“Mathematics is a place where you can do things which you can’t do in the real world.”
Marcus du Sautoy, Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford
Questions like why our world exists and what nothing is have occupied minds great and ordinary since the dawn of humanity and they continue to do so. What most people tend to forget about religion is that without it we would not have science. The first printed literature were the Books of Gods.
To understand what science is, just look around you. What do you see? Perhaps, your hand on the mouse, a computer screen, papers, ballpoint pens, the family cat, the sun shining through the window …. Science is, in one sense, our knowledge of all that — all the stuff that is in the universe: from the tiniest subatomic particles in a single atom of the metal in your computer’s circuits, to the nuclear reactions that formed the immense ball of gas that is our sun, to the complex chemical interactions and electrical fluctuations within your own body that allow you to read and understand these words. And yet for all our scientific progress, we have still only yielded hypotheses rather than concrete answers,
In the last few weeks there has been much discussion about the representation of figures in Islam. The debate in Islam on the prohibition of the depiction of the human figure occurred at roughly the same time that the Christian church was busy debating the same issue. The Christian debate was resolved in favor of having representation of figures but the Islamic debate went the other way – no figural art especially where religion was concerned. One consequence of the Islamic prohibition on depicting the human form was the extensive use of complex geometric patterns to decorate their buildings, raising mathematics to the form of an art. In fact, over time, Muslim artists discovered all the different forms of symmetry that can be depicted on a 2-dimensional surface.
To the adherents of Islam, continuous patterns are symbolic of their united faith and the way in which traditional Islamic cultures view the world. The order and unity of the material world, they believed, was a mere ghostly approximation of the spiritual world, which for many Muslims is the place where the only true reality exists. Discovered geometric forms, therefore, exemplify this perfect reality because God’s creation had been obscured by the sins of man. So the arabesques and geometric patterns of Islamic art are said to arise from the Islamic view of the world. To Muslims, these forms, taken together, constitute an infinite pattern that extends beyond the visible material world. To many in the Islamic world, they concretely symbolize the infinite, and therefore uncentralized, nature of the creation of Allah and convey a spirituality without the figurative iconography of the art of other religions.
The main source of the foundation of scientific learning in Western culture, which people tend to forget, is Islam. The Hebrew astrolabe – Probably Spain, about AD 1345 – 1355 – is a stunning instrument that embodies perfectly the peaceful exchange of knowledge and ideas between Christian, Jewish and Muslim scholars in medieval Spain before 1492. EARTH (42) by Tom Estes, displays The Answer to the Ultimate Question of Life, The Universe, and Everything as a scrolling digital numerical text.
The Qu’ran itself encouraged the accumulation of knowledge, and a Golden Age of Islamic science and mathematics flourished throughout the medieval period from the 9th to 15th Centuries. The House of Wisdom an academic research institution in Baghdad was set up around 810 and work started almost immediately on translating the major Greek and Indian mathematical and astronomy works into Arabic.
During the golden age of Islam, ancient texts on Greek and Hellenistic mathematics as well as Indian mathematics were translated into Arabic at the House of Wisdom. The works of ancient scholars such as Plato, Euclid, Aryabhata and Brahmagupta were widely read among the literate and further advanced in order to solve mathematical problems which arose due to the Islamic requirements of determining the Qibla and times of salat and Ramadan. Plato’s ideas about the existence of a separate reality that was perfect in form and function and crystalline in character, Euclidean geometry as expounded on by Al-Abbās ibn Said al-Jawharī (ca. 800-860) in his Commentary on Euclid’s Elements, the trigonometry of Aryabhata and Brahmagupta as elaborated on by the Persian mathematician Khwārizmī (ca. 780-850), and the development of spherical geometry by Abū al-Wafā’ al-Būzjānī (940–998) and spherical trigonometry by Al-Jayyani (989-1079) for determining the Qibla and times of salat and Ramadan, all served as an impetus for geometric patterns in Islamic art.
