Geeks Rule: Quasicrystalline Patterns in Mediaeval Islamic Architecture

Illustration above: Cathy Wilcox

I fell madly in love with Archimedean solids and stellated icosahedra at an early age, and never recovered. Like architecture, geometry is a bridge between art and science. Plato exhorted all philosophers to be geometers for good reason: spatial and social logic go hand-in-glove. Geometry is revelatory. To ancient Greeks—and now we are discovering, probably to early Muslims as well—the ontological truth of geometry meant that issues of politics and social logic were questions of symmetry, proportion and harmony.

M.C. Escher said, ‘Mathematicians go to the garden gate but they never venture through to appreciate the delights within.’ An observant pair of scientists discovered that medieval Muslim mathematicians not only ventured through the gates, they artistically expressed the enchantments of the garden.

Five Centuries Ahead of the West

In Bukhara, Dr. Peter Lu was astonished by the beauty and complex geometry in the tiling in Abdullah Khan madrasah, and decided to further investigate.

Dr. Lu (Harvard) and Dr. Steinhardt (Princeton) state that Islamic designers had mastered techniques 'to construct nearly perfect quasicrystalline Penrose patterns, five centuries before their discovery in the West.'

Dr Lu with his cousin in Bukhara, where he saw this tiling. 'It was the one that first caused me to get curious about the whole issue of decagonal geometry in Islamic patterns.'
Photo courtesy of Peter J. Lu

'This would be a hitherto undiscovered episode in the spectacular developments of geometry in central Islamic lands...achieved by artisans probably inspired by theoretical mathematicians', said Islamic art specialist Oleg Grabar.

Esfahan nesf-jahan (Esfahan is half the world)
Sheikh Lutfallah Mosque, Esfahan, (1602-1619) built much later than the Bukhara mosque above. Architect: Muhammad Reza ibn Ustad Isfahani. Photo by Tile Driessen

Above, another view of Lutfallah by Mahdi Asheghvatan

Photo by Dietrich Meyer, above.

‘The study of sensible [i.e. of the senses] geometry leads to skill in all the practical arts, while the study of intelligible geometry leads to skill in the intellectual arts, because this science is one of the gates through which we move to the knowledge of the essence of the soul, and that is the root of all knowledge.’ The Rasai’il ikhwan as-safa (Encyclopedia of the Brethren of Purity, a vast compendium of knowledge compiled c. 950 CE.)

Another sample of what geometer-artists can do!

Analysing Patterns

Above, Darb-i Imam shrine in Iran. Courtesy of K. Dudley and M. Elliff

A computer reconstruction of the quasicrystalline patterns shown above, courtesy of Peter J. Lu

Above, further analysis

Pattern from a Turkish mausoleum, c. 1200 C.E, above A reconstruction of the tile templates is overlaid at the bottom.

These tiles may have been used to generate a wide range of complex tiling patterns on medieval buildings, including mosques in Isfahan, Iran, and Bursa, Turkey; madrasahs in Baghdad; and shrines in Herat, Afghanistan, and Agra, India. This approach produces infinite patterns with decagonal symmetry that never repeat. (Harvard University Gazette)

Photo from Professor N. Rabat, above, Aga Khan Program at MIT

Photo by Nima Mehrabany, above

Penrose Tiling in Architecture

Johannes Kepler (1571-1630) may have been the earliest European who explored non-repeating symmetries. But it was not until the 1970’s that Sir Roger Penrose described the mathematics of interlocking polygons whose pattern never repeats: quasicrystal geometry. "It shows us a culture that we often don't credit enough was far more advanced than we ever thought," said Peter Lu. “They made tilings that reflect mathematics that were so sophisticated that we didn't figure it out until the last 20 or 30 years.”

Penrose tiling is made of just two kinds of tiles, kites and darts. A kite is made from two acute golden triangles and a dart from two obtuse golden triangles.

