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Old Islamic Tile Patterns Show Modern Math Insight
arbitraryaardvark writes "Reuters reports that medieval Muslims made a mega math marvel. Tile patterns on middle eastern mosques display a kind of quasicrystalline effect that was unknown in the west until rediscovered by Penrose in the 1970s. 'Quasicrystalline patterns comprise a set of interlocking units whose pattern never repeats, even when extended infinitely in all directions, and possess a special form of symmetry.' It isn't known if the mosque designers understood the math behind the patterns or not."
It seems fairly self important to assume that they didn't understand the math behind the tiles. They generated them, didn't they? Islamic culture was well considered to be centuries ahead of Europe during that time period. They had access to some of the ancient Greek writings that Europe only rediscovered years later. My question is, and I don't mean to troll, what happened? From my perspective, it seems that many people almost disdain the idea of progress in culture and arts now.
Rhymes that keep their secrets will unfold behind the clouds.There upon the rainbow is the answer to a neverending story
Since it isn't known (as TFA points out) if they fully understood the mathematics behind the designs, we could have a bit of fun speculating, yes?
I am no expert on Islam but I really like to read and study up on various forms of encryption. I'm not a crypto genius by any means, I don't endeavour to break codes, I just like to be able to recognize them.
If I am not mistaken (flog me if I am), the mural depicted could in effect be a key to a cipher, and one's starting point applying that mural as a key would be very important. In fact, perhaps a key with infinite grooves and landings that fits a lock with only a few tumblers.
Now, if that structure was destoryed during war (many were), and that key easily re-created from mathematical notes, that would be something. The notes themselves would be useless to pretty much anyone else at the time.
I don't think they understood the math behind it was we do (or better wording would be the significance of the math beyond their application of it) but I do think they understood quite a bit more about cryptography than we previously thought.
Of course, it could just be that the design held some spiritual significance. A lot of trouble to go through, however.
Well it's pretty, I'll give it that. TFA's a bit light on details though, and "tantalizingly close to having the structure that Penrose discovered in the mid-70s" isn't exactly awe-inspiring; maybe a few more examples would have been in order before they published?
See Penrose tiling on Wikipedia.
They really have cool properties - you can tile an infinite plane with just two different tiles, in such a way that the pattern never repeats; the ratio of the frequencies of both types is exactly the golden ratio. There's a lot more, see the article.
Apparently they found actual Penrose tiles, hundreds of years old.
I believe posters are recognized by their sig. So I made one.
The Arabs got zero from the Indians through their trading contacts actually. See the Wikipedia entry: History of Zero.
The Rise and Fall of Online Community
The recent documentary by Oxford historian Brittany Hughes, When the Moors Ruled in Europe , revealed (among many other very surprising findings) that the strikingly gorgeous Alhambra Palace also contains a very interesting mathematical curiousity within the design of all of its walls and floor patterns. (I won't spoil it for people who want to watch the documentary, which is available in its entirety on Google Video.) Also, many more Islamic patterns from throughout the Muslim world are available on flickr's Muslim Cultures group for those intrigued by the sort of artwork mentioned in the article.
Unfortunately their civilisation was destroyed by a European power under the aegis of the Catholic Church. For much of recent history, Christian societies have attempted to control and dominate Islamic societies. Since the socially mobile tend to follow the ways of the dominant power, Islam has become increasingly a religion of the poor and ill educated. (I know this is a simplification, but it is a useful simplification.) We are now seeing the effects of creating a society of poor and ill-educated people with ready access to cheap weapons.
On the broader point, I tend to disagree. It is easy to blame television, the movies and the music industry for the destruction of "high" culture, but of course we don't know what "low" culture was like in largely preliterate societies. I suspect the reality is that high culture is more disseminated and understood than ever before, but whereas in the Middle Ages it might have been available to 0.1% of the population, now it is available to, say, 2%. Because mass culture now has access to the media, this fact is concealed in the sheer noise of low culture.
A genuine example, from the 1500s. A footnote to an edition of Rabelais reveals that at one public fair in France, the prostitutes wanting to operate their trade had to take part at the start of the fair in a naked public footrace. This operates on a number of levels. It would tend to discourage unhealthy or diseased prostitutes. It constituted a form of advertising. And it provided entertainment. But it also shows that, no matter what you think of current entertainment standards, they were just as bad in the 1500s.