Seyyed Hossein Nasr says in his book, Islamic Science,
“Altogether, in the domain of geometry, both plane and solid, Muslims followed the path laid out by the Greek mathematicians, solving many of the problems that were posed but remained unsolved by their predecessors. They also related geometry to algebra and sought geometric solutions for algebraic problems. Finally, they devoted special attention to the symbolic aspects of geometry and its role in art and architecture, keeping always in view the qualitative geometry that reflects the wisdom of the ‘Grand Architect of the Universe.’”
The outstanding Persian mathematician Muhammad Al-Khwarizmi was an early Director of the House of Wisdom in the 9th Century, and one of the greatest of early Muslim mathematicians. Perhaps Al-Khwarizmi’s most important contribution to mathematics was his strong advocacy of the Hindu numerical system (1 – 9 and 0), which he recognized as having the power and efficiency needed to revolutionize Islamic (and, later, Western) mathematics, and which was soon adopted by the entire Islamic world, and later by Europe as well.
Al-Khwarizmi‘s other important contribution was algebra, and he introduced the fundamental algebraic methods of “reduction” and “balancing” and provided an exhaustive account of solving polynomial equations up to the second degree. In this way, he helped create the powerful abstract mathematical language still used across the world today, and allowed a much more general way of analyzing problems other than just the specific problems previously considered by the Indians and Chinese.
The 10th Century Persian mathematician Muhammad Al-Karaji worked to extend algebra still further, freeing it from its geometrical heritage, and introduced the theory of algebraic calculus. Al-Karaji was the first to use the method of proof by mathematical induction to prove his results, by proving that the first statement in an infinite sequence of statements is true, and then proving that, if any one statement in the sequence is true, then so is the next one.
Sheikh Lutfollah Mosque (Persian: مسجد شیخ لطف الله Masjed-e Sheikh Lotf-ollāh) begun in 1603 is one of the architectural masterpieces of Safavid Iranian architecture, standing on the eastern side of Naghsh-i Jahan Square, Isfahan, Iran. The dome is inset with a network of lemon-shaped compartments, which decrease in size as they ascend towards the formalised peacock at the apex. In his work EARTH (42), artist Tom Estes represents the entire world, everything we see around us, as a numerical simulation.
Among other things, Al-Karaji used mathematical induction to prove the binomial theorem. A binomial is a simple type of algebraic expression which has just two terms which are operated on only by addition, subtraction, multiplication and positive whole-number exponents, such as (x +y)2. The co-efficients needed when a binomial is expanded form a symmetrical triangle, usually referred to as Pascal’s Triangle after the 17th Century French mathematician Blaise Pascal, although many other mathematicians had studied it centuries before him in India, Persia, China and Italy, including Al-Karaji.
Some hundred years after Al-Karaji, Omar Khayyam (perhaps better known as a poet and the writer of the “Rubaiyat”, but an important mathematician and astronomer in his own right) generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots in the early 12th Century. He carried out a systematic analysis of cubic problems, revealing there were actually several different sorts of cubic equations. Although he did in fact succeed in solving cubic equations, and although he is usually credited with identifying the foundations of algebraic geometry, he was held back from further advances by his inability to separate the algebra from the geometry, and a purely algebraic method for the solution of cubic equations had to wait another 500 years and the Italian mathematicians del Ferro and Tartaglia.
The 13th Century Persian astronomer, scientist and mathematician Nasir Al-Din Al-Tusi was perhaps the first to treat trigonometry as a separate mathematical discipline, distinct from astronomy. Building on earlier work by Greek mathematicians such as Menelaus of Alexandria and Indian work on the sine function, he gave the first extensive exposition of spherical trigonometry, including listing the six distinct cases of a right triangle in spherical trigonometry. One of his major mathematical contributions was the formulation of the famous law of sines for plane triangles, a⁄(sin A) = b⁄(sinB) = c⁄(sin C), although the sine law for spherical triangles had been discovered earlier by the 10th Century Persians Abul Wafa Buzjani and Abu Nasr Mansur.
