Buduh the Arabic Magic Square

Buduh:

Magic square in Islamic Civilization

The BUDUH tradition is believed to predate the Quran. Among its other attributes, the Islamic luoshu is credited with insuring that one finds love; helping one secure a mate; preventing childhood fears; curing headaches, stomach ailments, fevers, and epilepsy; preventing theft, attacks by bandits, and poisonings; and helping one find lost objects. These different applications may have required writing the square on a specific surface: a part of the human body, for example, the forehead, palms, or fingernails; using a specific substance, such as blood, to draw the square, and going through a physical ritual or performing an accompanying chant.

Variations of this square have served Muslims as religious mandalas, meditative devices, and occult talismans and amulets. The square’s magic sum of 15, which for the Chinese represented “human being perfected,” has been culturally transformed in the Islamic context to reflect the phrase “O man” or “O Perfect Man” referring to the Prophet Muhammad. A configuration formed from the square’s central row and column—a cross of the odd numbers—is considered a harbinger of bad luck or evil, whereas a configuration of the four corner numbers—the even numbers—is thought to be a powerful talisman for good luck.

So popular was this magic square that the name buduh itself was assigned talismanic properties. In subsequent years Islamic writers developed a variety of methods for forming larger magic squares, in which no numeral was repeated and the sums of each row and each column and the two diagonals were the same. Magic squares with cells 4x4 or 6x6 or 7x7 were particularly popular, with 10x10 squares being produced by the 13th century. By the 19th century 100x100 magic squares, with 10,000 individual cells, were being produced.

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Magic squares were known to Islamic mathematicians in Arabia as early as the seventh century. They may have learned about them when the Arabs came into contact with Indian culture and learned Indian astronomy and mathematics – including other aspects of combinatorial mathematics. Alternatively, the idea may have come to them from China.'

Jabir ibn Hayyan

The first appearance of a magic square (known in Arabic as wafq) in Islamic literature occurs in the group of writings attributed to Jabir ibn Hayyan   , known in Europe as Geber, and generally thought to have been compiled at the end of the 9th or early 10th century A.D. The magic square, given as a charm of easing childbirth in the Jabirean corpus, is thought to be of Chinese origin. It consisted of nine cells with the numbers 1 to 9 arranged with 5 in the center so that the contents of each row, column and the two diagonals added up to 15.

The numbers were written in the abjad letter-numerals, and because the four corners of this square contained the letters ba', dal, waw [or u], and ha', this particular square became known as the buduh square.

The Arabic letters for the four corners, if read from right to left, form the word BEDUH, the “word of power” as handed down by Adam. This word itself, or its abbreviations—just the letter B or the numerical equivalents 2, 4, 6, 8—are believed to be powerful talismans that protect travelers, babies, and postal letters and packages in transit. In some Islamic countries today, one finds packages with “2,” “4,” “6,” “8” written in their corners or postal letters bearing an extraneous “B” written under the address as added postal insurance.

Ikhwan As-Safa

The first magic squares of order 5 and 6 known to have been devised by Arab mathematicians appear in an encyclopedia from Baghdad circa 983, the Rasa'il Ikhwan al-Safa (the Encyclopedia of the Brethren of Purity) ;   simpler magic squares were known to several earlier Arab mathematicians.^[2] 

The Arabic term for a magic square is wafq al-a’dad, “the harmonious disposition of numbers,” and the Brethren considered their magic squares “small models of a harmonious universe.” Their magic square of order three is presented as the luoshu rotated 90 degrees clockwise.

Its construction is described in terms of chess moves—“First, two knight’s leaps, then a pawn step”—which reveals that, by this time, Islamic scholars had experimented with the structures of magic squares and had devised their own techniques for developing them.

However, an examination of the Rasa’il magic squares reveals no dominating technique for their construction but rather an eclectic mix of methods. Perhaps the most interesting and innovative square in this series is the one of order seven, a concentric bordered magic square.

Its author apparently took the middle numbers from the sequence 1 to 49, namely, 21 to 29, and arranged them into a magic square of order three which he then used as a core for further construction.40 Next, he chose the sequences 30 to 37 and 13 to 20 and paired elements of each to form complementary pairs whose sum was 50. He distributed these as a border around the central core.

Finally, he employed the remaining numbers in twelve complementary pairs to form the outer border. The resulting square was comprised of a core with two concentric borders—a magic square within a magic square within a magic square!

 After Ikhwan As-Safa

1. The versatile scientist-philosopher Thabit ibn Qurra (836–901) wrote on magic squares. Thabit ibn Qurra is known for his formula for amicable numbers.

2.  Ibn Sina (980–1037)  , an Isma’ili, known in the west as Avicenna and respected for his medical knowledge, used magic squares.

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3.  Al-Ghazali of Tus (1058–1111)  , primarily known as a theologian, mentions them in his work  الاوفاق للغزالى

4. Ibn al-Lubudi (b. 1210)  , a Syrian physician, astronomer, and mathematician, while considered a medical author, wrote an essay on magic squares.41

5. Abu’l-Abbas al-Buni (d. 1225), a north African occultist, wrote three books on the use of magic squares as talismans. His most comprehensive but cryptic writings on the subject are found in Shams alma’arif al-Kubra in which he labels the luoshu “Izra’il,” “Angel of Death.” In this book, Ahmed al-Buni showed how to construct magic squares using a simple bordering technique, but he may not have discovered the method himself.

al-Buni in North Africa had published a complete description of such amulets in which 

1.        Saturn was represented by a third-order magic square, the luoshu;
2. ·         Jupiter, a fourth-order square;
3. ·         Mars, a fifth-order square;
4. ·         the Sun, a sixth-order square;
5. ·         Venus, a seventh-order square;
6. ·         Mercury, an eighth-order square; and
7. ·         the Moon, a ninth-order magic square.

These amulets were each to be made of a different metal as per the Harranian tradition. Lead, the “father of all metals,” was to be the substance upon which the luoshu was inscribed.

 "Magic square 4x4 Shams Al-maarif3 (Arabic magics)"
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It appears that magic squares were introduced to Europe through Spain. Indeed, Abraham ben Meir ibn Ezra (c. 1090-1167), an Hispano-Jewish philosopher and astrologer, translated many Arabic works into Hebrew and had a deep interest in magic squares and numerology in general. He travelled widely throughout Italy and beyond, and may have been the one of the people responsible for the introduction of magic squares into Europe.