Symbols and the Magic Square

This is a fantastic magic square and truly magical.

This is a magic square because all the rows, columns and major diagonals add up to 175.  In addition, this is a symmetrical magic square because any pair of numbers equidistant from the center cell of 25 add up to 50 or 2(25).  Finally, this is an ultra-magic square because in addition to being a magic and symmetrical square, this is also a pandiagonal square.

Triplets of numbers of the same cell color add up to 75 or (3x25).  There are eight gnomon shaped triplets and eight linear groups of numbers, all revolving around the central number, 25.

This square features several important symbols seen in sacred geometry and early Christian art (for example, book covers of illuminated manuscripts):

·         The cross in square pattern

·         the cruciform shape

·         several gnomon or right angles

·         the axis mundi

·         the quincunx

·         perfect mathematical symmetry

·         the swastika

The Swastika

The swastika was a common solar symbol to the Buddhist, early Chinese, Greek, Roman, and Native American cultures.  The swastika was used on Greek coins contemporary with Pythagoras and was commonly used in Roman mosaics around the time of Christ. 

Two coins from Knossos, c. 350 B.C., distribution map of swastika symbol usage 

The swastika was a powerful symbol to all cultures as the swastika incorporated these familiar Chinese/Pythagorean concepts:  the path and cycle of the sun, the gnomon, math, the calendar, and magic squares.

The theory of the swastika being connected to the Luo Shu (or the TLV bronze mirror) has been discussed by many including Marcel Granet (La Pensee Chinoise) and also touched upon in Sir Joseph Needham’s epic book, Science and Civilisation in China, Vol. III.

The Quincunx

The cross-in-square creates four quadrants with each quadrant containing a 3x3 grid of numbers.  The four quadrants form a quincunx around the center number, five.

As noted earlier, this is a symmetrical square, the sum total of the center number plus any four numbers equidistant from the center number will sum to 5³.  In addition, any group of four numbers equidistant from the center are all odd or all even!

The Center or axis mundi

The numbers  5, 13, and 41 correspond to the center numbers of the order 3, 5 and 9 magic squares respectively and these are the odd components of their own Pythagorean triplet (3-4-5, 5-12-13, 9-40-41).

These numbers occur along the same diagonal.  In addition, the number to the left of each one of these centered magic square numbers is a square number.  Furthermore, the 13/25 connection points to the 5x5 magic square and this 7x7 square is linked to five and powers of five.

Magic Squares in the Luo Shu format
magic square	center no.	total numbers in square	Pythagorean triplet	Planet
3x3	5	9	3 - 4 - 5	Saturn
5x5	13	25	5 - 12 - 13	Mars
7x7	25	49	7 - 24 - 25	Venus
9x9	41	81	9 - 40 - 41	Moon
27x27	365	729	27 - 364 - 365	Sun

In other words, numbers are a language and magic squares help to unravel the code. Some of the keys to this "language of numbers": 

• powers of numbers, 
• Pythagorean triplets, and 
• the center numbers of other magic squares.

Example:  Locate the 5-12-13  Pythagorean right angle and notice the remaining numbers within the triangle are powers of 3, 6, and 7.  The hypotenuse is on the same diagonal as the centered numbers for three other magic squares (in the Luo Shu format).

Symbols for Time and Space

The Gnomon and Magic Squares played  significant roles in the development of symbols such as the carpenter’s square, the cruciform shape, and the swastika.  Number represented the first language to humankind and was believed to be a gift from Heaven in order for humankind to evolve and prosper through math and the ordering of the cosmos.  The carpenter’s square, the cruciform, and the swastika could very well be mathematical and solar symbols representing the connection of humankind to the Heavens and earth.

This is called the Knigts Tour. The numbering includes the numbers 1 thru 49 but not including the five numbers: 23, 24, 25, 26, and 27 (which add to 125) so the Star of David is more evident. 

This square was discovered by Walter Trump, a leading researcher and world authority in magic squares.

His website can be found in the right margin of this blog under “links” (scroll to top of page).  Among other discoveries, Trump has figured out how many possible arrangements of numbers will generate a magic square for some orders of magic squares.  For instance, there is one possible arrangement of numbers for the 3x3 Magic Square.  There are 880 possible arrangements for the 4x4 magic square and there are over 2.75 million arrangements for the 5x5 magic square!