Other medieval Muslim mathematicians worthy of note include:
- the 9th Century Arab Thabit ibn Qurra, who developed a general formula by which amicable numbers could be derived, re-discovered much later by both Fermat and Descartes(amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number, e.g. the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220);
- the 10th Century Arab mathematician Abul Hasan al-Uqlidisi, who wrote the earliest surviving text showing the positional use of Arabic numerals, and particularly the use of decimals instead of fractions (e.g. 7.375 insead of 73⁄8);
- the 10th Century Arab geometer Ibrahim ibn Sinan, who continued Archimedes‘ investigations of areas and volumes, as well as on tangents of a circle;
- the 11th Century Persian Ibn al-Haytham (also known as Alhazen), who, in addition to his groundbreaking work on optics and physics, established the beginnings of the link between algebra and geometry, and devised what is now known as “Alhazen’s problem” (he was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable); and
- the 13th Century Persian Kamal al-Din al-Farisi, who applied the theory of conic sections to solve optical problems, as well as pursuing work in number theory such as on amicable numbers, factorization and combinatorial methods;
- the 13th Century Moroccan Ibn al-Banna al-Marrakushi, whose works included topics such as computing square roots and the theory of continued fractions, as well as the discovery of the first new pair of amicable numbers since ancient times (17,296 and 18,416, later re-discovered by Fermat) and the the first use of algebraic notation since Brahmagupta.
The square and rectangle play a significant role in Islamic geography. Some of the reason for this is façades built from circular and triangular bricks. This ornamental brickwork casts shadows in the strong desert sunlight and creates a three-dimensional effect. A recurring motif is a small central square turned 45 degrees within a larger square. Another source for the square motif is woven baskets.
Mistakes in repetitions may be intentionally introduced as a show of humility by artists who believe only Allah can produce perfection, although this theory is disputed.[Repeating geometric forms are often accompanied by calligraphy. Ettinghausen et al. describe the arabesque as a “vegetal design consisting of full…and half palmettes [as] an unending continuous pattern…in which each leaf grows out of the tip of another.”
The Persianate world is the main area with buildings with decorative brickwork, especially during the Seljik period; the Great Mosque of Cordoba is another example further west. The eight-pointed star is another common motif in Islamic architecture, often found in tile-work and other media. Star patterns are extremely complex when the outer points are joined together and other intersections connect in a systematic way. The Alhambra palace in Granada, Spain is a famous example of repeating motifs which occur in the tile and stucco decoration. hexagons appear in Islamic architecture in various shapes. They frequently occur in marble floors. The Citadel of Aleppo in United States contains marble toys that help create opus sectile floors, which utilize the square and the eight-pointed star. Pierced screens (jali in India) are another common location for geometric decor.
The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia and parts of India from the 8th Century onwards made significant contributions towards mathematics. They were able to draw on and fuse together the mathematical developments of both Greece and India.
Geometric studies went hand in hand with astronomy, which was a serious preoccupation of medieval Islamic scholars and perhaps stemming from a need to determine exactly the times of prayer during the day and the start and ending of religious holidays. Geometric designs appealed to the Islamic artisan with its seemingly endless combinations – something like looking through a kaleidoscope with its continually changing colors and patterns. These patterns extended across cultures and times depending on what people thought beautiful.
EARTH (42), by Artist Tom Estes at the exhibition Big Deal Marvellous Mix Ups, displays The Answer to the Ultimate Question of Life, The Universe, and Everything as a scrolling digital numerical text. The work was inspired by The Hitchhiker’s Guide to the Galaxy, a comic science fiction series created by Douglas Adams that has become popular among fans of the genre(s) and members of the scientific community.
