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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."
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.
The patterns shown in the article are not true penrose patterns, it exhibits two lines of reflection, horizontal and vertical and the pattern does not repeat indefinitely.
There are four sorts of people in the world: fools, lunatics, idiots and morons. - Umberto Eco, Foucaut's pendulum.
It is no coincidence that Algebra comes from the "Al-jabr" the Arabic word for reunion, and Algorithm comes from al-Khwarizmi a Persian mathematician living in Baghdad(!). Kind of makes TFA seem a bit patronising.
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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?
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.
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?
I think he was referring to Islam in Spain as the civilisation that got destroyed, because Spanish Muslims had diverged significantly from those in Africa and the Middle East (or more correctly, African and Middle Eastern Muslims had diverged from them), so they can justifiably be regarded as a distinct civilisation with a unique culture. Unfortunately for them this meant that they were basically caught between a rock and a hard place, with Catholic Europe on one side who regarded them as enemies, and a stricter, more fundamentalist Islamic culture in Africa (i.e. the other side of Spain) that had also regarded them as enemies since at least the 11th century. What's surprising therefore is not that they ended up getting destroyed, as that was obviously inevitable, but rather that they lasted as long as they did when surrounded by such powerful and fanatical opponents.
NB: although it ended up being Christian rulers who destroyed the Spanish Muslim civilisation, the original poster's claim that this was done "under the Aegis of the Catholic Church" is unjust. As has often been the case, the Catholic Kings used religion as a political and propaganda tool very effectively, but the conquest of the Muslim kingdoms in Spain was really about territory, and their subsequent persecution of Jews and Muslims had a lot more to do with eliminating possible sources of dissent together with jealousy (jews in particular occupied important administrative positions that Spanish nobles wanted for themselves) than real religious differences. There's no better evidence of this than the fact that many Spanish Jews fled to Catholic Italy, home of The Vatican, where they not only managed to live without many problems (i.e. some people had personal prejudices against them, but there was ittle if any persecution by either the Italian political authorities or the Church), but were also able to obtain important administrative posts and teach in universities, where their translation of ancient Greek works that had been preserved by Spanish Muslims into Latin became a key factor in the subsequent European Renaissance.
I'm not going to change your sheets again, Mr. Hastings.
The interesting thing about quasicrystal (and 2d mapping thereof) tilings is that they are emphatically *not* regular -- so at any given vertex you can put in any number of available tile shapes, and all might appear to work, until you get 100 tiles further along and find you've backed yourself into a corner where nothing fits.
there are rules (now) that Steinhardt and his colleagues (including my wife...which is why I know something here...heh) have developed which can tell you what shape should come next to prevent backing yourself into the corner, but they have taken years to develop -- not because they're mathematically complex, but just because it takes a looooong time to try all the possible combinations, and then recognize what happened at each vertex.
my assumption is the same as the parent's -- that the Muslim artists simply "brute forced" these -- that is to say, put down random tiles, took them back up when they created bad spots, and patted themselves on the back when it all worked and looked pretty -- and then jotted down what the pattern looked like. having helped my wife do the same thing early in her thesis work -- let me tell you, it's a pain in the arse with these shapes -- but by no means impossible, and the results are always impressive.
You read the wrong FA. Typical slashdot, using Reuters or the Wall Street Journal or Fox or some such nonsense for a science or mathmatecal FA. Here's a FA that actually answers your question and explains why they had to have had advanced math to construct these things.
The dome shape was explained in (of all places) an undergraduate art history class I took thirty years ago. Those domes are imposible to construct without advanced geometry (and other advanced disciplines as well).
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