In East Anatolia

 

 

 

 

 

 

 

 

 

 

 

Rebel Land [1] is the story about the ethnic and religious conflicts in East Anatolia in the 20^th century. It is remarkable as it blends known and largely unknown historical facts with personal experiences of an investigative journalist and very emotional descriptions of a largely forgotten, harsh and dramatic region and its mainly inhospitable people(s). It is from the borders between civilization and notorious unrest, West and East, Europe and, well, something in-between. Let’s describe it as a country gradually developing into (or from) Central Asia, from where the various Turkish tribes have once colonized the Anatolian highlands. Willingly or not, Rebel Land provides excellent evidence for denying modern Turkey’s desire of eventually joining the European Union in the near future.

The author, Christopher de Bellaigue, is a sympathetic writer. He has lived in Turkey in the 1990s for five years and speaks the language fluently. At first encounter he is usually considered a Turk, as he writes not without some pride. I have become curious after having read his two books about Iran [2], a country which also fascinates me for a long time. He went farther. He has got married to an Iranian woman and even converted to the Shi’a branch of Islam. Rebel Land is written in a century-long tradition of a traveling reporter who wants to tell a true story about history.

De Bellaigue’s first and main intention to travel to Varto in the East Anatolian province of Mus was definitely to figure out the truth about the Turkish genocide of the Armenian people in 1915. It is the most disturbing part of the book and portrays well the problems of modern Turkey, which officially denies the very facts and threatens with prosecution everybody who is telling what actually had happened. It is also about Turkish ‘historians’ counterfeiting the dark chapters of Turkish history in the last century [3]. There is an unfortunate melting pot there made of Armenians, Ottoman Turks and Kurds, Sunnis and Alevis. The struggle for forced modernity in remote regions hopelessly stuck in medieval traditions, numerous military coups, etc.

De Bellaigue visited and interviewed also Turks from Eastern Anatolia now living in by and large xenophobic Germany, who seemed to have lost their real identity as Armenians, or Alevis, even Kurds. Lost identity, another sad aspect of this book. It may culminate in the epilogue, when de Bellaigue describes a visit in Armenia’s capital Yerevan and an afternoon in an unforgiving Armenian friend’s home. Armen, so his name, told him that once he met, in a tea house in Anatolia, a Kurdish man wearing a silver belt heavily embossed with detachable sections and with Armenian inscriptions of 1902. He managed to buy the belt after some bargaining. Men are not wearing this kind of belts, he said. Armenian girls are given these belts when they got married. For his friend, still full of hatred, it was clear that this belt had been stolen from an Armenian family which had been killed in the massacres. Now, ridiculously, a man was wearing it! De Bellaigue confesses:

“I think these things in a neat, well-ordered terraced house in London, where I have belts of my own – my family; the nice reassuring things that I inherited from my mother. Supposing these people, these things, were wrenched away from me by an ancestral enemy, supposing that I were robbed of everything in a matter of minutes – I suppose that I too would disregard those principles, of love and forgiveness, that were instilled in me painlessly as a child, and abandon myself to insatiable rage.”

The book is at best when its author entertains with sad and poetic stories about the people there, somewhat disclosing their soul. “Tell me about the Armenians”, de Bellaigue asks an Alevi from Varto, who narrates the following story about the pepukh, the yellow-winged cuckoo.

“There were once a sister and a brother. Their mother had died and their father had married again. The stepmother was wicked and she was cruel to the children, who were scared of her. When spring came, and the cardoon started to sprout across the meadows, the stepmother gave the children a saddlebag and told them to fill it with cardoon. When they had filled it, they set out for home, the little boy carrying the saddlebag over his back. As they approached home, the girl noticed that the saddlebag was empty and she accused her brother of eating the cardoon. ‘It’s almost dark! What will our stepmother do to us now?’ Her brother was distraught. ‘I didn’t eat the cardoon. I only took one stalk, and that was with your permission. Open up my stomach and look; you’ll find one stalk inside.’ So the girl split open her brother’s stomach and saw that he was telling the truth; there was only one cardoon stalk inside. Then she was filled with remorse, for her brother would never rise again, and after washing and burying him she prayed: ‘God! Turn me into a bird that will forever mourn my brother.’ And this is what God did. And she sang:

‘Pepukh! Oh woe! Who slew him? I slew him! Who washed him? I washed him! Who buried him? I buried him!’”