Pining for the fjords
The Arabs got zero from the Indians through their trading contacts actually. See the Wikipedia entry: History of Zero.
Nah, that was just one of the first examples of outsourcing.
A more interesting link.
It would be nice if the article actually identified why these patterns have to be based on a complicated mathematical principle, and if they're not - how they could have been made and still represent that mathematical principle. According to the article, the patterns aren't even exact but quasi-crystalline-structures.
I can do a quasi-fractal-pattern by accident if I have enough time to create random patterns, like say an entire country's worth of structures covered in patterns.
Can some statistics-guru figure out the odds of this being a random accident, considering how few examples they have, and how the examples aren't even exact representations of the mysterious mathematical formula(s) they mention? I really don't get why this is believable based on the article.
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Pre-Roman Crystalline Structure Dance
Ace
I suppose it's not really surprising that Muslim architecture is going to uncover these sorts of complex patterns. As I recall, the Quran prohibits art depicting humans (or possibly anything created by Allah, I can't recall exactly), and as a result, Islamic art tends to the more abstract. Without the devotion to realism that characterised Western art through much of history, it makes sense that they'd develop the more abstract art to a greater complexity.
Just because you're paranoid doesn't mean there isn't an invisible demon about to eat your face
Presumably in this case it actually means resdiscovered-by-western-academics since presumably these patterns have been looked at by thousands of people everyday for hundreds of years as they went to pray. I can't think of any other reason why despite millions of arabs looking at these patterns over the years they were considered "lost" to mankind until "rediscovered" by an english professor.
Recently I visited the Escher museum (http://www.escherinhetpaleis.nl/) in The Hague. They have a statement of Escher on the wall in which he expresses his he expresses his sadness that Islam didnt allow depiction of anything else other than abstract patterns. Apparently Escher works of interlocking creatures were inspired by his visits to the mosques in Spain (?)... Guess Penrose wasn't the only one in "the west" to have discovered those mathematical qualities.
I have an ongoing debate with a friend who is both a philosopher of science and a mathematics teacher.
..
Suffice it to say that I wish he had taugh me mathematics (and algebra, geometry, calculus) rather than the teachers I had
One of the things that come up in our discussions is the idea the the Ancient Egyptians knew about PHI and PI - as can be seen from the structure of their architecture - and that the builders of Stonehenge also had working knowledge of trigonometry.
But as a mathematician - he denies that the there was any knowledge of "mathematics" because the principles were never described "mathematically" - just used in an "intuitive way".
"Without the maths", he said, "You can't argue that they understood the maths" and, he continued, "if they never expressed their finding in mathematical terms (i.e. in formulas with proofs) - then it isn't maths anyway - its just architecture"
During those 8 centuries Moors and Christian and Jew people lived together. They had their spaces, but also had interaction, trade, ... . Christian were not obligated to convert to Islam, etc. After Christian re-conquest Moors and Jew were ejected from the territory (or obligated to convert to Christianism- nevertheless I'm not sure they had the same rights than Christians after doing that)
Tell me, who did write those books?
Yes. And obviously the spanish did too.
Ah, yes. BTW, i hereby call all native americans to reclaim their country from the US-Invaders!
Mmmh, if this is true, maybe it counts as prior art in his patent dispute with the makers of Kleenex. They were using Penrose tiles because the quasi-periodic structure makes it less likely that the overlapping of the pattern will cause ridges to form. Math patents!!
Deconstruct the State
So you wouldn't mind if the indians decided to drive the white invaders out of America? After all, the whites have been living as invaders there for less than 5 centuries, much less than the muslims in medieval Spain.
