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!
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
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.)
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!