“We and the Armenians were like brother and sister,” the Alevi said sadly. “Only we didn’t have the decency to bury them.”

                                                        

Notes

[1] Christopher de Bellaigue C. Rebel Land. Among Turkey’s Forgotten Peoples. Bloomsbury Publishing Plc. London 2009.

[2] C. de Bellaigue. In the Rose Garden of the Martyrs. A Memoir of Iran, HarperCollins Publishers 2005; and The Struggle for Iran, NYRB collections 2007

[3] De Bellaigue describes with bitterness his discussions in the 1990s with Professor Yusuf Halacoglu, author of Ermeni Tehciri, or Armenian Deportation, who estimates a ridiculous 30’000 casualties among the Armenian people during the 1915 deportations, rather than the one or one and a half million commonly assumed.

Mr. Biden’s Green Light?

Exactly one year ago, in July 2008, former President G. W. Bush had more or less encouraged Israel for going ahead with its preparations of a military strike against Iran in order to stop Tehran’s assumed illicit nuclear weapons program. It was actually one week after Iran had aired fabricated pictures of missile tests in the Dasht-e Kavir desert near Qom. Bush’s remarks had widely been interpreted as ‘amber’, not ‘green’, light for former Prime Minister Ehud Olmert who, after all, then only organized some war games in the Mediterranean but was more than reluctant of pursuing an attack. While it had been leaked during the weekend that according to a diplomatic source “the Saudis have tacitly agreed to the Israeli air force flying through their airspace on a mission which is supposed to be in the common interests of both Israel and Saudi Arabia,” a statement which has been denied by Saudi officials, the American Vice President Joe Biden, in an ABC interview said yesterday:

“Israel can determine for itself — it’s a sovereign nation — what’s in their interest and what they decide to do relative to Iran and anyone else.

“Whether we agree or not. They’re entitled to do that. Any sovereign nation is entitled to that. But there is no pressure from any nation that’s going to alter our behavior as to how proceed.

“If Netanyahu’s (Israel’s Prime Minister) government decides to take a course of action different than the one being pursued now, that is their sovereign right to do that. That is not our choice.”

At least, it is not their (the Obama Administration’s) choice. What Biden actually does, namely publicly declaring that Israel is ‘entitled’ to attack Iran’s nuclear facilities which are run, at least officially, and surveyed by the International Atomic Energy Agency (IAEA) under the Nuclear Non-proliferation treaty (NPT), having been signed by Iran but not by Israel, which many believe possesses up to 200 nuclear warheads, is highly questionable. Maybe it is even illicit, and he should better know. While his President is presently in Moscow in order to convince his Russian counterpart Dmitry Medvedev of the advantages of a world-wide denuclearization, his Vice President may have put the Middle East another step further into the direction of a real disaster, as Admiral Mullen has warned on Fox News yesterday.

Meanwhile, the designated new Director General of the IAEA, Yukiya Amano, horrified the Israeli Administration when telling Reuters Friday last week that he has not seen “any evidence in IAEA official documents” that Iran was trying to gain the ability to develop nuclear arms.

The last four weeks have seen an almost struggling regime in Tehran after serious and probably qualified accusations inside the country and by foreign media of massive election fraud. Dozens have been killed in riots after the largest demonstrations the country has seen since the Islamic Revolution. And probably hundreds detained, maybe tortured. Show trials are expected, maybe even executions.

The so-called Green Revolution which has never been a revolution since the silent majority, Iran’s have-nots, did not try a rebellion but rather supported ‘their’ demagogic hardliner President, has failed for the time being. What we have seen is understandable dissatisfaction of Tehran’s upper class, students, and intellectuals, with an oppressive regime which clearly violates human rights, a true dictatorship. It indicates a deep disruption of the whole Iranian nation, not only within the complex and unusual framework of political, economic and social powers. We have also learned about the immense power of the internet and the profound fears of regimes all over the world of the new ways of communication and ease of spreading information.

