No science teaches the harmonies of nature more clearly than mathematics. In arithmetic we explore a universe of figures by counting; in geometry we discover another universe by drawing lines and laying down definite directions in the conceptual field of the imagination; in algebra we produce magnitudes of a still more abstract nature that are expressed by letters. In all of these cases the first step is to establish the general conditions that lay down the rules which all further steps are subject to. Every one of these “universes” is dominated by a consistency that produces an excellent symmetry.

Certainly the world, the universe, and nature can be reliably understood using mathematics. In other words, we can say that nature is mathematics. For instance, the arrangement of seeds in a sunflower can be understood using Fibonacci numbers. Every number in the Fibonacci sequence equals the sum of the two previous except for the first two numbers. Sunflower heads, like other flowers, contain two families of interlaced spirals—one winding clockwise, the other counterclockwise. The shape assumed by a delicate spider web suspended from fixed points, or the cross section of sails bellying in the wind, is a catenary—a simple curve defined by a simple formula. Seashells, animal horns, and the cochlea of the ear are logarithmic spirals that can be generated using a mathematical constant known as the golden ratio. Mountains and the branching patterns of blood vessels and plants are fractals, a class of shapes that exhibit similar structures at different magnifications. Einstein’s E = mc² defines the fundamental relationship between energy, matter, and the speed of light. Additionally, a few simple constants—the gravitational constant, Planck’s constant, and the speed of light—are profound examples for better understanding our universe as a whole.

Magic squares are conspicuous instances of the intrinsic harmony of numbers. They are also noticeable instances of the inherent harmony of numbers that some consider as an interpreter of the cosmic order that dominates all existence. Can they? Maybe. Though they are a mere intellectual play, they not only illustrate the nature of mathematics but also, incidentally, the nature of reality dominated by mathematical regularity. This is incidental, because its discovery has a very extraordinary story. Magic Squares are square grids with a unique arrangement of numbers in them. These numbers are exceptional because every row, column, and diagonal adds up to the same number. They are understood from every part in their overall square shape, the sum of which gives the same number – the sum of rows, columns, and diagonals. Magic Squares, one of the Sudoku-style question types, may vary in the numbers to be placed in the square according to its size. For example, let’s say there is a 3×3 square and ask: how can you put the numbers 1 to 9 each in their own square so that all of the the rows, columns, and cross sums are always the same? I recommend you stop here for a while and work on the question. You can see that it is not as easy as it sounds. Otherwise, the answer is in Figure 1.

The historical development of magic squares is also very fascinating. The earliest record of magic squares is from China in about 2200 BC and is called Lo-Shu. It tells that the first magic square was incidentally discovered by the Chinese emperor Yu 4,200 years ago. Emperor Yu discovered a tortoise while sauntering along the Yellow River (Lo River). The tortoise’s shell had a magic square shape, and this square divided into grids contained dots instead of numbers. Imagine having the same quantity as spots instead of numbers (Figure 2).

According to the myth, Emperor Yu took the tortoise and kept caring for it as a valuable guest in his palace. Its fame spread all over the world and extends beyond time to today’s modern world. Perhaps, it deserves the title of the world’s most famous mathematical tortoise with an outstanding mark on him. After the tortoise, the magic squares have become a very significant mystical symbol in China and in the Karma philosophy. In Karma, it is said that the world is surrounded by four essential elements and this is symbolized with magic squares. If we consider the number 5 in the center as the world, the followers of this philosophy surrounded the world with numbers that were supposed to be the four elements of Yin Yang. 4 and 9 symbolize metal, 2 and 7 symbolize fire, 1 and 6 symbolize water, and 3 and 8 signify wood, as shown in Figure 3.

We see that magic squares have mystical meanings not only in China but also in many places. For example, in pre-Islamic Arabia, people hung magic squares in their homes as amulets that they believe protected their houses from evil spirits. One explanation to this superstition could be that Arab astrologers used magic squares to cast horoscopes.

