Art and the Magic Square, Part Three

THE AMBASSADORS DEPART BY VITTORE CARPACCIO, 1498

Numbers represented a language that helped to connect humankind to the Heavens.  The magic square helps to unravel this language.

Another use of the magic square in art was in the form of geometric symbolism rather than an explicit square of numbers.  This was a subtle representation as the cognitive use of the magic square was not intended for the common person.

The magic square was used in art in the form of the quincunx, known as sacred geometry, a pattern that was symbolic of a universal cosmology.

This universal cosmology does not have a known origin.  The early Chinese of more than three thousand years ago developed the Luo Shu into a model that represented their cosmology, as documented in the Yi Jing. 

The Luo Shu represented a perfect balance of Yin and Yang as the even numbers symbolized female and yin energy, the earth, and the four intermediate directions.  The number four symbolized the four elements: air, water, earth, and fire.  The four quadrants of the magic square symbolized the earth.  The “square” earth was a reference to measuring the earth with a carpenter’s square and the Pythagorean Theorem. 

The cross of odd numbers symbolized male and yang energy, Heaven, and the four cardinal directions.  The circular nature of Heaven encapsulates the earth and is represented by the compass and circle.  In temple and church design this concept is illustrated as the circular dome (omphalos) is placed over a square base. 

The Center

The center is the most important concept in Chinese cosmology; also known as the axis mundi (Latin).  The center number five is surrounded by four odd and four even numbers for a perfect balance.  In the ground plan of early church design the axis mundi symbolized the convergence of Heaven, earth, the transcended human, and corresponded to the central dome and altar (and sometimes a tomb). 

The odd and even numbers (sans the number five) are in a quincuncial relationship with the number five, i.e., the center.   This quincunx of numbers can be translated into a geometric representation using the square and circle:

Detail of the Ambassadors Depart (c. 1498) by Vittore Carapaccio

This sacred geometrical pattern is modeled after the Luo Shu and represents a cosmology that explains the universe.

This pattern is known as the quincunx and was used to identify places or things of political or religious importance.

The Cosmology of the Quincunx: The Circle and Square

The quincunx was the most popular ground plan for churches in the middle Byzantine era.  The quincunx pattern has been commonly used on the covers of the most sacred books, both in church art and for real.  And the quincunx was a favorite pattern of the Cosmati pavements in dozens of the oldest Italian churches.  The pattern was used with intention to mark the most revered places or items of religious or political significance because the quincunx encompassed all the elements that could explain the universe.

The quincunx church

"The center symbolizes the beginning, the origin, the starting point, the pure being, the absolute, the transcendent, in three dimensions, the center corresponds to the axis, which unites a point with the zenith (the North Star), indicating verticality.  The circle - in space, the sphere - represents the infinite, transcendent, and complete, in sum, the divine, god.  The square – in space, the cube – is the symbol of the earth, connected in its order to the four cardinal points.  The cross marks the four points of the compass; it stems from joining the center with each one of the points, establishing the orientation of the point in space and in time.  The cross is the mediating symbol that connects heaven and earth."

Paloma Pajares Ayuela, Cosmatesque Ornament  (2001)

QUINCUNX EXAMPLES

1.  Altar cloth, c. 450 AD 

2.  Lindau Gospels, c. 875 AD

3.  Cosmati Pavement, c. 1150 - 1350 AD

4.  Church drawing, Da Vinci, c. 1485 AD

5.  The Ambassadors Depart  c.  1498  AD

ART AND THE MAGIC SQUARE, PART TWO

ATHANASIUS KIRCHER, ARITHMOLOGIA

The second known use of the magic square in Western art is the front piece to Arithmologia, a book authored by the great German Jesuit, Athanasius Kircher.

Kircher was a most unusual scholar of the seventeenth century; he was fluent in eleven languages including Latin, Hebrew, and ancient Egyptian.  He was a mathematician, a scientist, a geologist, an expert on Egyptian hieroglyphs, and was one of the first people to observe a micro-organism with a microscope.  