However, none of this would justify ‘green light’ for Israel to preemptively attack the country right now. Insofar the remarks of Biden are irresponsible and highly counterproductive.

Benford’s Law

Astronomer Boudewijn F. Roukema at the Centre for Astronomy of Nicolaus Copernicus University Torun in Poland has launched an analysis of vote counts of 366 voting areas, which had been published by the Iranian Ministry of Interior, and has applied Benford’s Law in order to detect election fraud. According to these calculations, “the null hypothesis that the vote count distributions satisfy these distributions is rejected at a significance of p < 0.007, based on the presence of 41 vote counts for candidate K (Mehdi Karroubi) that starts with the digit 7, compared to an expected 21.2-22 occurrences expected for the null hypothesis. A less significant anomaly suggested by Benford’s Law could be interpreted as an overestimate of candidate A’s (Mahmoud Ahmadinejad’s) total vote count by several million votes.” 

The study is not completely convincing. Still, the observed anomalies may be explained by chance alone. Moreover, figures 5 and 6 (pp. 5, 6) may have erroneously been exchanged. The motivation of conducting such an analysis is definitely driven by the assumption that the incumbent Iranian president was in urgent need for massive manipulation to become re-elected. That might not even be the case, as I have argued before. The brutal abolition of demonstrations in the previous two weeks have shown the true face of this regime which cannot easily been overthrown, in particular if one has to assume massive western support of  a “Green Revolution” with now rather deleterious outcome, as we experience these days. Sad to say but Iran may be on its way to a police state.

The online manuscript by Dr. Roukema (as of June 16, 2009, i.e., 4 days after the election!) which has been submitted to the Annals of Applied Statistics can be found here.

Martyrs

Somewhat hidden on Iranian government-funded presstv’s webpage you may find this Monday’s sensation. Ten or 13 killed demonstrators on Tehran’s streets, among them 19-year-old Neda, whose shocking death millions around the world have watched in an extremely graphic video, have possibly not died in vain. Iran’s Guardian Council has to admit today that probes into ballots boxes have proved a massive election fraud. In 50 cities, they contained more votes than people eligible. So, rumors that the 11 million additional ballot papers had been in the boxes before the day of election have had all reason. The remarks made by the Council’s spokesman, Abbas-Ali Kadkhodaei, that “[s]tatistics provided by the candidates, who claim more than 100% of those eligible have cast their ballot in 80-170 cities are not accurate — the incident has happened in only 50 cities” really won’t matter. He should know that election fraud is independent of magnitude. And who believes authorities in Iran anymore? And who cares?

It is quite amazing that the more than 646 ‘irregularities’ (emphasis by presstv) reported by the three defeated rivals of the incumbent Mahmoud Ahmadinejad, Messrs Mousavi, Rezai, and Karroubi, were partly acknowledged today. Former president Mohammad Khatami has proposed, in the meantime, an impartial committee to investigate the complaints about the results of the presidential vote.

Although nullification of the June 12 election as urged by the latter might in fact be at hand, power struggles within the system might have moved the Islamic Republic of Iran closer to what everybody fears most, a Pasdaran-controlled police or military dictatorship. Four-hundred-and-fifty-seven people had been detained after Saturday’s riots in a massive crackdown.

Does It Matter?

So far, President Obama had been reluctant to comment on the results of last week’s presidential election in Iran, the possibility of massive election fraud and the continuing demonstrations in Tehran and other larger cities. It won’t matter who will be the president, we read, the incumbent, Mahmoud Ahmadinejad, or his main rival, Mir Hossein Mousavi. “It’s important to understand that although there is amazing ferment taking place in Iran, that the difference between Ahmadinejad and Mousavi in terms of their actual policies may not be as great as has been advertised,” Obama told CNBC News. Obviously, the American President doesn’t want to weaken the opposition movement by showing overt partisanship.