And saying that the spanish wanted them out is misleading, the catholic kings and the church wanted them out, what the people wanted is anybody's guess. Spain didn't exist at that point, the christian part was divided into three parts, the kingdom of Navarra, the kingdom of Castilia and the kingdom of Aragon. And although the Kingdom of Navarra came under the control of the catholic kings (Ferdinand and Isabella) it wasn't until the 19th century it became officially a part of Spain. And when the Moors came to the Iberian peninsula, it was under the control of the Visigoths and they didn't put up much of a fight, so "invasion" is maybe stretching it a bit. Besides, it was at a time when the people of Europe were wildly "invading" each other, none of the nations we know today actually existed at that time. You're obviously prejudiced against the muslims, but the truth is that Al-Andaluz was the most civilized part of Europe at that time.
Odd. A document released recently by a Muslim group in Britain said schools shouldn't force Muslims to draw pictures of humans:
"In Islam the creation of three dimensional figurative imagery of humans is generally regarded as unacceptable because of the risk of idolatress practices and some pupils and parents may raise objections to this. The school should avoid encouraging Muslim pupils from producing three dimensional imagery of humans and focus on other forms of art, calligraphy, textile art, ceramic glass, metal/woodwork, landscape drawing, paintings, architectural representations, geometric figures, photography and mosaic art."
Muslim Council of Britain
Just because you're paranoid doesn't mean there isn't an invisible demon about to eat your face
Umm, ACTUALLY their civilisation was destroyed by the Spanish reclaiming their own country that had been conquered by the Moops.
But I suppose "tantalisingly close" isn't enough to prove prior art on Penrose's U.S. Patent 4133152.
If I recall correctly, the proof that Penrose tiling is aperiodic depends on projection of a line marked out in intervals representing an irrational number onto a line marked out in uniform intervals. According to Wikipedia (hey, this isn't an academic paper, so I can cite Wikipedia, right?) the first reference for irrational numbers was in the Indian Sulba Sutras composed between 800-500 BC, so the fundamental knowledge was available in plenty of time for these tilings. And because irrational numbers were arrived at geometrically I can imagine that the ancients could indeed have understood the math.
There's more information about the ancient tilings here, which shows that the Islamic tilings break down into five basic tiles, and that each of those five tiles can be broken down into Penrose tiles. So it looks as if they beat the first modern aperiodic tiling, Berger's initial one, which needed 20426 tiles, but didn't get as far as cutting it down to Penrose's two.
Quidnam Latine loqui modo coepi?
It is little know that Robert Amman co-discovered one of Penroses aperiodic tiles. Amman was am amatuer mathematician in the United States. See his wiki page.
http://en.wikipedia.org/wiki/Robert_Ammann
I knew Bob Amman. I shared an office with him in my first job out of college. He was doing minor programming work for a small network/modem company in the early 80s. His white board always had tiled diagrams on it. I graduated from MIT but he was probably the best example I knew of a prodigy.
The curious thing about Amman was how poorly he dealt with life. A man of his genius should not have ended up at the post office.
I never knew he was famous until years later when something must have happen to Penrose (quasicrystals?) and Amman was in the local paper. I couldn't believe the guy I worked with traveled in these circles. One of the scientists I worked with at Kodak had a book on tiles. I checked the index, Amman was all over it, using cited by other mathematicians "unpublished personal correspondence."
It makes one wonder what other geniuses are out there sorting mail.
Paul
Nobody knows, but the step after that will be Profit!
Free as in mason.
Al Ghazali was indeed a very influential philosopher who brought in umpteen damage to the scientific inquiry of the Islamic world. But the real damage was done by another person called Ahmed Sirhindi. In simple words, what he has said was that human brain is futile. Any effort to understand nature/God through reasoning and thought is a waste of time. The only way salvation could be obtained is through studying the Kuran (the unmorphed message from God) and the Hadith (stories about the life of Mohammed). Without the use of Mohammed, man is inherently powerless to understand Nature or God ! In his philosophy, the biggest evil were the Greek & Hindu philosophers. His philosophy sounded the death to the movement of Sufism (mysticism and philosophy) in Islam. At the same time, it put an end to the systematic enquiry of science. Ahmed Sirhindi became the Mujaddid (the equivalent of the pope in Islam) and he convinced the Ottoman empire to use his methods. He convinced the Mughal empire in India to use his methods. Consequently, India and Arabia were mired in dark ages ever since 1000 AD.His influence is strongly felt in the later and the final Mujaddid - Wahhab of Saudi Arabia. The major school of Islam in Pakistan and India is the Deoband school, which is drawn from the ideas of Wahhab & Sirhindi. These are the seeds of Islamic fundamentalism. It is no wonder that all glories of Islamic mathematics, medicine and astronomy were reached before 1000 AD.
