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

RETENTION MAGIC SQUARE CONTEST

Late last night, Lady GaGa telephoned to let me know that there has been another update to Al Zimmerman's Magic Square Contest.

Walter Trump has won!

Hermann Jurksch finished second.

MAGIC SQUARE CONTEST

There seems to be some confusion with the final results of Al Zimmerman's Magic Squre Contest - perhaps the results are still being tabulated because of a technical glitch.

Apparently, Walter Trump and Hermann Jurksch have finished in a tie.

How can this be? One would think that computer programmers could figure out a way to score entries so a tie would be an unlikely event.

Anyway, the final results may still be in limbo and a change in the standings quite possible.

STAY TUNED TO THIS WEB SITE FOR THE LATEST RESULTS