Even in his latest remarks on demonstrations in Iran, made in an interview with CBS, he only stressed his hope that “ultimately the Iranian people will obtain justice.” “Now, what we can do is bear witness and say to the world that the, you know, incredible demonstrations that we’ve seen is a testimony to, I think what Dr. King called the arc of the moral universe. It’s long but it bends towards justice.”

While yesterday’s Friday prayer sermon by Ayatollah Ali Khamenei in Tehran had been interpreted as threat of future violence if demonstrations would go on, little may be found in this regard in the transcript of the speech by the Islamic Republic of Iran News Network Television (IRINN):

“In these elections the people truly performed their duty. Their duty was to come to the polls and this duty was performed in the best possible way. However, we and you have more serious duties to attend to. Those — from politicians, heads of parties and directors of political currents — who can exert some influence on the public and are listened to by some groups, should be very careful of their conduct. They should be very careful of what they say. If they show the slightest extremism, the repercussions will be felt by the body of the people and can lead to a very sensitive and dangerous situation which even they cannot control. We have seen examples of it. When extremism starts in a society, any extremist move can fuel other’s extremism. If the political elite ignore the law, or cut off their noses to spite the face, whether they want or not, they will be responsible for the bloodshed, violence, and chaos (that will follow).” [emphasis added]

In addressing the people of Iran, he made very clear that all four candidates who had been allowed to run for presidency by the Guardian Council (more than 450 had been rejected) had been campaigning within the system.

“Of course, they have different opinions. They have different plans. They have a range of differences in their various political stances. But all of them belong to the system. All of these four people are members of the system. This race took place within the system. […] [It] was not a race between the insiders and outsiders of the system. It was a race inside the system between the members affiliated to the system, who have such records. I know all of them closely. I am familiar with their thoughts, ideas, and traits. I have worked with them closely. I know all of them.”

So, in principle, Obama is right. It wouldn’t matter. Can it be that hundreds of thousands if not millions of Iranians have not really understood that simple fact?

The Lost Election

Mass demonstrations in particular in Tehran these days, even with casualties, give the strong impression that the majority of the Iranian people will not accept the results of last week’s presidential election when the incumbent president Mahmoud Ahmadinejad yielded allegedly a landslide victory over his main rival Mir Hossein Mousavi. Western media spread claims of fraud and rigged results. One especially ridiculous example of how to manipulate the lay public opinion was the use of regression analysis of cumulative counting results after polling stations had been closed which has been launched by tehranbureau, an exile Iranian organization located in the US. The mere nonsense of such an analysis was quickly debunked by mathematician Nate Silver of FiveThirtyEight. But the risks of manipulating the public by fraudulent reporting are still present. 

Detailed results of the election have been released in the meantime by Iran’s Interior Ministry and can be found here. Although fraud cannot be proved or disproved at the moment and most probably never, claims of fraud are still around and growing. But do the demonstrators on Tehran’s street who are in favor of the largely unknown, rather colorless Mousavi whose political program would not, in the West, really been regarded as that of a reformer, represent the majority of the Iranian people? On election day, FiveThirtyEight had published some opinion polling results showing that Ahmadinejad always led since April. Unfortunately, even these polls were largely confined to Tehran and, if either conducted or monitored by the government, must be considered untrustworthy.  

A nationwide public opinion poll had been conducted by Terror Free Tomorrow. Its conclusions, published on Monday this week by Ken Ballen and Patrick Doherty in the Washington Post, indicate that the 2 to 1 margin in favor of Ahmadinejad was already visible in mid-May. It was based on telephone interviews conducted by a neighboring country. In that survey, the breadth of support for Ahmadinejad was apparent. Even Azeris, the second largest ethnical group in Iran favored Ahmadinejad by a 2 to 1 margin, although Mousavi had stressed his identity as an Azeri. Some analysts have taken the mere fact that Mousavi lost the election even in the two Azerbaijan provinces as strong evidence that the election results had been rigged.

Ballen and Doherty write: “Allegations of fraud and electoral manipulation will serve to further isolate Iran and are likely to increase its belligerence and intransigence against the outside world. Before other countries, including the United States, jump to the conclusion that the Iranian presidential elections were fraudulent, with the grave consequences such charges could bring, they should consider all independent information. The fact may simply be that the reelection of President Ahmadinejad is what the Iranian people wanted.”

