ART AND THE MAGIC SQUARE, PART ONE

ALBRECHT DURER, MELENCOLIA I

The first known use of magic squares in Western art was the 4x4 magic square in Albrecht Durer’s Melencolia I (1514).  

This was during the embryonic phase of the Italian Renaissance with legends like Brammante, Da Vinci, and Pacioli introducing math into art and architecture – an ancient concept which was enjoying a rejuvenation of sorts.*

Plato was an important influence to the philosophies of the great mathematicians (artists and architects) during the Italian Renaissance as his books were just coming onto the market after being out of print for about a thousand years (the "Dark Ages").

At the same time, books on alchemy and magic were introducing magic squares to the public.  Luca Pacioli and Cornelius Agrippa would publish books with a set of magic squares and their magical correspondences.  

The symbolic significance of the 4x4 magic square corresponded to related Pythagorean themes:

1. Plato’s mantra that “number” was essential for the evolution or destiny of humankind as Number could provide the way to measure.
   
   
2. An alchemical or "magical" correspondence to the planet Jupiter; that is, part of a complicated system where powers of numbers related to magic squares and a mathematical language that could explain the universe.  

The 4x4 magic square corresponded to Jupiter and could be used to invoke the spirits of Jupiter to counter the effects of Saturn, that is whatever affect Saturn has on the planet earth and its inhabitants - which in this case could be the psychological affects of the mind and its physical  impact on the body.  What was referred to as "Melancholia" in the sixteenth century may be commonly known today as Major Depression, which can be associated with chronic widespread pain and Fibromyalgia.

Plato, from the Republic

“And haven’t measuring, counting, and weighing come to light as the most charming helpers in these cases?  As a result of them, we are not ruled by a thing’s looking bigger or smaller or more or heavier; rather we are ruled by that which was calculated, measured, or, if you please, weighed.”

Other Platonic (measuring) symbols in Durer’s Melencolia I include:

• The carpenter’s T-square
•  The compass
•  The ruler
• The hourglass
• The scales

Additional symbols include a bell, a pot of gold, keys, a purse, a ladder with seven rungs, a polyhedron, a brooding Goddess with her muse, a brooding dog, carpenter’s tools and materials, and more.

There are 880 different ways to arrange the numbers 1 thru 16 in a 4x4 magic square, Durer chose the one square that appeared in Fra. Luca  Pacioli's book De Viribus Quantitatis (1509) with these particulars:.

• The numbers fourteen and fifteen are arranged to identify the date of this masterpiece:  1514.
• Each row, square, and major diagonal add up to 34 (the conventional definition of a magic square)
• Each quadrant of four numbers of the square add up to 34
• The middle four numbers add up to 34
•  The numbers in the four corners add up to 34
• The middle two numbers of the first and last row add up to 34
•  The middle two numbers of the first and last column add up to 34

The use of magic squares in art was a reference to a Platonic or Pythagorean world view that emphasized number as the most integral part of human existence.**

This paralleled the ancient Chinese reverence for number (the Luo Shu) as represented in their art (the TLV Bronze Mirror, Jade Bi disc) and architecture and should be considered as an influence on Platonic, Neoplatonic, Pythagorean, and early Christian philosophy and numerology.  

*  The magic square was part of an elaborate "sacred geometry" that was incorporated into the design of metal and jeweled covers of Illuminated Manuscripts during the Carolingian era (example: upper cover of the Lindau Gospels, c. 975 AD).  Monks from monastaries (such as Hrabanus Maurus) incorporated math (numerology and magic squares) into poetry, art, and the design of several bookcovers during this period.

**  The use of  magic squares in art can also make alchemical correspondences to the planetary spheres that was part of a complicated system connected to Renaissance magic.

SOME GOOD WORDS ON THE NUMBER 13

THE 13X13 MAGIC SQUARE

If numbers are in fact a language, then the most important feature for numbers is to provide functional information about Time and Space. Magic squares in the Luo Shu format do exactly this.

THE NUMBER 13

The most important thing about the number 13 is that it is the seventh odd number.

The second most important thing about the number 13 is that it is part of the 13 – 84 – 85 Pythagorean triplet.
 
The most intriguing number is the number 7. The most important thing about the number 7 is its connection to the calendar (seven days in a week and cycles of seven).