Magic squares were introduced to the Western world in the work of Theon of Smyrna. Theon’s most important work is “Expositio rerum mathematicarum ad legendum Platonem utilium.” This work is a handbook for philosophy students to show how prime numbers, geometrical numbers such as squares, progressions, music, and astronomy are interrelated. Its rather curious title means that it was intended as an introduction to a study of the works of Plato, which sounds rather fanciful. In the section on numbers, Theon adopts a Pythagorean approach and writes about all kinds of numbers: odd, even, prime, composite, square, oblong, triangular, polygonal, circular, spherical, solid numbers with three factors, pyramidal, perfect, deficient, and abundant numbers. The work of the Greek mathematician Moschopoulos in 1300 AD helped to spread knowledge about magic squares. So here we are now, more than 700 years later, and teachers are using them in class for problem-solving and practicing addition.

In the 16^th century the Catholic physician, astrologer, and theologian Cornelius Agrippa (1486–1535) constructed squares of orders from 3, 4, 5, 6, 7, 8, and 9 which he associated with the seven known astrological planets: Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon (the Sun and Moon were considered to be planets at that time.) Agrippa had a colorful life that included various dangerous run-ins with the Church and jobs as an occult scholar, lawyer, and military strategist. Agrippa’s De Occulta Philosophia stimulated Renaissance study of magic and got his name into early Faust legends. Agrippa believed that a magic square containing the digit 1—which exhibits the magic constant of 1 in all directions—represented God’s eternal perfection.

I do not think that anyone goes beyond the Indian people in terms of expecting help from magic squares for healing diseases and even finding a spouse. They also took first place in producing magic squares known as the Jaina inscriptions. The first 4×4 square was discovered on a door in Khajuraho, India, around 1100 AD (Figure 4). Jainism is an Indian philosophy that was established in the 6th century BC, which may imply how important magic squares are for Indians who follow this tradition.

India brought the usage of these squares to daily life. One superstition still practiced today is to write a 3×3 magic square to find someone missing and hang it everywhere like a notice. Strangely, the name of the missing person is not in the notice! The magic square’s merits are not limited to this. It is believed that a couple can save their marriage if they can get a copy of a square that is drawn on a Wednesday or Friday.

For mathematicians, it makes sense if the sums of squares across each row, column, or diagonal are equal. But Indians have different expectations from the squares. For example, the square used by young girls to get married is shown in Figure 5. This square is painted on a plate with pastels. The plate is washed in the Ganges River. The girl who is seeking her desired husband drinks the pastel-colored water from the river. This tradition is still being practiced today.

There are also squares used by Muslims in the Islamic period. One of the most beautiful examples of these squares is one designed to express God’s name as shown in Figure 6. The square with 66 letters in total on each side evokes the sum of the Arabic word Allah in Abjad calculation (In Arabic, every letter is assigned a numerical value, and this calculation is called Abjad).

Throughout history, humanity has been influenced by mathematics and numbers and has tried to reveal the unknowns by investigating the mysteries of numbers. Humanity’s long-time fascination with mathematics has arisen because the universe is constructed from a mathematical fabric. In 1623, Galileo Galilei reinforced this belief by stating his credo: “Nature’s great book is written in mathematical symbols.” Plato’s doctrine was that God is a geometer, and Sir James Jeans believed God experimented with arithmetic. Newton supposed that the planets were originally thrown into orbit by God, but even after God decreed the law of gravitation, the planets required continual adjustments to their orbits. Likewise, magic squares have been one such mysterious dimension of mathematics that have influenced human life in various ways.

Figure 1 – 3×3 Magic Square

Figure 2 – Spots on the tortoise

Figure 3 – Five Elements in Magic Square

Figure 4. Magic square at the Parshvanatha temple, Khajuraho

Figure 5 — Magic Square Drawn on a Plate with Pastel Paint

Figure 6- The sum of Allah’s words with Abjad calculation is 66. The grid is formed by the letters in the word “Allah,” whose numerical value is also 66 (nineteenth century, Damascus).

Just as there is a perfect order that allows life to exist and function, there are incredible numerical symmetries and orders within the Qur’an. Exploring them can deepen our appreciation of both the Qur’an and the world.