The newspaper article hardly tells the story. Here is the abstract from Science
The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in medieval Islamic architecture were conceived by their designers as a network of zigzagging lines, where the lines were drafted directly with a straightedge and a compass. We show that by 1200 C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations of a special set of equilateral polygons ("girih tiles") decorated with lines. These tiles enabled the creation of increasingly complex periodic girih patterns, and by the 15th century, the tessellation approach was combined with self-similar transformations to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before their discovery in the West.
If you care to look at the article, it has some very interesting pictures and explanations in the "supplement". Peter Lu, et.al. Science 315, 1106 (2007)
"If all the American people want is security, let them live in prisons." Eisenhower
And so he did. But he did it 500 years later.
a phies/Al-Sijzi.html
Your argument was that Muslim architecture was in most part due to their policy of leaving societal elites in place, including architects, during the conquest. You go on to support this argument by pointing to the careers of two Ottoman architects Sinan and Mehmet Aga, and claiming that most of the mosques in Istanbul were built therefore by Christians. From this you draw the conclusion that muslims in the 11th century did not have the mathematical skills required to build large domes in the 11th century.
Your argument fails in a number of respects:
* The architects you reference flourished some 700 years after the first Muslim empire
* They were both active in a country that was not part of the Muslim Empire
* The Ottoman empire had been established for 200 years by the time of Sinan's birth, so he was not conquered
* Both architects were trained by the Ottoman army, so their skills were not acquired before a Muslim conquest
* Both architects were converts to Islam, not Christian as you state.
In addition, Islamic mathematicians were intrigued by the properties of Spheres, see, for example the work of Al Sijzi, who was active in exactly the time frame you claim Muslims were mathematically ignorant: http://www-groups.dcs.st-and.ac.uk/~history/Biogr
Their life is that way because of their own choices. Sure you can go back a hundred years or so and say this or that happened. But today, they have the same opertunities as anyone else. Even moreso then most, especialy the iner-city folks we refere to as the welfare clans. (notice, I'm not refering to blacks, hispanics or anyone because of race. You will always have poor people living next to poor people in some of the worst neighborhoods availible).
While there has been challenges in the past, the bigest limiting factor is a culture that refuses to be part of the success around it. And this goes to the inner-city clans too. Far too many people fail to shine because it just isn't hip! Those who do succed get ignored while everyone concentrates on those losing at life (and no i didn't call anyone a "loser" I said they weren't succeeding). But thats what happens when the government "keeps" people dependent on them, too few look at taking care of themselves.
I haven't visited the Arabic world but my encounters with Moorish craftsmanship in Spain have been awe-inspiring.
Don't miss Granada's Alhambra, a breath-taking treasure and not just for the intricate artwork.
There are a number of books, I'll let you browse Amazon at your leisure, on the beauty of Islamic art. One I purchased explores the mathematics behind the designs, Keith Critchlow's "Islamic Patterns: An Analytical and Cosmological Approach". It explains how patterns emerge from arcs and intersections of polygons. Further, Critchlow argues that for the Muslim these patterns displayed a spiritual aspect, that the wonder experienced at looking at these patterns pales in comparison to the complex thoughts behind their creation.
Alas, if only there were more hours in the day I'd try reproducing them via Java2D or OpenGL. Fascinating stuff.
Except for one fact: Europe LOST the crusades. Yes, they held an area of land approximately equivalent to modern day Israel for a short period of time, but most of Arabia was still dominated by Islam. Yes, "The Caliphate" as Mohammad's original empire was known was gone, but it had been in serious decline for some time due to internal strife, the slow march towards religious extremism and traditional tribalism for years by that point.
The only real "advanced" Islam was the one destroyed years earlier in the Grenada area. The only reason they were advanced was their rejection of Fundamentalist Islam, and the creation of a more modern more egalitarian society that viewed Christians and Jews as, if not equals, valuable citizens. Most of the advances IN that society were brought to it by the Jews and Christians living within it. Not the Muslims themselves. Of course, all that was gone by the time of the crusades due to the destruction of that society by greedy Kings using Christianity as an excuse to take land.
