Faith & mathematics

10 Jul 2020


IN 1079 AD, Omar Khayyam calculated the length of the year to be 365.24219858156 days, astonishingly accurate to the sixth decimal, an inaccuracy of a fraction of a second of the true length calculated in the 20th century using atomic clocks, computers and the Hubble telescope. Yet, more than 900 years later, some of the greatest academic works chronicling the contributions of Muslims to science, mathematics, society, law, and the arts have been carried out by non-Muslims.

Most Muslims admire Islamic art. However, it was only recently that a young physicist made a discovery that many of the beautiful patterns in ancient mosques were based on complex mathematics thought to have been first developed in the mid-1970s.

In 2005, a Harvard doctoral student Peter Lu, was visiting Uzbekistan. He noticed the intricate patterns in some historic mosques. Lu recognised the tiling’s patterns which were known by mathematicians as Penrose Tiles. At first, he could not believe that the tiling was 500 years old since he knew that the mathematics behind these was believed to have been first developed in 1974 by the distinguished Oxford mathematician and physicist, Roger Penrose.

After returning to Harvard, Lu collaborated with a world-renowned Princeton physicist Paul Steinhardt, who in 1984 had proposed the existence of three-dimensional analogs of Penrose Tiles, shrunk to the atomic level. Steinhardt had named such structures ‘quasi-crystals’. After meticulously reviewing historic Islamic mosaics and manuscripts, they published a paper in February 2007 in Science, a journal published by American Academy for the Advancement of Science (AAAS) in which they showed that Muslim mathematicians had made the geometric breakthrough behind Penrose Tiles about 500 years before Penrose did. They wrote in their paper, “ the 15th century, the tessellation approach was combined with self-similar transformations to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before their discovery in the West”.

Previously, Muslims treated learning as an act of faith.

Tessellations are repeating patterns made of one or more shapes without gaps or overlaps. They generally tend to be tilings in which patterns are repeated (‘periodic’), meaning one can easily predict the next pattern. It was well known that surfaces could be tiled with tiles with three, four or six sides but not with only five-sided tiles.

Penrose had shown that it was possible to build geometric patterns using a small set of tile shapes that may have both fivefold rotational symmetry, ie shapes that look the same if turned one-fifth of a circle such as a five-pointed star, and reflectional symmetry, ie shapes whose reflection is the same as the pattern. Penrose Tiles can also form a non-repeating (‘aperiodic’) pattern. Regardless of how long one walks on tiles with aperiodic patterns, the next pattern cannot be predicted.

In 1982, Daniel Shechtman, a scientist at the US National Bureau of Standards announced that he had discovered a crystal whose patterns at the atomic level seemed to show fivefold symmetry similar to Penrose Tiles and its pattern was non-repetitive. However, at that time, scientists had assumed that arrangement of atoms in a crystal must have a repetitive pattern. After announcing his discovery of crystals against the established scientific belief, Shechtman was asked to leave his position. Nevertheless, he was able to publish his paper in 1984.

In 1987, Shechtman’s discovery was verified by scientists using X-rays. He was awarded the 2011 Nobel Prize in chemistry. The announcement of his Nobel Prize, said, “Aperiodic mosaics, such as those found in the mediaeval Islamic mosaics of the Alhambra Palace in Spain and the Darb-i-Imam Shrine in Iran, have helped scientists understand what quasicrystals look like at the atomic level”.

Muslims in Islam’s golden age knew that a surface cannot be tiled with just five-sided tiles. Why then were they not content with designing elegant patterns using tiles with three, four or six sides whose patterns are periodic, that is, whose next patterns are predictable? Why was it important for them to use complex mathematics for tilings with fivefold symmetry and which were aperiodic, ie their next pattern could not be predicted? Perhaps, to them, fivefold symmetry represented Islam’s five pillars of faith and the five daily prayers. And, perhaps aperiodic tiling represented human beings’ limited ability, or even an inability, to predict the future which was in the realm of God.

Designing tiles which were in harmony with their faith was imperative, even if it required the use of complex mathematics. It was possible in that era because two-thirds of the world’s most renowned mathematicians who lived between 650AD and 1300AD were Muslims. Such symbiosis between faith and learning existed for centuries when Muslims treated learning as an act of faith.

The writer is a finance professional based in the US.

Published in Dawn, July 10th, 2020