ART AND THE MAGIC SQUARE, PART FOUR

HRABANUS MAURUS 

Early Christianity had recognized what the early Chinese revered for thousands of years prior to Christ:

The gnomon and the art of numbering is what links Heaven and earth.  Secondly, magic squares were considered to be divine arrangements of numbers that could be used in magic, prognostication, and divination.

Examples of incorporating calendrical numerology and magic squares into the Christian ideology are demonstrated by the works of Hrabanus Maurus (780 – 856 AD), the archbishop of Mainz, Germany.  Hrabanus was a Frankish Benedictine monk during the Carolingian period known primarily for De Laudibus Sanctae Crucis (In Praise of the Holy Cross), a masterful collection of 28 poems that feature his unusual and enduring style of poetry that incorporated imagery onto the manuscript.  The letters that are enclosed within the image(s) form an additional miniature poem, that is, poems with in a poem. 

Hrabanus adhered to the Church’s formula for sacred geometry as his imagery incorporates:

• A cross – in – square pattern 
• The four quadrants
• An axis mundi
• Numerology associated with the four seasons, lunar or solar cycles

This formula for sacred geometry generates the quincunx, a geometric symbol that is used to identify the most sacred of objects, most commonly, illuminated manuscripts. 

Poem 9 from De Laudibus Sanctae Crucis

Poem number 9 from De Laudibus Sanctae Crucis is composed of 41 lines and columns of letters.  These numbers, 9 and 41, correspond to the 9x9 magic square (in the Luo Shu format) that has 41 as its center.  This is only noteworthy because the 9x9 magic square corresponds to the Moon in alchemy and magic.

Therefore, it may be possible for an illiterate person to somewhat understand the message of the poem through the (numerical) imagery and the association of the numerology with the Moon and the four seasons.

Poem number nine mentions time, the revolution of the sun and the moon around earth, order and prosperity, the crucifixion;  but mostly the poem is about the glorification of the cross. 

What is noteworthy in the prose is the role of the light of the sun and moon upon the cross and the resultant shadow of information.  Hrabanus is describing the tradition of the gnomon.

The poem can be read by substituting the gnomon for the cross.

Example:  Lines 1 – 9:

1.   Sun and moon, praise god Jesus Christ.

2 .  The cross (gnomon) is your honor, enduring light, peace-bringing order,

3 .  glory, goodness, succession, and light for all ages.

4 .  Within this space of time (i.e. 365 days) you (the gnomons) measure night and      day

5 .  And ply your (the gnomons) course according to its perfect rhythm.

6 .  The four branches of the stem (the gnomon) contain in its

7 .  Thickness six times ten, five times one, and ten times thirty.

8 .  For indeed it is recognized that this is the way the sun and the moon are  restricted in their circular   motion;

9.   and the whole year goes around once through all the seasons;

And lines :
               20.  Altar and scepter of the high god,

21.  you (the gnomons) link in a strict order the underworld and the lofty sky;

22.  for the cross (gnomon) is the glory of the world.

The most intriguing feature of the poem is the imagery of four hexagonal shaped figures that are inclusive of some of the letters.  It has been determined that the number of letters included in three of the hexagons numbers 91, while the fourth numbers 92 for a total of 365.  Therefore, Hrabanus has managed to incorporate the following calendrical numerations:

The lunar cycle of 28 days

The solar cycle of 365 days

The four seasons

The number of days in each season, 91

The number of weeks in a year, 52

The four parts of a day

John North in his The Ambassadors Depart (2002, p. 284 – 287) comments:

Some of the doctrines developed by Rabanus here help to explain an even more remarkable exercise in mysticism for which he is responsible.  Praise of the Holy Cross (De Laudibus Sanctae Crucis) combines theology with mystical mathematical symbolism.  While in outward form it is a literary work, it has a strong figurative element…….The verses made up from the letters inside the hexagons refer to 365 as the “universe of time”, the 3 times 91, and 91 plus one.  Just how intricate the scheme is only becomes obvious when we count the numbers of letters inside the hexagons, for they represent those same numbers, the days in a season……

Hrabanus incorporates the same Chinese/Platonic/Pythagorean principles that we have seen before:  the quincunx, the use of magic squares that correspond to a planetary sphere, calendar numerology, the gnomon, the solar and lunar cycle, prosperity and agriculture.  All of these themes are referred to in this forty one line poem. 