The studies on numerical and gematrical codes in the Qur’an show great variety in their results. Some present such an approach as major evidence for the divine origin of the Qur’an, and others render such an approach meaningless or absurd. Such studies have produced interesting connections and verifiable facts, as well as unacceptable claims and abuses. This article is a meek attempt to point to a middle way between the extremes.

There are thirty distinct whole numbers that are mentioned in the Qur’an: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 19, 20, 30, 40, 50, 60, 70, 80, 99, 100, 200, 300, 1000, 2000, 3000, 5000, 50000 and 100000. The numbers that are made italic in the list occur only once. The eight distinct fractions used in the Qur’an are 1/10, 1/8, 1/6, 1/5, 1/4, 1/3, 1/2 and 2/3, some of which occur more than once. The Qur’an made general references to all even and odd integers (whole numbers) as stated in, “And [by] the even [number] and the odd…” (89:3). Besides the set of integers that define what are called countable numbers, it may even be possible to find general references to uncountable numbers in the Qur’an (namely the reals, which also include the rationals as a subset): “And should you attempt to count God’s blessings, you could not compute them. God is indeed All-Forgiving, All-Compassionate” (16:18).

The Arabic gematrical value system, also known as /abjad/, assigns a whole number to every letter in the Arabic alphabet [1]. It was in use long before Islam, and has been used by many scholars and saints after Islam. There are essentially six /abjad/ systems: major, big, small (standard), minor, marked, and unmarked. The practice of predicting future events using /abjad/ systems on sacred texts is sometimes called /hasab al-jifr/, /hasab al-jummal/, or gematria. The numerical values for the standard /abjad/ system are given as follows:

Before discussing the numerical and gematrical codes in the Qur’an, it may be worthwhile to recall another mysterious aspect of whole numbers. Associated with most small integers, there is some symbolism that is often made use of in Sufi traditions. In order to understand the basis of mystic symbolism, the verses where these numbers occur should be studied carefully. In addition, the Arabic letter corresponding to each number in the gematrical (/abjad/) system should be considered. The verse 2:196, for example, states “these (make) ten (days) complete,” suggesting a sense of completeness and maturity for the number 10, as the word /kamilah/ in the original text denotes such meanings. Incidentally, the Arabic letter /ya/ has the /abjad/ value of 10; it is the final letter of the Arabic alphabet and it completes the verse. Likewise, the number 3 is seen to indicate gathering and strength. It is mentioned in a tradition [2] as the minimum number for a group of people to form a convoy or congregation. We also read verse 36:14 with a similar reference: “When We sent to them two but they denied them, so We strengthened them with a third, and they said: Indeed, we are messengers to you.” On the other hand, the letter /jim/ has the /abjad/ value 3, and is the first letter of the root verb /j-m-a/ which gave rise to most words indicating congregation and gathering, such as /jumua/, /jamaah/, etc. Most small integers are associated with a similar numerological symbolism.

In all creation, we see clear signs of precise measures and delicate proportions at the level of perfection. There are no faults or flaws in creation at all. If the created book of the universe presents such delicate measures, why should not the spoken book of the Qur’an be studied in this respect? Most meticulous scholars, indeed, have studied the wording of the Qur’an and the proportionality in its word composition, and have discovered many remarkable mysteries. In a (weak) tradition we read the Jewish scholar Huyayy bin Akhtab suggesting to the Prophet Muhammad, peace and blessings be upon him, that the life of his community will be short by pointing out the isolated letters /alef-lam-mim/ in the beginning of chapter al-Baqarah [3]. These letters add up to the value 71. The Prophet smiled and told him there are more like that, and recited /alef-lam-mim-sad/ (adds up to 161), and /alef-lam-ra/ (adds up to 231), /alef-lam-mim-ra/ (adds up to 271). This incident shows that the Prophet did not reject the validity of such calculations outright, and cited a few more examples for his listener.