Today’s situation in Tehran and other large cities might in fact escalate. If it is being settled and President Ahmadinejad is resuming his second term, crackdown and terror will definitely increase and another round of oppression of the opposition in Iran commence. Insofar the outcome of this election is a true tragedy for the country, but it had to be expected. It might even reflect the will of the silent majority of have-nots in the vast country. Ahmadinejad’s competitors in the campaign were not really alternatives of the ideological hardliner who still (but how long?) enjoys sympathy and full support by the real leader of the country, Ayatollah Ali Khamenei.

Fraudulent Reporting

According to official announcements, the incumbent Iranian president Mahmoud Ahmadinejad had won the election with 62.6% of the votes while his main challenger, former prime minister and reformist Mir Hossein Mousavi, yielded a disappointing 33.8%. The two other candidates, former speaker of the parliament Mehdi Karroubi and former Pasdaran commander Mohsen Rezai only became also-runs.

There have quickly been claims of rigged election results in Iran and massive fraud. While Mr. Mousavi did not use this particular word, he talked about a charade. In western media, a graph circulates which seem to provide indisputable evidence of rigged results. Data are derived from six announcements of incoming voters’ counts of the Ministry of Interior itself. The data (cumulative voters counts for Mr. Mousavi and cumulative counts for Mr. Ahmadinejad), if displayed in a scatter plot, show that the incumbent president always got about twice as many votes as his rival. A regression line is provided and an R² of 0.998 calculated for the correlation.

 

 

 

 

 

 

 

 

 

This analysis is of course nonsense. It cannot prove that the Iranian presidential election was fraudulent. A charming example from the recent U.S. American election by Nate Silver is provided here. He supposed that voters’ results are announced in a similar fashion as those from the Iranian Ministry of Interior, i.e., in six large waves. For example:

Wave 1: 23,564,027 for Obama; 18,862,194 for McCain

Wave 2: 36,772,827 for Obama; 30,199,247 for McCain

Wave 3: 49,333,482 vs. 39,911,405

Wave 4: 57,230,954 vs. 47,112,351

Wave 5: 63,176,968 vs. 54,474,198

Wave 6: 69,498,216 for Obama; 59,948,240 for McCain

See the graph below. An R² of 0.996 can be calculated which is, of course, complete nonsense. No proof for election fraud at all. Simply due to violating fundamental assumptions in linear regression.

 

 

 

 

 

 

 

 

 

What is dangerous here is that journalists, who should have some basic skills in mathematics and in particular statistics, spread, intentionally or unintentionally, wrong, in fact fraudulent, information.

There are other means of proving that the results of the Iranian presidential election were rigged. That the Iranian people re-elected the old foe with whom any collaboration with the West will not become easier does not mean that the Iranian people has been fooled. At least as long as the opposite has not been proven.

Re-election

A few notes on the results of the Iranian presidential election. It’s not surprising. It would have been quite a sensation if the incumbent populist hardliner president wouldn’t have been re-elected. Definitely sort of a political earthquake with imponderable consequences, i.e., risks, for the Islamic establishment. The widely spread, in western media, impression of low support of the Iranians for Mahmoud Ahmadinejad after four years of disastrous domestic and foreign policies with exploding prices, unemployment and general hopelessness among the youths may in fact have been heavily biased. Mr. Ahmadinejad is an exceptionally gifted, therefore immense dangerous, populist. In split countries like Iran with a voiceless, even silent, majority of underprivileged have-nots in the huge rural areas of the country liberal attitudes of well-off city dwellers, well-educated intellectuals heavily fearing loosing everything what they had achieved in years of oppression after the Islamic Revolution (don’t forget that Iran had been celebrating earlier this year its 30^th anniversary) do not really count although they might find their way easier into western media.