The point is, Islam as we know it today has brought nothing to the table to advance society. While I am all for giving people their due, Modern Islam is owed no credit for any discoveries (unless you consider suicide bombers a discovery), and trying to credit them for this smacks of Political Correctness gone awry.
Official Heretic from the "Church of Global Warming". Proven right thanks to whistle blowers. AGW = Flat Earth Theory
We all know it's pure chance they stumbled into these mathematical patterns. What can Arabs possibly know about algebra and numbers in general? Oh, wait...
The first usage of zero as an actual numerical quantity occurred in the Mayan vigesimal numbering system around 36 BCE. (see the Wikipedia article on Mayan numerals ) Some people postulate, although, that this was derived from the older Olmec numbering system, which could have gone as far back as 1200 BCE. If this is true, then it means that the Olmecs discovered zero 1828 years before Brahmagupta re-discovered and formalized it in 628 CE in India. jdb2
What drove you to that conclusion?
Here's a translation of the verse in Question (I also read it in the original Arabic): "It is not fitting for a Believer, man or woman, when a matter has been decided by Allah and His Messenger to have any option about their decision: if any one disobeys Allah and His Messenger, he is indeed on a clearly wrong Path"
It roughly means "believers are not to disobey Allah or his prophet". Why does that makes you think it prohibits independent thinking?
There are may verses of the Quran and many quotes of the prophet that encourage thinking and reasoning (for example Quran verses 4:82, 47:24, 16:11 to 16:13).
In fact, a complete branch of Islamic studies is called Ijtihad, which is all about independent thought.
to quote an online Islamic site: "A scientific approach has been encouraged in the Qur'an with the objective of ascertaining its truthfulness. It provides man with a chance to verify its authenticity." so in Islam, independent thinking is in fact an essential part of the religion.
People say that it's a coincidence. What Lu and Steinhardt are demonstrating is deep knowledge of advanced math. The best example of quasiperiodic tiling (as they are called), is the Fibonacci sequence. To build it you need to differnet tiles (say a long segment "L" and a short segment "S") and two combining rules:
1. at every S you change it with a L
2. at every L you change it with LS
so you build the different generations of the sequence as follow:
S
LS
LSL
LSLLS
LSLLSLSL
LSLLSLSLLSLLS
LSLLSLSLLSLLSLSLLSLSL
etc...
You can go at infinity with this. You won't find periodicity or a pattern that repeat itself. Now to the point: does this means that you take the two segments and you put them together randomly you get the F. sequence? No, by any chance. The rules are simple (and the Fibonacci sequence is old (~1200), so I would not be surprised if the Islamic mathematicians were aware of it, so they "ported" it in 2D (the Penrose tiling is the 2D version of the F. sequence).
By the way the story goes even back in time further: the ratio between the number of L and S for a significantly large sequence, is tau, the golden mean (again the same is true for the Penrose tiling). The golden mean was a key number (sqrt5+1)/2~1.6... in the greek world, where it was used as a proportion standard to build building and temples. It's also a key element in fractal growth, in key dimensions of our body, etc.
So the Islamic artists (scientists?) of the time were a bit like today's scientists. they gathered previous studies and assembled together using some new insights.
Why don't give them credits for it, instead of stupidly saying: "well they just got lucky?".
You can use brute force to form pseudo quasiperiodic tilings. However they are not really quasiperiodic, they are "approximants". So if are patient enough you can get to a point where you have a large pattern, but not necessarily being quasiperiodic. So to a degree of trial and error the Islamic artists must have developed a degree of knowledge which you seem to underestimate. point in case: the Fibonacci sequence and the golden mean. Simple mathematical rules at the base, not impossible to grasp with the arabic knowledge of math of the time. The 2D mapping may just have been their "next step". There have been long speculation of the fact that Islamic art was for long considered just a coincidence. The article here present the prove that this may indeed be not true. Disclaimer: I (not my wife) did my PhD in quasicrystal tiling.