We also find references in the Qur’an to the divine act of enumerating and appraisal. The verse 78:29 states, “But all things We have enumerated in writing [in a book],” and the verse 72:28 reads “and He has encompassed whatever is with them and has enumerated all things in number.” The enumeration of all things is related to the divine name al-Alim, the All-Knowing, and issues from the name al-Muhsiy, which means the Accountant, the Appraiser, and the Numberer of All. It is, therefore, a form of reflection to study numerical treasures of the Qur’an provided that one does not study it as an end in itself, does not make a mockery of it, and does not forget the primary purpose of pursuing its healing and guidance. As in all matters, we should seek a balanced approach between the two extremes. It is not appropriate to consider numerical codes in the scripture as the main evidence for its truths, and give too much weight to it. On the other hand, it is not proper to reject it, and claim it null and meaningless.   

It is known that there is a natural 19 year cycle between the moon and the sun. That is, the full moon appears at the same position in the sky for approximately 19 years [4]. We remarkably find, for example, that the words “sun” and “moon” have been mentioned together exactly 19 times in the Qur’an. It turns out that other miraculous numerical codes in regards to the number 19 exist. The first chapter revealed as a whole (chapter 1, the Opening) has 19 letters in its first verse, the /basmalah/. Likewise, the last revealed chapter (chapter 110, the Help) also has 19 letters in its first verse. (We note that /basmalah/ is considered as a verse from the Qur’an only in the beginning of the 1st and the middle of the 27th chapters.) It is interesting to also note that the first revelation (the first five verses of chapter 96, the Clot) has 19 words. In addition, it is the 19th chapter from the end of the Qur’an in the sequential ordering. The last revealed chapter (chapter 110, the Help) also has 19 words in it. Let me cite one more example about the number 19: the fundamental pillar of Islamic faith is the unity of God. The number of times the word /wahid/ (which means one) is used to refer to the unity of God is 19. Incidentally, the /abjad/ value of the word /wahid/ is also 19.

Other numbers besides 19 have similar mysterious connections. For example, the number of distinct letters used in the first chapter (al-Fatihah, the Opening) and the last chapter (al-Nas, the Mankind) with respect to the sequential ordering in the Qur’an are both 22. In addition, both chapters have 7 verses counting the /basmalah/. Chapter 112, titled Sincerity of Faith, describes the divine attribute of unity. It has 11 letters in the first verse, which matches the /abjad/ value of the divine name /huwa/ (He) that refers to the Divine Being alone, without any consideration of other names or attributes.

Even though there are different views among scholars, the sequential ordering and the names of all chapters in the Qur’an are commonly believed to be given by divine revelation. The lengthier chapters are usually located in the beginning and the shorter ones are towards the end; however their sequential ordering was not determined based on their lengths or chronological order, but the revelation. [5] We point out a few marvels in this regard to illustrate further evidence for the divine ordering and naming of the chapters of the Qur’an. The chapter titled the Night (/al-Layl/) is the 92nd chapter of the Qur’an. The word “night” is repeated 92 times in all of the Qur’an, including both singular and plural forms (/layl/ and /laylah/). For another example, consider the gematrical value of 57 for the word /al-hadid/, the iron. The chapter named al-Hadid, the Iron, turns out to be the 57th chapter in the Qur’an. The /abjad/ value of iron in the indefinite form without the article “the” in the beginning (that is, just the word /hadid/) is 26, which turns out to be precisely the atomic number of iron. In addition, the word “iron” occurs only once in that chapter, which is in the 26th verse. Such marvels of the Qur’an are so myriad that it is not possible to advance any arguments in favor of chance and coincidence.

A different type of numerical code found in the Qur’an is the distance between certain words: the distance that is measured in terms of the number of words or letters between two given words. In the story of “the people of the cave” in chapter al-Kahf, for example, the duration of their sleep in the cave is given to be 309 years (18:25). It turns out that their story in the Qur’an is narrated in 309 words. That is, there are precisely 309 words between the two words /labithu/ (they stayed) that occur in verses 18:12 and 18:26 from the beginning of the narration to the end, including these two words and counting the conjunction word /wa/ (and) as a separate word.

The letter statistics and word statistics in the Qur’an show interesting proportions. We see, for example, that the word /shahr/ (month) is repeated 12 times in its singular form. The words /rajul/ (man) and /imraah/ (women) are repeated 24 times each. Similarly, we find the words /malak/ (angel) and /shaitan/ (devil) repeated 88 times each. Even a modest list of such relations contains numerous items; we wanted to indicate only a few cases here.     