The presidential hopefuls who had been competing with Mr. Ahmadinejad didn’t have a chance. In particular former Prime Minister Mir Hossein Mousavi was too colorless, not well-known among the mainly young voters, too intellectual. The 30-something per cent he gained was what some observers had actually expected. That the former speaker of the majlis, Mehdi Karrouby, and former Commander of the Revolutionary Guards Mohsen Rezai were so weak might be considered the real sensation of this election, whose campaign had got a lot of momentum especially in the past few days. It was nice to see engagement of young people in the politics of their country. Sad to say, but another piece of hope for the future of this gorgeous country had been destroyed today.

Polygons

“He who knows not and knows not that he knows not, shun him. And he who knows not and knows that he knows not, awaken him. And he who knows and knows that he knows, follow him.”

Arabic saying

The swastika has nowadays a bad reputation but it has of course not been invented by German Nazis. Rather it is a positively connoted, sacred symbol in Hinduism and Buddhism, such as lucky charm. It is interesting to see that it has also found its way into Islamic Art, even as a sign of blessing. A famous square panel on the western iwan of Esfahan’s Great Mosque dating from the 17^th century (Shi’ite Safavid) resembles a Swastika, and its calligraphy mentions Ali [1]. It might be a beautiful example of “a simple design rotated 45 degrees which acquires two separate values, one as a carrier of geometric forms filled with (by the time of the panel) antiquarian writing, the other one as a violator of the sequence of both writing and architecture by forcing one into rare contortions to read the writing” [2]. The southern iwan which had got additional decorations by Sayyid Mahmud-e Naqash in 1475/76 sports a similar but definitely Timurid swastika-like panel, with its ample arabesque and floral motifs [3].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A Square from Five Squares

These examples are not strict swastikas. Rather, they represent a popular Islamic geometric pattern, a square composed of three squares. In the 10^th century, artisans were thoroughly taught in a distinct academic context by mathematicians in geometry. Alpay Özdural (d. 2003) describes [4] how, for instance, Abu’l-Wafā’ al-Būzjāni (940- ca. 998), in his famous treatise Kitāb fīmā yahtāju ilayhi al-sani’ min al-a’māl al-handasiya (On the Geometric Constructions Necessary for the Artisan) teaches the right way of constructing this very combination of squares and avoid often made mistakes of the carpenter whose job involved cutting single pieces of material into parts and arranging them skillfully in attractive patterns in mosaics. Abul’l-Wāfa explains that artisans and even geometers (muhandis) often err in the assembling of the pieces, the former since they do not know the scientific proof, the latter due to lack of practice. As Özural writes, Abu’l-Wāfa’s book on Geometric Constructions was apparently motivated by meetings with practitioners and aimed in the proper advancement of Islamic Art. As a true academic, he displayed, in his book “pure geometry, familiarity with practical applications, and skill in teaching theoretical subjects to practical-minded people.”  

The figure below (from Özdural’s article) shows how, by cutting and pasting two, five and nine squares, according to Abu’l Wāfa’s theoretical solutions [5], pretty attractive patterns are created. The earliest “square from five squares” can be seen on the wooden door of the mosque of Imām Ibrāhīm in Mosul which is dated 1104 CE. And Abu’l-Wāfa also explains patiently why some popular ‘practical solutions’ were essentially wrong.

 

 

 

 

 

 

 

 

 

While between the 11^th and 15^th centuries in Iran and Central Asia, Spain and elsewhere in the Islamic World, geometric tessellations became more and more ambitious, dazzling, breakneck artistic, it is not clear how much artisans actually knew about geometry and mathematics. Özdural’s paper convincingly shows how academics such as Abu’l-Wāfa in Baghdad or later Omar Khayyām in Esfahan and Jamshīd al-Kāshī in Samarqand frequently met with artisans, architects, masons and carpenters in what he calls conversazione, i.e., seminars and practical sessions, where the then popular cut and paste technique of dividing larger material into smaller pieces was exercised and got a sound theoretical foundation. While the Golden Age of Islamic Science and Art before and around 1000 CE, in particular Persia, was brutally brought to an end by Mongol invasions after 1220, with catastrophic destruction and by and large architectural inactivity for several decades, later-on, during Ilkhanid, Timurid, and even Ottoman periods, scholars again took over in assisting those who created the most incredible geometric and arabesque tessellations. But they still noted lack of knowledge and unwillingness of master-builders to entirely rely on geometric proof but rather dealt “with geometry in their unmethodological and incorrect way three centuries after Abu’l-Wāfa.” “Yes, we have heard of it, but in essence we have not heard how science of geometry works and what it deals with.”