Thirteen is the result of the difference of the squares of two consecutive numbers.*

Eg: 7*7 - 6*6 = 49 – 36 = 13 and adding the squares generates:
     7*7 + 6*6 = 49 + 36 = 85

These numbers, 13 and 85, represent the odd numbers of the 13 – 84 – 85 Pythagorean triplet.

The sum of all the numbers one through 13 equals 91. This is a most important number that corresponds to the calendar as there are 91 days in a season.

All of the odd numbers mentioned above are the numbers that describe Time and Space. Time as in the numbers of the calendar and Space as in the Pythagorean Theorem

•  7 – the number of days in a week
• 13 – the number of weeks in a season and part of the 13, 84, 85 Pythagorean triplet
•  85 – part of the 13, 84, 85 Pythagorean triplet
•  91 – the number of days in a season
•  169 - part of the 13, 84, 85 Pythagorean theorem
• These numbers add up to 365 - the number of days in the solar cycle

These same numbers are the significant numbers of the 13x13 magic square.

•  7 – is the odd component of the square, which is based on 6 & 7
•  13 – is the size or order of the square
•  85 – is the center number of 1 thru 169 and is the center number of the square
•  91 – is the sum of the numbers 1 thru 13 
• 169 - the 13x13 magic squares in the Luo Shu format is the arrangement of the first 169 numbers in this square

Now is where it gets interesting.

7 + 13 + 85 + 91 = 196 or 14*14

Add this to 169 or 13*13 and one gets 365, the most important number of the calendar, the solar cycle.

13 and 14 are the components of the 27x27 magic square which has 365 at its center.  In other words, the seventh odd and the seventh even numbers are the components of the 27x27 magic square with 365 at its center.

Therefore, the 13x13 magic square is connected to the 27x27 magic square.

These five significant numbers of the 13x13 magic square (7, 13, 85, 91, and 169) also form a triabolo in the 27x27 magic square.

THE TRIABOLO

Conclusion

1. The 13x13 and 27x27 magic squares in the Luo Shu format are related to Time as in the numbers of the calendar, and Space as in the Pythagorean Theorem

2. The number seven plays a key role.

3. Odd numbers have a higher status than even numbers.

*  Odd numbers are created by taking the difference of the squares of two consecutive numbers, eg:

1*1 - 0*0 = 1
2*2 - 1*1 = 3

3*3 - 2*2 = 5

4*4 - 3*3 = 7

THE TIBETAN MANDALA AND THE 3X3 MAGIC SQUARE, aka the LUO SHU (or the Philosophy of Architecture)

The Tibetan mandala was a visual expression of the macrocosm of the universe using a symbolic microcosm that included:

• The square
• The circle
• The four cardinal directions
• The four seasons
• The axis mundi
• Numerology, in this case the Luo Shu or 3x3 Magic Square
• The quincunx or quinary grouping of images      
• Color

The map of the universe, Time and Space, and transfiguration are all concepts of the mandala.   The mandala represents an elaborate palace or temple with a geometry that assists in the attainment of  “Supreme Illumination”.

In a mandala, the quinary grouping of images and symbols is psychologically significant as well as indicating the four cardinal points that revolve round a center which conditions them, thereby evolving a succession in time and space round itself.     
     (Guiseppe Tucci, The Theory and Practice of the Mandala)

The Luo Shu shares some of these very characteristics with the mandala:

• The Luo Shu represented the cosmology of the early Chinese, the macro expressed by the micro
• The quincunx, the four cardinal directions, and Heaven are represented by the odd numbers (the cross in square)
• The Luo Shu was a model for the Ming Tang Temple, or Temple of Illumination (Sir Joseph Needham, Science and Civilisation in China, Vol. III)
• The square and circle relationship is well represented by the two dimensional magic square and it's three dimensional torus (see post below)
• The magic square is the basis of Chinese (and Pythagorean) numerology
• The Luo Shu is an ideal model for Time and Space because:

1. a unique Pythagorean triplet occurs in the heart of every Luo Shu magic square,
2. the grand daddy of all magic squares, the 27x27 MS was referred to as the square of the Sun because it contains the "numbers of the calendar".