The ratio of kites to darts is the Golden Ratio. The fact that the ratio is irrational is the underlying principle in the proof that the tiling is non-periodic, since the ratio would be rational in a periodic tiling. (With some notable exceptions, many European mathematicians did not recognise irrational numbers until the 19th century)

Because aperiodic (non-repeating) tiling is more challenging to design and install than ordinary repeating tiles, we seldom see it in modern buildings. Below are a few examples on both modest and grand scales.

Bench in Amsterdam by Javier Lopez

Above, Helsinki Science Museum, Finland

Above, Kuilema Pottery: http://www.stonewaretile.com

Above, interior of the Royal Melbourne Institute of Technology's Storey Hall, designed by Ashton Raggatt McDougall and constructed in 1994. The building won the Royal Australian Institute of Architects' National Architecture Award (Interior). Photo credits: Adam Dimech

Storey Hall entrance, above.

Storey Hall facade, above

Properties of Penrose Tiling

Penrose Tiling is extraordinarily dense. It has a number of remarkable properties. One is that every finite portion of any tiling is contained infinitely often in every other tiling. Therefore, no finite portion of tiles can determine the rest of the tiling, and it is impossible to tell from any patch of tile which tiling it is on.

The full article by Peter J. Lu and Paul J. Steinhardt, “Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture,” Science 315, 1106 (2007) is here. Science_315_1106_2007---.pdf

Architectural Scrolls

Until Peter Lu announced his findings to Western audiences, it was widely believed that medieval artists and architects were limited to the basic compass and straightedge. It is now known that the intricate accuracy of these designs is impossible with only compass and straightedge.

Professional mathematicians and master architects wrote architectural scrolls, important instructional manuals of the day, which were disseminated throughout the Islamic world. Inventive architects and artists kept improving the sophistication of their work until, by 1453, they achieved the quasicrystal.

Above photo by Mehrdad Tadjdini

‘This cannot be solved with plane geometry, since it has a cube in it. For the solution we need conic sections.’ —Omar Khayyam (who not only not only put his toes into non-Euclidean waters, he is also given credit by some art historians for the design of the North Dome chamber of Esfahan’s Friday Mosque, built in 1088.)

Tessellation (Tiling)

Repeating tiling is far easier to play with than the sets of five contiguous polygons (decagon, pentagon, diamond, bowtie, and hexagon) that medieval Muslim artists used to generate mosaic patterns on large surfaces with exquisite accuracy.

One of the joys of tessellation is, of course, venturing into non-Euclidean space, entering the hyperbolic plane and leaving 2D and 3D far behind. In my view, this transcendence is exactly what medieval artists—guided by theoretical mathematicians—achieved.

Perhaps the best way to appreciate such accomplishments is to try it oneself. The following is an example of what one can make on Quasi-G, an easy-to-use freeware program for geeky art explorations:

Not all patterns one creates with this software will conform to the Penrose rules - QuasiG allow clusters of thin tiles where more than 2 thin tiles share a vertex, and are all adjoining.

Make your own tessellations

One can fool around and make one’s own (somewhat primitive) designs here by distorting the hexagon, and clicking ‘tessellate’.

Sometimes it’s more satisfactory to play in 3 dimensions. Here is a set of Lucite tiles to make some well known patterns

One of the masters of tiling, M.C. Escher, considered himself a poor mathematician, but he has inspired generations of both scientists and artists. Tessellation.info has a large data base of artist-geometers.

Here are just a few samples of people having fun. I hope you have fun with tiling, too!