 

Pentagons and Decagons

Especially fascinating may be the way, artisans had tried to use pentagons and decagons in their tessellations. There have even been speculations, at least since the late 1980s, whether medieval Islamic artists had been able to create aperiodic tiling, such as those which had been described by Roger Penrose in the 1970s.

 

 

 

 

 

 

 

 

 

In studying the probably 13^th century manuscript by an anonymous author, Fī tadhākul al-ashkāl al-mutashābihah aw al-mutawāfiqa (On Interlocking Similar or Congruent Figures), which is now located in the Bibliotheque Nationale in Paris, Wasma’a K. Chorbachi and Arthur L. Loeb [6] point to the similarity of the here described problem of interlocking convex decagons and pentagonal stars (the Islamic Pentagonal Seal) with those being now popularly known as aperiodic Penrose Tiling [7].

 

 

 

 

 

 

 

 

 

In this manuscript one may find an interesting ‘practical’, albeit incorrect, solution for creating regular decagons and pentagons by cutting and pasting the kunya-5 triangle, a right-angled triangle with one angle equal to 36°. The approximation differs from 36° by only 12’22’’, i.e., 0.5% [8].

 

 

 

 

 

In particular in the 13^th century, the golden triangle (an isosceles triangle having angles of 36°, 72° and 72°; its base length equals f times its side-length, where f is the golden fraction defined by the equation phi = 1/(1+phi)), was used by Muslim scientists for the construction of regular pentagons and decagons [9]. The golden triangle can be subdivided in such a way that another golden triangle and a golden gnomon results, i.e., a isosceles triangle having angles 108°, 36° and 36°. As Chorbachi and Loeb write, artisans may actually have created the 36° angle using the (incorrect) method of constructing kunya-5.

The construction of the Pentagonal Seal in the Paris manuscript is, according to Chorbachi and Loeb, a very particular one, with its five-pointed star constituted by ten golden gnomons which exactly match the ten golden triangles which constitute the decagon. “It is historically significant that as early as the thirteenth century A.D., it was known that what we presently call the golden triangle and golden gnomon are together capable of tessellating the Euclidean plane, and that during the Middle Ages, Islamic design continued in the tradition of the Alexandrian and other eastern Mediterranean schools of mathematics. The use of this five-pointed star appears to have stimulated mathematicians to work on these practical problems in design. The importance of this problem to the Muslim scientists may be inferred by the fact that they tried over the course of several centuries to find the perfect solution.”

According to Wasma’a K. Chorbachi in “The Tower of Babel” [5], “[t]he true patron of the scientists who wrote these ancient manuscript was art. It was the artisans and the architects who called for the services of science and scientists to assist them solving the design problems that they were facing. And as in the case of Islamic art in the past, science must come to the service of the arts, whether we are talking today of Islamic art, of Western art or of art generally, today more than ever before […].” “[I]slamic tradition is so strong that, if we are in touch with the language of the present time and ground ourselves in this strong old tradition, we can arrive at an expression that is not only contemporary but could be meaningful and valid in the coming century.”

 

Notes

[1] According to Oleg Grabar in his fine book The Great Mosque of Isfahan (New York University Press 1990, p. 34) it contains in the four corners the pious quatrain: “As the letter of our crime became entwined [i.e., grew so long], [they] took it and weighed it in the balance against action. Our sin was greater than that of anyone else, but we were forgiven out of the kindness of Ali.” Grabar notes that the central part of the panel is nothing else than the plug of the artisan who was diligently involved in restoring the mosque in the 17^th century, Muhammad ibn Mu’min Muhammad Amin.

[2] Ibid.

[3] Decorative brickwork on the northern iwan of the mosques also shows clockwise and counterclockwise swastikas in one of the circumferential bands.