The above mandala, Vajradhatu Mandala, Central Tibet, ca. 14th century, demonstrates the influence of the Chinese Magic Square.

Many scholars have reported the Luo Shu as the basis of temple design (the Chinese Ming Tang, the Indian Stupa, and  the Lama Dagoba) as well as being the foundation of the mandala. (Cammann, Snodgrass, Schinz, Wheatley, Granet)

Examples of the Magic Square and Temple Design

 A. The Villa Capri  - Andrea Palladio, 1566

Ground Plan

B. The Pantheon, 1790

C.  Brammante, Da Vinci, and the Quincunx,  975 - 1506

MAGIC SQUARES AND TORIC RELATIONSHIPS

THE TORUS: HOW A MAGIC SQUARE BECOMES A CIRCLE


Magic squares in the Luo Shu format describe a torus. The definition of a magic square in the Luo Shu format can be found here: LINK

The 9x9 magic square in the Luo Shu format can be transcribed so all the numbers are reduced to their Pythagorean root number.

For instance, 37 = 3 + 7 = 10 = 1 + 0 = 1

This creates a new magic square using only the numbers one thru nine but still maintain some features of a Luo Shu magic square, i.e., the constant of the square is the center number times the size of the square or 5 x 9. And the total sum of all the numbers in the square is the size of the square, squared, times the center number or 81 x 5.

There are some interesting features to this new magic square:

1. The numbers in common to the two squares demonstrate how a magic square in the Luo Shu format is constructed. Follow the weave of numbers begining with the "1" below the center cell and one can see how the numbers wrap around the square in continuum.

2. The number five is the center number of the square as well as one of the major diagonals. The number five is the center number of the 3x3 Magic Square known as the Luo Shu.

3. Beginning in the upper left corner cell, the numerical pattern in each column, row, and one diagonal direction is: 1, 6, 2, 7, 3, 8, 4, 9, 5, 1 - similar to the He Tu pattern.

4. The magic square forms a three dimensional torus. That is, the numbers continue in order from one edge to the opposite edge.

5. The diagonal in one direction consists of one integer only, and the diagonal in the opposite direction is always a continuous progression of numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9....

6. The 9x9 Magic Square and the 27x27 Magic Square in the Luo Shu format can have their numbers transcribed into their Pythagorean root numbers and still maintain a magic square as well as define a torus. The next magic square that could do this would be the 45x45 Magic Square or any order magic square whose root number = 9.

7. At the heart of every magic square in the Luo Shu format lies a Pythagorean triplet. In the 9x9 Magic Square it is the triplet of 9, 40, and 41.

This new magic square has a modified Pythagorean triplet: 3x3, 4 and 5, very close to the 3 - 4 - 5 classic Pythagorean triplet found in the 3x3 Luo Shu Magic Square.

THE CIRCLE AND THE SQUARE

The symbolism of the circle and the square has millenniums of tradition.  The gnomon, the sun, the calendar, the four cardinal directions, the right angle triangle and magic squares in the Luo Shu format are part of the ingredients of the symbolism of the circle and the square.  

This so called "sacred geometry" was used as a model for city layouts, temple design, and in places of religious / political importance in many cultures including early China (Shang dynasty), southeast Asia, India, Iran, Armenia, Turkey and elsewhere.

The long tradition of this symbolism continued throughout history and influenced church design and architecture of the Byzantine era and the early Italian Renaissance.

ARABIC MAGIC SQUARE

This is an iron plate inscribed with the 6x6 Magic Square in Arabic numerals from the 14th century. The 6x6 Magic Square was found in the palace foundation of Mongolian Prince Anxi to ward off the evil spirits.

This represents primary evidence of magic square usage in the daily lives of the royal family. The set of higher order Magic Squares were a part of the cosmological system of the early Chinese. The 6x6 Magic Square represents the Sun, perhaps the most important magic square in alchemy, with the exception of the 27x27 Magic Square, which was also known as the square of the Sun.

The significant numbers of the 6x6 Magic Square: the magic constant of the square is 111, and the sum total of all the numbers in the square is 666, the Mark of the Beast. This square has been referred to in the Bible, but most likely has Chinese origins.

More information on the above iron inscribed magic square: Arabic 6x6 Magic Square