Pure maths: algebra and Islam

From the perfect symmetry of a dome to the curvature of a parabola, the application of mathematics is evident in much of the architecture of the Islamic world. Rob Flemming traces the history of the advances made in mathematics by Islamic scholars, and their legacy in both the study and the practice of modern maths

The Oxford English dictionary defines mathematics as “the abstract deductive science of number, quantity, arrangement and space”. Which might itself appear to be a phrase of abstraction with too many variables. But mathematicians revel in testing those relative juxtapositions, praising aesthetic outcomes and the elegance of proof.

Standing in a mosque one might gaze in awe at the infinite tracery and patterns of multi-coloured tiles that decorate the perfect symmetry of the dome. We might wonder how man could build such a place but rarely consider the ellipse of the dome in relation to its base circumference or the angle of incidence of ray to tile – considerations that were of great importance to the medieval Islamic scholars: the search for synthesis between art and science, to reflect the spiritual by the temporal.

Religious observance

Hand-in-hand with this search for synthesis came the specific application of mathematics for religious purposes. Calculations were needed for religious observance such as determining the Qibla (the direction faced when a Muslim prays) and the positions of the sun from pre-dawn to dusk to calculate the times for the Salah (formal prayers). In his essay The Principles of Islam, the philosopher Dr Seyyed Hossein Nasr suggests that the abstract nature of mathematics appealed to Muslims, who saw it as a bridge between multiplicity and unity.

“It [mathematics] provided a fitting texture of symbols for the universe; symbols that were like keys to open the cosmic text… However important its uses may have been in calendarial work, in irrigation, in architecture, its ultimate aim has always been to relate the corporeal world to its basic spiritual principle, through the knowledge of those symbols which unite the various orders of reality. It can only be understood, and should only be judged, in terms of its own aims and its own perspectives.”

Grasping the philosophical aspects of this search for knowledge may be difficult; the application of mathematical sciences and their tangible outcomes are far more accessible. To a point. The concept of a rectangular area measuring 15 metres by 20 metres is reasonably easy to visualise – it’s a rectangle that might represent the measurements for a floor. Understanding the fact that the ratio between width and length is 1:1.33 is not too bad until that has to be applied to every other measurement in the building. When one learns that the applied ratio in most Islamic architecture was 1:√2, which algebraically might look like x(√2y) and that the square root of two is an irrational number, it might be time to take a deep breath.

The advances made in mathematics by Islamic scholars between the 7th and 14th centuries were truly extraordinary, bringing new concepts and developments to the various disciplines of the science.

This Golden Age truly began with the studies of Muhammad ibn Musa Al-Khwarizmi in the late-8th century at the House of Wisdom in Baghdad under the patronage of the sixth Caliph of the Abbasid dynasty, al-Ma’mun. Al-Khwarizmi’s  Hisab al-jabr w’al-muqabala, from which the word algebra derives, is considered to be the first book written on the subject. His treatise on Hindu-Arabic numerals, the Hindu Art of Reckoning, expanded on the use of the numbers one through to nine and introduced zero as a placeholder in notations. ‘Algorithm’ is a derivation of his name. Al-Khwarizmi’s discoveries in algebra were the springboard that enabled his successors’ great progression.

Solutions to problems

Simplistically, algebra was a way of finding solutions to problems. But its importance lay in its power of unification, to allow geometry, trigonometry, rational or irrational numbers, quantity and volume to all be treated as algebraic objects. It even allowed mathematics to be applied to itself.

Only 40 years later, al-Mahani devised the concept of reducing geometrical problems such as duplicating the cube to ones that were algebraic. More than 100 years later, al-Karaji managed to strip away the geometrical operands that still clung to the early written forms, replacing them with the arithmetical types of operation that are still current in today’s algebra.

Algebraic studies coincided with the advances made in numerical methodology. Al-Kashi, a later scholar, made a significant contribution in the use of decimal fractions, approximating both algebraic and real numbers. Aspects of his treatise, The Key to Arithmetic, leads us back to the specific use of maths in relation to architecture. For example, he used decimal fractions to calculate the total surface areas of muqarnas, (decorative corbels used to disguise edges in Islamic architecture). Often appended to domes or vaulted ceilings, the muqarna resembles a stalactite composed of three-dimensional polygons.

Although Islamic scholars are most noted for their contributions to algebra, just as important were the advances they made in more overtly practical disciplines such as geometry and trigonometry. Al-Haytham made a serious study of optics. In addition to poetry, Omar Khayyam is known for combined trigonometry with approximation theory and the application of algebra to solve cubic equations.

The need to make astronomical calculations was a prime driver for many of these mathematicians; thus the need for in-depth studies of line and curve in relation to the stars, sun, moon and earth.

Specific calculations were needed for religious observance so that Muslims could find the direction of the Qibla and calculate exact prayer times. While geometry encompassed length, area and volume in squares, circles, triangles and cylinders, trigonometry looked at the relationships between the sides and angles of triangles, the curvature of parabolas and ellipses. This, in turn, led to the development of measuring tools to include the quadrant, sextant and spherical astrolabe.

The results of the synthesis of all these aspects can clearly be seen in the great 15th-century scholar and astronomer Ulugh Beg’s Observatory in Samarkand (in what is now Uzbekistan). Under Beg’s direction, the team of scientists at his Observatory produced accurate approximate solutions for cubic equations, accurate tables of sines and tangents, formulae for spherical trigonometry and a comprehensive catalogue of the stars.
Sacred aspect

Science and the study of natural phenomena were considered to be linked to the concept of Tawhid, the Oneness of God. The pursuit of the mathematical sciences therefore had a sacred aspect that pointed to the Divine. The application of pure mathematics, algebra, geometry and trigonometry together with their sub-disciplines are clearly in evidence in Islamic architecture. One starts to comprehend the principles of the use of algebraic formulae to calculate the parabola of a dome or to achieve the symmetrical ratios of column to pier to arch and niche. Dr Nasr brings those ideas simply together.

“The arts and sciences in Islam are based on the idea of unity, which is the heart of the Muslim revelation,” he says.

In as much as algebra and mathematics might be perceived as reflecting the Divine in architecture, the reverse is equally true. Dispossessed of its material form, a structure translates to a multiplicity of simple and complex equations in which there is clear unity. Pure maths.