thought this was rather interesting

Islamic architects and mathematicians were creating quasi-crystalline patterns some 500 years before similar patterns were described in the West, claim two physicists in the US. Peter J Lu of Harvard University and Paul Steinhardt of Princeton University say that sets of special tiles developed around the 13th century allowed artisans to use complex mathematics to create the fantastic geometric patterns that adorn mosques, palaces and other buildings in the Muslim world. These patterns include "nearly perfect" Penrose patterns, which the researchers claim are similar to the first quasicrystals described in 1974 by the British mathematical physicist Roger Penrose (Science 315 1106).


http://www.sciencenews.org/articles/20070224/mathtrek.asp

thank you Iran

http://www.sciencenews.org/articles/20070224/f8196_3634.gif

I wonder at the artisans who had to join these tiles to create that perfect symmetry.

Could such dedication be found today?

http://www.sciencenews.org/articles/20070224/f8196_1743.jpg

http://www.sciencenews.org/articles/20070303/f8196_6249.jpg

sowhatifit'sdark: Thanks :)

My math is far too weak to fully appreciate the patterns except aesthetically, which I do, but I found the properties of these tiles very interesting.

Remarkable Properties of Penrose Tilings
The most remarkable property of Penrose Tilings is that every finite portion of any tiling is contained infinitely often in every other tiling. This, of course, is true of all periodic tilings, but it's not at all obvious that it should be true of a non-periodic tiling. This property has several consequences:

No finite patch of tiles can force a tiling (determine the rest of the tiling).
It is impossible to tell from any patch of tile which tiling it is on.
Only at their infinite limits are the different patterns distinguishable. A finite patch of an Infinite Star pattern might only be a local piece of some other pattern, but there is also an Infinite Star pattern that has five-fold symmetry to infinity. Only if you know the characteristics of the pattern to infinity can you tell.
This property is both less and more remarkable than it seems. For example, consider the numbers pi (3.14159265358979326433....), e (2.718281828459045235360287...) and the square root of 2 (1.4142135623730950488...). All of them contain the number sequences ..23.. and ..35.. . In fact, it is widely believed (though not formally proven) that any finite sequence of digits will be contained infinitely often in all three numbers, and no finite sequence of digits will enable you to tell which number you are looking at (except, of course, for the integer and decimal point).

What's remarkable about the Penrose Tilings is how dense the patterns are. The sequence ...89793... occurs infinitely often in pi, e, and the square root of 2, but only every 100,000 digits on the average, and the actual spacing could be vastly greater. In fact, there is no known upper limit. If a patch of tiles in a Penrose tiling has a diameter d, there will be an identical patch within a distance of at most 2d and most likely within d. (See how close to the center of the cartwheel above you can find another Batman.)

http://www.uwgb.edu/DutchS/symmetry/penrose.htm

aww.. thats nice!! Could we have a pig in the middle of it!!

Could such dedication be found today?


Sadly, I don't think so.

Sadly, I don't think so.

the intrique designs within engineering control systems are way more dedicated to the goal than these tile designs.

http://www.icmm.csic.es/spmage07/img/fourth_prize_w.jpg

Control System engineering with aerospace sector are way more complicated and devoted to one systematic goal...by ways of matrix linear algebra calculations embedded within control system designs.

http://www.seaeye.com/images/falcon_block.gif

Art within math within systems applicable to real world

the intrique designs within engineering control systems are way more dedicated to the goal than these tile designs.

Yeah, but these are etched and not hand-laid.

Yeah, but these are etched and not hand-laid.

and difference is? :bugeye:

and difference is? :bugeye:

Ever done any tiling?:)

At first glance, they appear to be the work of masterful artists. However, they were only forced to make patterns like these because depiction of the human form is forbidden. Every time I see these examples, I can only think of how it must be, as an artist, to live in such a totalitarian state.

At first glance, they appear to be the work of masterful artists. However, they were only forced to make patterns like these because depiction of the human form is forbidden. Every time I see these examples, I can only think of how it must be, as an artist, to live in such a totalitarian state.

In some articles on these tiles I see 500 years ago, in others 800. Shall we peek over in Europe and see what some of the totalitarian states were doing there and how free artists were?

Or what started, about 500 years ago, in the New World?

Come on. Oddly enough, there are people today in the west who make abstract patterns in tile.

In a general way I do agree with you. I dislike how nearly all cultures have limited humans and artists unnecessarily, but in this context yours seemed an odd contribution.

Shall we peek over in Europe and see what some of the totalitarian states were doing there and how free artists were?

http://www.nga.gov/image/a00004/a0000498.jpg

Quentin Massys
Ill-Matched Lovers, c. 1520/1525

At first glance, they appear to be the work of masterful artists. However, they were only forced to make patterns like these because depiction of the human form is forbidden. Every time I see these examples, I can only think of how it must be, as an artist, to live in such a totalitarian state.