Tags: Islamic architecture, Penrose, geometry, islamic tiles, medieval, quasicrystal

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Comment by oz on October 15, 2011 at 9:52pm: wow

Comment by Attilio Taverna on January 2, 2010 at 7:32pm: Dearest Elizabeth,
I would like you see ( if it is possible, of course.. ) Scientific American n. 17 July 1974.
The article :" Perception of Transparence " by Fabio Metelli. He was a teacher of Psycology at Padova University. The same Departement years later he died published in 1997 an official book titled : " Between Perception and Art ". A chapter of this book is dedicated to my aesthetic-experience, as first example in Italy - but not only in Italy, of use into abstract art of originating space concepts by " perceptive transparence visual laws ". Why I say to you this ? Because as I told you before I have studied a lot Penrose tiling tessellation and have discussed for many times about Escher 's works. So, this your thought which I totally agree with : " one of the joys of tessellation is, of course, venturing into non-Euclidean space, entering the hyperbolic plane and leaving 2D and 3D far behind. In my view, this transcendence is exactly what medieval artists—guided by theoretical mathematicians—achieved "...this your conclusion to my eyes is really a strong and beautiful theoretical way to comprehend trascendence into abstract art. More : it is a clear mind' assertion to comprehend the true " optical opening ways of abstract painting " to force the visual spaces of all the geometrical systems....already known ...by humanity.. well.
Question : are we sure that geometrical abstract art ( painting) has not invented another way of " breaking " the visual space dimensions of every infinite possible geometries - except the faboulous and beautiful islamic ghiri quasi- crystal-vision ? ( about which I will send to you a little text written by me..when you will give to me the mail address...)
An help to you : perceptive transparence visual law concepts ( obtained by Metelli at Padova University 1974 ) and breakage of symmetry concept obtained by physics around 60' are two
fundamental roots ineluctably existing into every possible concepts of spacetime visualization...
If you desire only : it exactly is at this level of discussion that I want to speak with you about what does it mean to create Art & Architeture today.
Please dont answer for any reasons : sorry, I cannot at this level to speak with you about this...because it would be the perfect only one answer that I would not believe in all entire the ...universe... ciao, attilio

Comment by Attilio Taverna on January 2, 2010 at 12:46am: Dearest Elizabeth,
another terrific book was the one by ( now, unfortunally died..) Imre Toth titled : " Aristotele and the assiomatic fundaments of Geometry ". 1997 Ed. Centre of metaphysic researches " Vita & Pensiero "- University of Milan.
Mine is 700 pages of a very strong battle....with underlines,comments, writings at limit of pages ecc. as all my books after all are. Why I say to you this ? Because, this book created some years ago a very strong debate into many philosophy of math...department...and to many aesthetics philosophy departments and philosophy of physics too...all over Europe....Why ? because Imre Toth prouved by his incredible book that " Non -euclidean Geometries " were discussed by four or five arguments at least by Aristotele himself into " Corpus Aristotelicum " of 2500 years ago. And nobody have understood this...during 2500 years ! In fact, " Not-euclidean Geometries " have been descovered all together by Riemann, Bolay, Lobatchewskj, 1821 - 1826. Without these geometries on curve space, Einstein
couldnt descover his theory of " restrict relativity " first - and 15 years later his " General Relativity ". All of this to say what ? To say this : are we sure that Architecture ( generally speaking..logical not speaking about the well knowns " archi - stars ") is not still in love with " Euclidean Geometry " -only ?
Alain Connes, the french mathematician winner of " Field Medail " years ago, invented a very new body of Geometry called " Not-commutative Geometry ". Poincarè has demonstrated (last century ) that infinite geometries can exist into an infinite concept of space...if they are compact, axiomatic and not-contradictory. Conclusion : if physics says to us that spacetime close to " singolarities " become intractable mathematically....well from this math..evidence ineluctably derived that our formal dreams - doesnt matter if made by stones or painted images - has to strong confront themselves with all the dimensions of knowledgment by physics...math...and geometry....because Art & Architecture have to dream ( and realized )what never has been dreamed and realized before their own dreams....
Question : who can " see " in the vision that intractable math...spacetime concept of which physics is speaking about..?
I have a very great and stupendous story to talck you about who really descovered the " form " of not-periodic crystal, better : a- ( greek alpha privative )- periodic crystal......
P.S. recently, I have had a short mail communication with Oleg Grabar....( about ?..)
ciao, attilio

