Contemplating the infinity, muslim architects visualized and rendered impossible geometric patterns …

2011 Nobel prize in Chemistry was awarded to Israeli scientist Daniel Shechtman for his discovery of quasicrystals, metallic alloys with atoms arranged in orderly, infinite, aperiodic, crystal-like patterns with theoretically forbidden (typically 5 fold) symmetry.  This form of matter was believed to be impossible to create. Schectman made his discovery while on sabbatical at the U.S. National Bureau of Standards April 8, 1982 . He was laughed at.

"I told everyone who was ready to listen that I had material with pentagonal symmetry. People just laughed at me,"
Today, Dr Shectman has won Nobel prize in chemistry, 2011.

What the scientific community has failed to grasp, the muslim architects produced wonderful sacred art throughout the muslim world centuries ago in their contemplation of infinity ...

A woman standing in front of an illuminated quasicrystal ... 

Penrose tiling that produced the X-ray diffraction pattern above.

Remarkable Properties of Penrose Tilings
The most remarkable property of Penrose Tilings is that every finite portion of any tiling is contained infinitely often in every other tiling. This, of course, is true of all periodic tilings, but it's not at all obvious that it should be true of a non-periodic tiling. This property has several consequences:

    • No finite patch of tiles can force a tiling (determine the rest of the tiling).
    • It is impossible to tell from any patch of tile which tiling it is on.
    • Only at their infinite limits are the different patterns distinguishable. A finite patch of an Infinite Star pattern might only be a local piece of some other pattern, but there is also an Infinite Star pattern that has five-fold symmetry to infinity. Only if you know the characteristics of the pattern to infinity can you tell.

Quasicrystal type pattern in decagonal strapwork above an arch in the Abbasid al-Mustansiriyya Madrasa in Baghdad, Iraq, which dates to between 1227 and 1234.

DAVID JAMES: QUR’ANS OF THE MAMLUKS (THAMES & HUDSON)
A decagon, surrounded by bowties and hexagons, forms the basis of this cover of a Mamluk copy of the Qur’an that dates to the early 14th century.

Peter J. Lu, a physics graduate student at Harvard University, noticed a striking similarity between certain medieval mosque mosaics and a geometric pattern known as a quasi crystal—an infinite tiling pattern that doesn’t regularly repeat itself and has symmetries not found in normal crystals (see video below). Lu teamed up with physicist Paul Steinhardt of Princeton University to test the similarity: If the patterns repeated when extended infinitely, they couldn’t be true quasi crystals.

On this panel in the Shah Mosque in Isfahan, Iran, ten-pointed stars—none of them shown completely —anchor the edges of a pattern in which four of the five girih shapes can be found. Both the stars and the scallop-edged hexagons are placed according to an underlying design of still more decagons and pentagons. The stars’ incompleteness reminds the viewer that the pattern actually extends into infinity.

Most of the patterns examined failed the test, but one passed: a pattern found in the Darb-i Imam shrine (seen in the image above), built in 1453 in Isfahan, Iran. Not only does it never repeat when infinitely extended, its pattern maps onto Penrose tiles—components for making quasi crystals discovered by Oxford University mathematician Roger Penrose in the 1970s—in a way that is consistent with the quasi crystal pattern.

The quasicrystalline structured tilings were used by religious mystics, including Sufis, to contemplate infinity.

Girih pattern of a decorated arch in the Sultan’s Loge of the Ottoman-era Green Mosque in Bursa, Turkey, which was completed in 1424. The girih tiles make possible large-scale patterns because each edge has the same length, allowing different combinations to be aligned. What is more, every edge is intersected at its midpoint by two decorating lines at fixed angles, which ensures that the lines continue across the edges from one tile onto another. A further innovation was achieved by dividing girih tiles into smaller ones to create overlaid patterns at two different scales, a method mathematicians call “self-similarity transformation.” This kind of subdivision, combined with the symmetry imposed by the shapes of the girih tiles, creates non-periodic tiling, just like the Penrose patterns.

Wrapping a column in the early–19th-century Tash Hauli palace in Khiva, Uzbekistan, a strapwork pattern of decagons and pentagons is filled with vegetal arabesques that maintain five-fold and ten-fold symmetry.