 

 

 

 

 

 

 

 

 

[4] Özdural A. Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World. Historia Mathematica 2000; 27: 171-201.

[5] Ibid. It is the Islamic proof of the Pythagorean Theorem, which is closer to the Indian method of Bhāskara Achārya (d. 1185) than to the Greek method in Euclid’s Propositions, as is beautifully explained by Wasma’a K. Chorbachi in her eye-opening article “In the Tower of Babel: Beyond Symmetry in Islamic Design. Computers Math Applic 1989; 17: 751-789.

[6] Chorbachi WK, Loeb AL. An Islamic pentagonal seal (from scientific manuscripts of the geometry of design). In Hargittai I (ed) Fivefold symmetry. World Scientific, Singapore 1992, pp. 283-305

[7] Ibid., p. 284: “Although the approach to the generation of this pattern in the Paris manuscript is quite different from that taken by Penrose, it is notable that these ‘quasi-periodic’ patterns were already of interest at least in the thirteenth century A.D. The manuscript stresses the uniqueness of the fivefold center of rotational symmetry in the pentagonal seal, thus implying the lack of translational symmetry in the pattern, but does not explicitly deal with the matter of non-periodicity.”

[8] Ibid., p. 286f: “The construction was therefore remarkably accurate, though not correct. Kamal ad-Din Musa Ibn Yunus Ibn Man’a in his thirteenth-century commentary on Abu’l Wafa’ al Buzjani’s book on the geometry of construction, with whom this construction may well have originated, actually was quite explicit in cautioning that some of his constructions, in particular of the heptagon, were practical, but not mathematically exact. They can be used in small-scale designs without noticeable discrepancies, which however become manifest on a larger scale.” 

[9] Ibid., p. 293: “[I]n the second half of the thirteenth century (ca. 1259) in the town of Marāgha, which became a center of scientific activities and contained the famous observatory, another illustrious mathematician, Nasir ad-Din at-Tusi, wrote commentaries on Euclid, in which he made obvious use of the golden triangle. … [H]is commentaries on Euclid included a short treatise dealing with the inscription and circumscription of polygons within the circle: Sittat Maqalat min Kitab Tahrir Uqlidis: Six Books/Articles from Euclid’s Book of Elements.” As an example, see the construction below, which had been created with some guidance from Eric Broug’s booklet Islamic Geometric Patterns, Thames & Hudson, New York 2008.

 

 

 

 

 

 

 

 

See also on this blog

About difficulties of the Western perception of Islamic abstraction which might easily result in fundamental misconceptions

About decagonal tessellations on the west iwan of Esfahan’s famous Friday Mosque

About Alpay Özdural’s proof that the mysterious North Dome of Esfahan’s Great Mosque is based on Omar Khayyām’s triangle

A review of a booklet which makes complicated Islamic geometric patterns easy to reproduce

New Centrifuges

One day after President Barack Obama’s historical speech in Cairo addressing the Muslim World, where he as the first sitting American president admitted to the U.S. contribution on the 1953 coup d’état toppling the democratically elected Iranian Prime Minister Mohammed Mossadeq (another sensation was the note that Iran, as any other country, has the right to use nuclear energy for peaceful purposes) and a couple of days before an unusual presidential election in Iran, the Institute of Science and International Security (ISIS), an NGO directed by David Albright, has again leaked a report of the International Atomic Energy Agency (IAEA) on Iran’s nuclear issue. The report states that Iran has produced now about 1400 kg low enriched uranium (less than 5%) in its almost 5000 centrifuges at the enrichment facility in the vicinity of the Central Iranian village of Natanz. Another 2300 centrifuges have been installed but not yet activated. Iran has not responded to demands of the IAEA to comment on its previous clandestine nuclear program.

The incumbent Iranian president is right now struggling in an unprecedented re-election campaign. His nuclear programs has come under attack since it has internationally isolated the country in recent years and Iranians are suffering from harsh sanctions due to ignoring four U.N. security council resolutions 1737 (2006), 1747 (2007), 1803 (2008), and 1835 (2008) calling for a stop of any enrichment activities.