There are 29 chapters in the Qur’an starting with disjointed, isolated letters called /huruf-u muqattaa/. They are believed to be miraculous ciphers [6] that contain many secrets: They point out certain future events, and have relations to the invisible realm of existence, the overall content of the chapter, and the divine names of God. These letters also present many interesting numerical features. The letter /qaf/, for example, appears as a disconnected letter in the beginning of two chapters, namely 42 (al-Shura, the Consultation) and 50 (the Qaf). It is amazing to note that the letter /qaf/ is repeated 57 times in both these chapters. This is indeed remarkable as chapter 42 is more than twice as long as chapter 50.

Bediuzzaman Said Nursi listed several miraculous examples where a phrase or a verse in the Qur’an contains references to certain significant events in our times through its /abjad/ value. He, for example, explained that the phrase /tarmihim bihijaratin/, “striking them with stones” in verse 105:4 points out to the use of warplanes for the bombings of World War II that wiped away much of Europe [7]. The /abjad/ value of that phrase turns out to be 1359, which gives the year 1941 when converted to solar calendar years. Chapter 105, the Elephant, gives a brief account of the people of the elephant, and how their strong army was utterly destroyed by flocks of birds carrying stones in their beaks and talons. The content of the chapter as well as its wording point to the mass killings during World War II. Nursi’s treatise, called the First Ray, lists several other instances where the /abjad/ value of a verse foretold the date of a future event in a figurative way.

The geometric features of the beautiful /mushaf/ (the scrolls of the Qur’an) inscribed by Ahmed Husrev Efendi, a student of Bediuzzaman, should also be mentioned here [8]. Bediuzzaman Said Nursi noticed that the divine names “Allah” are all perfectly aligned beneath one another (called /tawafuq/) in all pages of Qur’an scrolls inscribed by Hattat Hafiz Uthman in the 19th century. Hafiz Uthman took the longest verse (/ayat al-mudayanah/, the verse of loaning, 2:282) as the size for one page and used chapter 112 for the size of one line, and thus inscribed the entire Qur’an in a beautiful form in such a way that none of the 6000 plus verses will extend into the next page. However, the features Nursi pointed out were not noted before his time. Bediuzzaman asked his students to write a new copy where such features are made manifestly clear, which resulted in the lovely /mushaf/ we read today. Not only the name Allah, but other divine names, and the names of the prophets, as well as certain ordinary words that are derived from the same root, are lined up perfectly, with a few exceptions.

I would like to conclude by pointing out again that these extraordinary numerological marvels should not be seen as an end in itself, and should not be studied just for the sake of information or for amusement: “Indeed, the Qur’an is a decisive statement; and it is not a thing for amusement (merriment or pleasantry)…” (86:13-14). Such numerical marvels should be seen as an invitation to study the sublime wisdom and deep meanings of the Qur’an. In addition, the few instances listed here make it clear that altering any parts of the divine writ, its wording or ordering, would break the codes in their respective chapters, or the entire scripture. The Qur’an is the final testament, and all its features are determined by divine revelation, and are preserved under divine protection as it was originally revealed.

References

1. Yakit, I. 2003. Abjad Calculation and Date Deduction in Turkish-Islamic Culture.
2. Abu Dawud, Jihad, 79.
3. Tafsir ibn Kathir, al-Baqarah, Verse 2:1.
4. Meeus, Jean. 1997. Mathematical Astronomy Morsels, Willmann-Bell.
5. Yazır, Elmalili Hamdi. 1979. Hak Dini Kur’ân Dili, Istanbul: Eser Yayınları, IX, 6431.
6. Nursi, Bediuzzaman Said. 2010. The Words, 25th Word, NJ: The Light, Inc. p. 395.
7. —. Risale-i Nur Külliyatı, Sikke-i Tasdik-i Gaybi, p. 55.
8. —. 2010. The Rays, 11th Ray, NJ: The Light, Inc.