So there is example of Penrose tiling from the Western artists of the time?

Mughal miniature art :

http://www.exoticindiaart.com/artimages/mf43.jpg http://www.exoticindiaart.com/artimages/royalatelier_sm.jpg

Islamic metalwork

http://www.islamicarchitecture.org/art/images/metalwork/islamic.metal.work.tbar.101.gif

More Penrose tiling in arabesque:

http://www.victorynewsmagazine.com/images/ArabesqueArch5.gif

More Islamic metalwork:

The axe has calligraphy on it spelling out the name Ali (علي) forwards and backwards

http://www.islamicarchitecture.org/art/images/metalwork/islamic.metal.ali-sword.gif

OK, perhaps I was exaggerating about the lack of human forms, but they certainly seem rather 2-dimensional.

OK, perhaps I was exaggerating about the lack of human forms, but they certainly seem rather 2-dimensional.

I think they were into more detail work than brushstrokes. Much of Islamic art is highly characterised by attention to minutae.

Like calligraphy:

http://lh5.google.com/RHavarti/R0yB8-g9IAI/AAAAAAAAAhY/O1XrA8islOY/IMG_0730.jpg?imgmax=512

http://www.victorynewsmagazine.com/images/Ivory_Sculpture.jpg


Or marble latticework:

http://members.pcug.org.au/~alanlevy/Thumbnails/Images/Delhi-Agra-Jaipur/Marble.JPG
And the tiling.

More calligraphy in Penrose in a niche (mihrab (http://en.wikipedia.org/wiki/Mihrab)) of a mosque:

http://faculty.evansville.edu/rl29/art105/img/islamic_mihrab.jpg

Its interesting how they must have worked out the precision to the nth decimal.

http://www.nga.gov/image/a00004/a0000498.jpg

Quentin Massys
Ill-Matched Lovers, c. 1520/1525

My point was not that European painters could not paint the human form. For the vast majority of them their subject matter was limited and so was the style. And the society around them was 1) totalitarian 2) highly restrictive 3) very dangerous to be 'different' in. The inquisition, persecution of jews, and the systematic abuse of everyone by the noble classes and the church. These patterns only became vastly more clear when medieval european cultures colonized or otherwise came in contact with non-europeans, slavery included in the mix.

Your first post seems as silly to me to feel sad for the architects of some of beautiful churches of Europe, or Medieval painters who focused on Jesus or other Christian themes. Or to assume that these tile and mosaicmakers REALLY wanted to be doing something else. I would guess some did. But it just seemed like you needed the tiles to be part of bad Islam.

I like the painting though.

It's true that European society was very strictly religious, but at least they could paint people.

Here is a study on the math of Islamic architecture:

"Straightedges and compasses work fine for the recurring symmetries of the simplest patterns we see," Lu says, "but it probably required far more powerful tools to fully explain the elaborate tilings with decagonal symmetry."

While it's possible to create these patterns individually with basic tools, they are incredibly difficult to replicate on a larger scale without generating extensive geometric distortions. The most complex medieval Islamic tilings have little such distortion, leading Lu to believe more is at play.

"Individually placing and drafting hundreds of decagons with a straightedge would have been exceedingly cumbersome," Lu says. "It's much more likely these artisans used particular tiles that we've found by decomposing the artwork."

These tiles, dubbed "girih tiles" by Lu and Steinhardt, consist of sets of five contiguous polygons (a decagon, pentagon, diamond, bowtie, and hexagon), each with a unique decorative line pattern. For medieval Islamic artisans, they may have represented a toolkit for generating huge numbers of distinctive tile patterns without the lengthy, painstaking, and often flawed process of creating each line segment individually.

These girih tiles may have been used to generate a wide range of complex tiling patterns on major buildings from medieval Islam, including mosques in Isfahan, Iran, and Bursa, Turkey; madrasas in Baghdad; and shrines in Herat, Afghanistan, and Agra, India.

In some cases, Lu found girih tiles used to create patterns of two distinct scales on medieval Islamic buildings. This approach generates infinite patterns with decagonal symmetry that never repeats -- also known as a quasicrystalline tiling, a phenomenon first described in the West in the 1970s by famed British mathematician Roger Penrose and more fully explained by Steinhardt and Dov Levine over the past 30 years.

In addition to examples on medieval structures that are still standing, Lu has been able to match his girih tiles with drawings in 15th-century Persian scrolls drafted by master architects to document their techniques.

"We're finding widespread evidence for the same approach being used for 500 years across the Islamic world," Lu says. "Again and again, girih tiles provide logical explanations for complicated designs."

http://www.sciencedaily.com/releases/2007/02/070222155706.htm


http://www.sciencemag.org/content/vol315/issue5815/images/small/315_1106_F1.gif

Direct strapwork and girih-tile construction of 10/3 decagonal patterns.


http://www.sciencemag.org/content/vol315/issue5815/images/small/315_1106_F2.gif

Periodic girih pattern from the Seljuk Mama Hatun Mausoleum in Tercan, Turkey (~1200 C.E.), where all lines are parallel to the sides of a regular pentagon, even though no decagon star is present

http://www.sciencemag.org/content/vol315/issue5815/images/small/315_1106_F3.gif

Girih-tile subdivision found in the decagonal girih pattern on a spandrel from the Darb-i Imam shrine, Isfahan, Iran (1453 C.E.).

Ref: Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture (http://www.sciencemag.org/cgi/content/figsonly/315/5815/1106)
Science 23 February 2007

Its fascinating stuff,
i remember watching a neat open university program about the Alhambra and the mathematics and patterns of the tiles.
http://findarticles.com/p/articles/mi_qn4158/is_19990401/ai_n14223717/pg_2

http://www.catnaps.org/islamic/geometry.html

Quoting from my text book, Symmetry, Shape, and Space by Kinsey and Moore:

"The most well-known set of aperiodic tiles are the Penrose tiles. The set was introduced by Roger Penrose in 1974 and contains only two tiles along with a set of rules for how these tiles must be put together. Penrose began his search for an aperiodic set of tiles by looking at pentagons. While it is true that the plane cannot be tiled by regular pentagons, Penrose studied the gaps left when one tried. He then took smaller pentagons and tried to fill in the holes. After several subdivisions, he found that the holes could have only a few shapes. These he called diamonds, paper boats, and stars. Significant insight and refinement led to Penrose's first aperiodic set of six tiles and finally to the set of two tiles most commonly known as the kite and dart, names suggested by John Conway, another mathematician who has contributed a great deal to what is known about Penrose tilings.

"Anyone who has studied stars and pentagons should expect to find the golden ratio somewhere in a situation involving these two shapes, and indeed it makes several appearances here. In any infinite tiling of the plane by kites and darts, the number of kites used is the golden ratio times the number of darts. In any tiling of a finite section of the plane, the ratio of kites to darts will approximate the golden ratio. The approximation improves as the area tiled increases. The area of a kite is the golden ratio times the area of a dart. With these facts in mind, it is probably not surprising that the golden ratio is part of the actual measurements of the tiles."

I think this defines the measurements of the angles in the tiles.

"The kite and dart are cut from a rhombus with side lengths equal to the golden ratio and main diagonal length equal to the golden ratio + 1. Connect the vertices at the obtuse angles to the main diagonal at a distance of the golden ratio from an acute angle (and hence a distance of one from the other acute angle). The large piece is the kite; the small piece is the dart. Notice that the kite is made of two isosceles triangles with the golden ratio for the length of the equal sides. The length of the third side of these triangles is equal to one. These triangles are sometimes called golden triangles. Consequently, the kite is also made of two isosceles triangles with equal sides of length one and a third side of length equal to the golden ratio.

"The rules for constructing a nonperiodic tiling with the Penrose tiles are simple. However, following them consistently is not necessarily easy. First, as with many tilings, only sides of the same length can be put together. This rule ensures that no vertex of one tile can occur in the middle of the side of another tile. The second rule requires a certain direction along the sides of the tiles, To enforce the direction rule, some people have put notches and bumps on the tiles, some use dots or holes, and some reshape the tiles to fit only in the correct way. Inspired by some of Escher's prints, Penrose reshaped the tiles as chickens. John Conway puts arcs on the tiles and requires that the arcs of the same color must meet to form continuous curves. Thus, in constructing a nonperiodic tiling, dark arcs must join dark arcs, and light arcs must join to light. Conway went on to prove a number of results involving the way the arcs connect."

There are 7 ways to put these tiles around a vertex. The star and the sun have fivefold rotational symmetry, and all of the groupings have at least one line of reflectional symmetry.

"As you work out from a vertex, sometimes there is only one way to place a tile at a certain point. Sometimes there are several choices. Sometimes it seems as if you have a choice, but farther out in your tiling you will find a place where no tile will fit and you will have to go back and change your original choice. If there is only one choice for fitting a tile around a vertex grouping, we say that tile is in the empire of the vertex."

The book goes on. To make things more simple for you, there is a Penrose tile applet: http://www.geocities.com/SiliconValley/Pines/1684/Penrose.html