Comment by Attilio Taverna on January 1, 2010 at 5:49pm: ..Also " Godel, Bach, Escher ... an Eternal Golden Braid " is a book terrible important on the subject. Two things to my eyes are as crucial -focus at this point : a) by Geometry - as formal language - as to say " language of Form" we are obliged to penetrate into an horizon of metaphysical problems involving reality, religions, philosphy, truth, science, art, architecture.
b) we have to recognize as a definitive prouved scientifical achievement the reality of " breakage of symmentry " as a fundamental axiome of the reality. As to say : the reality can appear like its appears just because a breakage of symmetry permits it...of appearing...exactly like its appears..
The range of all of these concepts oblige us to an incredible inter-disciplinary theoretical trajectory of mind and culture : not-commutative geometry, geometry of fractals, philosophy of math...philosophy of physics... aesthetics as philosphy of art....history of art....history of architecture....analisys of concept of Form ( ontologically understood )....we cannot escape from this intellectual condition. The ultimative reason is : we live as human being by an unsupressable tension to the Truth....the Truth of religion, of philosphy, of science of art....of architecture.....this is the last meaning of Heidegger 'sentence : " art is the last figure of truth "...
If this philosophical thougth is true..than.its meaning shows that art & architecture can be " last figure of truth " only after a strong confrontation with other truths of religion,philosophy,science....only after this strong confrontation...Art & Arch...they could be...last figure of truth...
Yes, the original bridge was Geometry - but now, ineluctably it became : what is Form -ontologically understood. This is the destiny of Art & Archietcture : to answer what Form - ontologically understood is !
Ciao, attilio

Comment by Attilio Taverna on January 1, 2010 at 5:20pm: Dearest Elizabeth, I have descovered just this evening your blog about a-periodic cristalline tiles.. It is more than 20 yaers that I am working in art with these concepts. Now it is really definitively important that I send to you two or three theretical analysis aorund these problems in Art & Architecture.
If the piece here below is yours...I have to say that I completely agree with....and in the same time I invite you to read my lecture at Udine museum modern art of 15 years ago...titled " Geometry as inner and outer space of man "..

I fell madly in love with Archimedean solids and stellated icosahedra at an early age, and never recovered. Like architecture, geometry is a bridge between art and science. Plato exhorted all philosophers to be geometers for good reason: spatial and social logic go hand-in-glove. Geometry is revelatory. To ancient Greeks—and now we are discovering, probably to early Muslims as well—the ontological truth of geometry meant that issues of politics and social logic were questions of symmetry, proportion and harmony.
Besides : I have studied all the books by Roger Penrose around 90' -2000 years. All very important books really. Also Hawkings's books are important about.. ( continue )..

Comment by Michael McKenzie on May 15, 2009 at 10:35pm: A fabulous collection of 'tiled arts' a universe removed from the everyday cut and paste craft of todays workingman who is so pressured by modern time/money constraints...and my own rudimentary math and eye often put to labours of bath and kitchen tile play...
Cheers E !

Comment by Talal Samarkandi on May 15, 2009 at 5:59pm: very good subject and hard effort, Thanks

Comment by cesur on May 13, 2009 at 4:59pm: thanks for sharing this detailed post. loved photos, especially Sheikh Lutfallah Mosque..

Comment by somasekhar on May 9, 2009 at 6:11am: shall i say " The best were already built and we try for the better"

Thank you for posting!

Comment by MFX Design and Consultancy on May 5, 2009 at 2:07am: I too think that geometry is a bridge between art and science. Somewhat obsessed with hexagons myself lately, I have looked a lot a tiles and tessellations as well. Geometry or 'Geometa' as I like to call it may also be the future of architecture. I think the hexagon is the much sought after 'squared circle', and may be the future template of architecture as a most basic polygonal formation.! The five pointed star has great resonance as well, or the pentagon. the golden ratio and many organic things are related to pentagonal geometry. Many crystalline and structural things as well as the flower of life formation and metatron's cube are related to hexagonal geometry. Ciao Elizabeth!