Furthermore, it was stated earlier that De Laudibus Sanctae Crucis was a collection of 28 poems.  The number 28 corresponds to the lunar cycle because:

• The synodic revolution of the moon, that is, from new to new or full to full is 29.53 days, and
• The sidereal revolution of the moon, that is, the time it takes the moon to return to its same place among the stars is 27.33 days.  

These periods are always out of step but the average of 28 served convenience.
And numerologists would take delight in the peculiarities of the number 28, that is:

The sum of all its divisors, 1 + 2 + 4 + 7 + 14 = 28

The sum of the first seven numbers, 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

Magic squares were obviously a great influence on the monks throughout Europe during the Carolingian age.  The Coptic Gospels from Fayum, Egypt (c. 800 AD), the works of Hrabanus Maurus (810 AD), and the upper cover of the Lindau Gospels (c. 875 AD) represent the primary evidence of the use of magic squares in Christianity during this period.  

KABBALA AND THE MAGIC SQUARE

Mathematics are most necessary in magic, for everything which is done through natural virtue is governed by number, weight, and measure.  Pythagoras said that numbers have more reality than natural things, hence the superiority of mathematical magic to natural magic.

Frances Yates  Giardano Bruno and the Hermetic Tradition (1964)

The above magic squares are from the book De Occulta Philosophia by Henry Cornelius Agrippa von Nettesheim (1486 – 1535), physician, magician, and Secretary to the Emperor Maximilian I. 

De Occulta Philosophia had the distinction of being the most authoritative book on magic and the occult arts during the early Italian Renaissance.  The book circulated in manuscript form since 1510, was published in 1531, and expanded to three volumes in 1533.  Albrecht Durer (1471 – 1528) and Hans Holbein (1497 – 1543) could have been influenced by such a work and the magical influence of numbers. 

The book lists the first seven magic squares and their corresponding Kabbalistic names, signs, and figures to be used in the practice of magic.  Each planet was assigned its own number and several divine names. 

The four magical figures that corresponded to the moon, its spirit, its daemon, and its position in the zodiac.

The above magic squares correspond to the moon, the metal silver, and the numbers 369 and 3,321.

Note of interest:  There are two mistakes in Agrippa's 9x9 magic square.  The numbers 43 and 45 appear twice in the square and we know this violates a basic tenet of magic squares. 

The most common magic square is the basic 3x3 magic square also known as the Luo Shu.  The corresponding planet is Saturn.  This square could be worn as an amulet for protection against the negative influences of Saturn during certain times of the year.  The number 45 corresponds to the planet Saturn.

The Luo Shu, Chinese Magic, and the Dance of Yu

The early Chinese practiced magic and alchemy with the aid of magic squares three thousand years prior to Agrippa's book.  The Dance of Yu was a ceremonial dance mentioned in texts dating from the second century   BC.  The above illustration demonstrates that the steps of Yu are based on the numerical sequence of the 3x3 Magic Square.

The Sigil of the Moon and Greek Coins

The Magic Square, the Moon, the Swastika, and the Quincunx

This example of a Greek coin from Knossos, c 350 BC, seems to incorporate the 3x3 magic square, the symbol for the moon, and the swastika.  

  

The Greek coin with the swastika has the number five as its center, or axis mundi.  The number five at the center of a cross – in – square pattern with four similar quadrants makes a strong correspondence to the 3x3 magic square.  The four quadrants have the symbol for the moon and the four cardinal directions correspond to the arms of the swastika.   The quincunx can be seen as the axis mundi as well as the four moons around the center or the four arms of the swastika around the center.  The coin also exhibits the circle and square relationship, that is, a square 3x3 grid inside a circular coin. 

Agrippa's magical sigil for the moon also has a quincunx pattern around its center or axis mundi.  The four arms pointing in the four cardinal directions also create the four quadrants where the symbol for the moon resides and just as in the Greek coin from Knossos the crescent moon is pointing in the same direction.  Four circles at the end of each arm form a quincunx with the center circle.  The sigil also demonstrates the circle and square relationship and corresponds with the 9x9 magic square.

The 9x9 magic square demonstrates a cross – in – square pattern as only odd numbers occupy the horizontal and vertical axis.  The center or axis mundi is represented by the number 41 as this is the center between 9 and 9² just as the number five is the center number between 3 and 3^2.

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?