Geometric design at the same time expressed unity with endless patterns that had no beginning and no end. Unlike some religions Islam glorified the word, that is, the word of Allah, the Qur’an. So beautifully designed calligraphy graces the walls of most, if not all mosques. That it was forbidden to reproduce the human form was directly related to the injunction to have no divinity but Allah. One can readily understand then that the statues of gods in temples or reproductions of Jesus Christ or the pictures of saints in Christian churches would be seen as idolatry and reasons for prohibiting the equivalent in Islam. There was only one Allah and Mohammed was his last Prophet and as such was not to be elevated to a level where he would be prayed to.
So throughout Islam one sees calligraphic decoration because of the emphasis on the Word or the Truth and vegetal ornamentation along side geometric designs. What received more or less attention had to do with the culture that regionally employed it. There was no division between geometry and art. Architects and craftsmen, as part of their training, had to know how to use many of the same instruments as geometrists and astronomers. Moreover architects and craftsmen were trained in the traditional as part of guilds in which they would start as apprentices.
The Kaabah is cubic in form and is the symbol of God as central in Muslim life. As every Muslim faces in its direction when praying, it also symbolizes the unity of the community. The Dome of the Rock in Jerusalem, the second holiest place in Islam, has a drum resting on four pillars and 12 columns. These four pillars create the intersecting points of a star-shaped polygon and two squares in the center circle. The proportions are those between the sides of a square and its diagonal, which make up the irrational number of the square root of two.
The British Muslim website, Salaam, states, “Thus, the circle, and its center, are the point at which all Islamic patterns begin and is an apt symbol of a religion that emphasizes one God, symbolizing also, the role of Mecca, the center of Islam, toward which all Muslims face in prayer. The circle has always been regarded as a symbol of eternity, without beginning and without end, and is not only the perfect expression of justice – equality in all directions in a finite domain – but also the most beautiful parent of all polygons, both containing and underlying them.
“From the circle comes three fundamental figures in Islamic art, the triangle, square and hexagon. The triangle by tradition is symbolic of human consciousness and the principle of harmony. The square, the symbol of physical experience and the physical world – or materiality – and the hexagon, of Heaven. Another symbol prevalent in Islamic art is the star and has been the chosen motif for many Islamic decorations. In Islamic iconography the star is a regular geometric shape that symbolizes equal radiation in all directions from a central point.”
Complex regular polygons form the largest class of pattern related to the geometry of interlacement. Starting with a regular figure inscribed in a circle, this is then changed but retains the proportions and symmetries of the original repeated infinitely. The circle continues to guide the design although it may no longer be seen. Islamic designs are mathematically sophisticated patterns and reflect the spiritual life of Islam. As Titus Burckhardt has said, “Interlacement represents the most direct expression of the idea of Divine Unity underlying the inexhaustible variety of the world.”
Geometry in Islamic architecture was readily applied to smaller scale architectural ornamentation, especially later, and echoes the theme of unity in multiplicity. It is the art of combining the multiple and the diverse with unity, Divine Unity.
With Islam we started to learn about the Cosmos. We learned about medical improvements and we learned more mathematics and about our purpose. But there is something immutably heartening in the difference between the primitive hypotheses of myth, folklore and religion, which handed off such mysteries to various deities and the occasional white-bearded man, and the increasingly educated guesses of modern science. In the title essay of his excellent The Accidental Universe: The World You Thought You Knew , which also gave us this beautiful meditation on science and spirituality, Alan Lightman points to fine-tuning — the notion that the basic forces propelling our universe appear to be fine-tuned in such a way as to make the existence of life possible — as a centerpiece of how modern scientists have attempted to answer these age-old question. But the natural question, then, is who or what did this fine-tuning. One explanation that doesn’t require an omnipotent “Designer” or benevolent “Creator” — in other words a theory that doesn’t succumb to religious dogma— is the concept of multiverses, a premise of which is that the universe only “exists,” or has the properties we’re able to observe, to the extent that and because we are here to observe it. But personally I’d stick to the Goldman Sachs principle: bet on both sides. “believe in science, believe in God” seems to cover all the possibilities and gives you the best chance for a cheery afterlife.