Kircher was also a prolific author having produced over forty works (including a book on China) during a period when official church censorship made it difficult to publish even one book.  Kircher had access to the great Vatican Library (perhaps old Chinese texts) and other church libraries that evaded other would be researches and authors.

The book Arithmologia is devoted to magic squares (the order 3 through the order 9 are demonstrated) as well as Pythagorean and other mathematical principles. 

Kircher and the Chinese

It seems plausible that Kircher was aware of the Chinese reverence for the Luo Shu as the 3x3 Magic Square is being held by a seraphin in Heaven as the model for the Pythagorean Theorem lay at Pythagoras’ feet on the terrestrial earth. 

Is Kircher demonstrating a connection with the Luo Shu and the Pythagorean Theorem? 

Although there is no proof that Pythagoras (or the Pythagoreans) knew of magic squares, the fact that a Pythagorean triplet of numbers occurs at the heart of any size magic square in the Luo Shu format and that these magic squares are excellent teachers of mathematics makes it plausible that Pythagoreans were fascinated with magic squares.  The Luo Shu was used in the Chinese civilization as early as one thousand years prior to the Pythagorean existence.

In addition, magic squares in the Luo Shu format fit Pythagorean numerology in the following ways:

1. Odd numbers are superior to even numbers
2. Powers of numbers play a significant role
3. Pythagorean triplets exist in any order magic square in the Luo Shu format
4. Both magic squares and Pythagorean numerology are great teachers of mathematics
5. These magic squares serve as a model for Time and Space

Chinese mythology maintains the Luo Shu was a gift from Heaven to assist humankind to evolve and prosper.  Kircher seems to understand this cosmology of the Chinese not only by connecting the Luo Shu with the Pythagorean Theorem but also by having the Luo Shu appear in Heaven.

Kircher would often write in a manner that was intentionally not easy to comprehend in order for his books to pass scrutiny during the times of the Inquisition; the Index of Prohibited Books listed the banned books and an inquisition could be ordered for any person associated with these books.   Giordano Bruno burned at the stake and Galileo was placed under house arrest for their literary crimes and philosophies during this era.

Kircher attempted to reconcile this confusion with his art and the use of symbols in order to clarify the message that he was unable to articulate in his writings.

The early Chinese of four thousand years ago believed that Number, that is the Luo Shu, was presented to humankind from Heaven as this pattern of numbers appeared on the back of a tortoise that had crawled out of the Luo River.     The great Fu Hsi recognized the pattern of numbers as sacred and soon temples and cities were being designed based on the model of the Luo Shu.

Pythagorean Symbolism

The art piece above demonstrates classic Pythagorean symbolism:

• The tetractys
• the proof of the Pythagorean Theorem
• The Luo Shu or 3x3 Magic Square
• Heaven and the terrestrial earth
• Numerology
• The enneagram
• The planetary spheres
• The pentagram and hexagram

Pythagorean Symbolism from the Royal Tombs of China

The legendary sage Fu Hsi and Nu Wa holding their customary symbols, the carpenter’s square (ju) and the compass (gui).  These symbols represented the yang / yin balance of male and female, Heaven and earth, odd and even.

 

The carpenter’s square also symbolizes humankind’s need for math, numbers, and measurement and in the Chinese language can be synonymous with the gnomon (ju). [gnomon. Gr., can mean interpreter, one who knows, rule of faith or conduct; canon or tenet]

The Compass and Square

The combination of the carpenter’s square and compass form a new word, gui ju and projects the important concept of establishing order (over chaos), the moral standard or the way things should be.   This is an example of how the early Chinese incorporated mathematical philosophy into the structure of their language.

The Luo Shu was considered a model for Time and Space and had functional use to humankind.  Together with the gnomon, the Luo Shu was a language gifted from above that helped to connect Heaven and earth via the observation of the Sun (rather the shadow of the gnomon cast by the Sun) and the understanding of math. 

Is this the message of Athanasius Kircher’s Arithmologia?

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