Thursday, December 2, 2010


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 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.

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

Monday, November 22, 2010

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)

     A. The Villa Capri  - Andrea Palladio, 1566
    Ground Plan

    B. The Pantheon, 1790

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

    Thursday, September 9, 2010



    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 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.

    Sunday, July 4, 2010


    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

    Monday, June 14, 2010


    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.

    Sunday, June 13, 2010


    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.


    Friday, June 11, 2010



    Al Zimmerman's contest is like a long and painful marathon. The computer programmers are dropping like a Lady Gaga lyric.

    If I could go to Las Vegas and bet everything in the world I owned, I would place that bet on Walter Trump. He has got this one locked up like a Lady Gaga hit song.

    Walter Trump is in second place and trails Hermann Jurksch by 0.001 points. Thats about as close as a paparazzi gets to Lady Gaga.

    For more results CLICK HERE



    As usual, I would have lost again. This is a lesson for all you young magic square enthusiasts - there is no future in gambling.

    Friday, April 30, 2010


    Here are the two latest winners in Christian Boyer's six enigmas Magic Square Contest

    Toshihiro Shirakawa of Japan is the winner of €1,100 for his efforts.


    Thursday, April 29, 2010


    A retention magic square is a magic square whose larger numbers surround smaller numbers forming retention lakes or ponds. Here is an example:

    Finally there has been a change in the standings in Al Zimmerman's Magic Square Contest. As predicted in the April 12 Blog, the Germans are favored to do well in this contest.

    The Germans are now one-two with Hermann Jurksch leading and Walter Trump trailing by only 0.004 points. Walter Trump is the favorite to win as he is one of the pioneers of Retention Magic Squares, a brilliant concept of combinatory mathematics.

    Lady Gaga likes retention magic squares!!

    Thursday, April 22, 2010


    Here we can see an order to the prime numbers in the 19x19 Magic Square.

    One can see that the numbers at the end of a column or row will continue in numerical sequence at the beginning of the next column or row. Therefore, a magic square in the Luo Shu format will also describe a three dimensional torus.

    Notes of interest:

    the number 2, the only even prime number stands alone as the only prime number in its row or column

    there are 72 prime numbers in the 19x19 Magic Square
    there are 71 odd prime numbers in the square, 71 is a prime number

    In this 19x19 Magic Square, the prime numbers will line up with one another in the three dimensional torus shape.

    The number 19 is the seventh odd prime number and the center number of 19x19 Magic Square is 181, also a prime number. This is why the 19x19 Magic Square Luo Shu would be a good place to look for an order of prime numbers within the trillions of magic square possiblities.

    Magic Squares constructed in the Luo Shu format
    offer the most balanced and harmonic way to arrange numbers. These arrangements were meant to assist humankind to evolve and prosper.

    Another interesting feature of the 19x19 Magic Square is the triabolo that links this magic square with the 9x9 Magic Square.

    See the below link:

    for more on the 19x19 Magic Square

    Monday, April 19, 2010


    Magic Square Contest #2

    This is a contest for the construction of magic squares with squares or cubes of pure integers as the major components of the square. The standard definition applies: all columns, rows, and diagonals must add up to the same number, known as the constant of the square.

    The smallest possible constant for a 4x4 magic square of squares is this square first constructed by Andrew Bremner, with the constant = 2823.

    There are six different competitions that involve different size squares and other variables.

    Does a 3x3 Magic Square composed only of squared integers exist?

    THIS IS CRAZY - WHO WOULD EVER HAVE THUNK THIS!!!!! I love magic square nerds, rock on!!!!


    Did you know that Lady GaGa is into magic squares?


    Thursday, April 15, 2010

    Leonardo Da Vinci and the Magic Square

    Leonardo da Vinci explored the incorporation of Math, architecture, and ground plans for church design in the late 15th century. Da Vinci used the Lo Shu as a model for the quincunx church design.

    Donato Brammante also incorporated the Magic Square in his ground plans for St. Peter’s Basilica. Brammante and da Vinci knew each other well and collaborated on projects. Both Brammante and da Vinci were greatly influenced by Luca Pacioli, who was known to possess a set of magic squares and reputedly knew about sacred geometry.

    Albrecth Durer specifically sought out Pacioli for this knowledge. Da Vinci lived with Pacioli and provided the illustrations to Pacioli’s book, The Divine Proportion.

    The concept of incorporating the square, circle, and right triangle with the four cardinal directions and sacred Math has always been intriguing to artists, architects, and Mathematicians for millennia.

    These concepts were employed in Southeast Asia and China for thousands of years prior to the Italian Renaissance to mark the cosmic center of the universe, the axis mundi.

    The mathematical symbolism of the Stupa and Chinese temple design may have had a great influence on early Byzantine church design that borrowed the quincunx pattern.

    Tuesday, April 13, 2010

    The Magic Square and Chinese Talismans

    This Chinese Talisman (approx. 2,000 years old) demonstrates the use of the Luo Shu Magic Square as a template for Chinese ritual talismans. The square and the circle represent earth and Heaven, female and male, yin and yang.

    The Luo Shu model was commonly incorporated in the fabrication of TLV Bronze mirrors and the Chinese bi disc, as well as the Ming Tang temple.

    This is also the ground plan for early Byzantine church design known as the quincunx or cross in square design. Leonardo da Vinci (church drawings), Brammante (St. Peter's Basilica), and Placidio (Villa Capri) used this design and were probably influenced by Luca Pacioli.

    There exists a school of thought that supports an influence of Chinese iconograpahy and cosmology (divining and the Yi Jing) with early Christian art and architecture.

    Monday, April 12, 2010

    Magic Square Contest


    James Youlton from Victorville, California has mostly been leading this contest thus far. However, he is now getting a serious challenge from Marcin Mucha (Poland) and Hermann Jurksch (Germany). The Germans are favored to do well here. Garr Godrey (Seattle) is fourth and all four contestants are within 0.04 points of one another. WOW!!!

    One reader of the blog has informed me that some of my reporting is incorrect concerning the languages/algorithms/environments that are receiving mention.

    Brainf*** has been talked about - it is a programming language known for its extreme minimalism and would be difficult to use in this contest. Brainf*** should be removed from the preferred list.

    PowerShell is a scripting language for Windows and probably should not have been on the preferred list.

    What seems to be clear is that the key to this contest is the algorithm selected and the diligence of the operator. Most languages should be able to implement a good algorithm.

    Sunday, April 11, 2010


    Water Retention Magic Square contest at:

    What is a magic square:

    What is a water retention magic square:

    The contest has attracted 130 participants from 29 countries. The discussion board gives an interesting look into the minds of some brilliant computer programmers. One enjoyable debate is which programming language is the best suited for the contest.

    What I have been able to glean so far, though not a programmer; is that there is no perfect language or development environment. An individual's genius and focus is what yield results. Some computer languages appear more suitable for this competition. From the discussion board commentary:

    Prefered languages/environments/algorithm:
    Turbo Pascal (outdated?)
    C #
    Brainfuck: no kidding, that is the name: reputed to be one of the simplest languages.
    zetagrid algorithm

    Languages/environments lacking: