## INTRODUCTION TO THE MAGIC SQUARE

FORMULA FOR THE LUO SHU

## Tuesday, December 6, 2011

### Magic Squares and Prime Numbers, Part Three

Taking a closer look at the 5x5 magic square, Part One

The fact that there are nine ( a square number) prime numbers in the 5x5 magic square that sum to another square number (100) and that the sum of the prime numbers in just the cruciform portion of the magic square is 61 (part of the 11-60-61 Pythagorean triplet) is cause to take a closer look at the 5x5 magic square.

Examining this square will reveal insights into the School of Pythagoras and what is referred to as the “secret math”.  Pythagoreans were understandably amused by Pythagorean triplets that satisfied the Pythagorean Theorem, dubbed the most important of all mathematical formulae as this was a formula that described space and was represented symbolically by the carpenter’s square.    The carpenter’s square was also known as the gnomon.

The gnomon established a long tradition for thousands of years as an astronomical instrument that could identify summer and winter solstice and is forever linked as a model for time.  Time, space, and math:  these were the concepts that allowed humankind to evolve and prosper (via agriculture) and thus were also of great importance to Plato, Pythagoras, the early Chinese, and the early Christian hierarchy.   Important symbols for these concepts have survived for thousands of years and are relevant, sometimes, to the understanding of early Christian art and architecture (see below).  [for more on the gnomon, see Websters Dictionary and Needham’s Science and Civilisation in China, Vol. III p. 19-22].

The 5x5 magic square

The first point of interest is that this square features two Pythagorean Triplets that are in a gnomonic or right angle relationship, 5-12-13 and 7-24-25.  This is the smallest order magic square in the Luo Shu format that features two Pythagorean triplets.  Note that all the odd components of the two Pythagorean triplets fall within the cruciform of odd numbers.

The second point of interest is the number 39.  The above 5x5 magic square that is colored shows four groups of numbers in the corners of the square shaded in beige that are also in the shape of a gnomon or right angle.  The total sum of the numbers shaded beige is 156; however, each gnomonic component sums to 39.

A closer look at the internal 3x3 grid within the 5x5 magic square
 12 25 8 5 13 21 18 1 14
demonstrates that each column, row, and diagonal that pass thru the central axis (13) sums to 39.

Concerning the blue colored numbers: 7, 14 and 18 sum to 39 as do 8, 12 and 19; 9, 12, and 18; and lastly 8, 14, and 17.  When two of these groups of three numbers are connected by lines, this is the image:

It was pointed out earlier that the total sum of all the prime numbers in the square is 100 and the total sum of the prime numbers in just the cruciform portion of the square is 61, the difference of these two numbers being….39.

The third point of interest concerns the sum total of each of the three groups of numbers identified by color.  The sum total of the numbers shaded beige is 156, when divided by 13 gives us 12, and there are indeed twelve numbers that comprise this sum.  The sum total of the numbers shaded light blue is 104, when divided by 13 gives us 8, and there are indeed eight numbers that comprise this sum.  Lastly, the sum total of the numbers shaded bronze is 65, when divided by 13 gives us 5, and indeed there are five numbers that comprise this sum.

Math can be fun, or as my nephew says “cool”.  (He did say weird at first but I corrected him).  More relevant, this is how math was taught thousands of years ago.

Pythagoreans as well as the Chinese thousands of years prior to Pythagoras were interested in functional math that explained Time and Space.  Things such as Pythagorean triplets, the gnomon, and calendrical numerology excited the Pythagoreans and the mathematically minded early Chinese; Luo Shu magic squares possessed all these characteristics and were part of the esoteric tradition of what some refer to as magic but was probably none other than math and wisdom.

More detailed  explanations than can be given here are available in my book, NUMBER TIME ARCHETYPE  Scroll to top of page, click on "a book for the library", and order for only \$36.50 for the hardcover and \$22.50 for the softcover.

The carpenter's square as a Christian Symbol
Jesus and the four apostles, Ravenna mosaic - 7th century
The use of the "gammadia" symbol in early Christian art has raised some controversial issues, mainly that art historians throughout the ages have refused to acknowledge any meaning to this symbol and that its usage is strictly ornamental.  The gammadia or carpenter's square is symbolic of the gnomon, time, space, and math.  The gammadia's use on several hundred mosaics throughout Italy but especially in Ravenna over the course of five hundred years (between 350AD and 850AD) has been very consistent: the symbol only appears on clothing, altar cloths, or altar curtains to identify people or things of religious significance.  The gammadia appears on the clothing of Jesus, the apostles, evangelists, and others who had status within the church.  Furthermore, it is well documented that Popes during the Italian Renaissance had an obsession for Egyptian obelisks, another representation of the venerated gnomon.  Therefore, the symbolic role of the gnomon and carpenter's square is consistent through out Christian art and architecture.  Subjects such as the Christian use of symbols that relates to the early Chinese culture and their veneration of the Luo Shu are discussed in detail in my book.

## Saturday, October 1, 2011

### Magic Squares and Prime Numbers, Part Two

Table 1: examples of prime numbers in magic squares

This discussion of the occurrence of prime numbers in magic squares will mostly be limited to magic squares that can be constructed with the following formula and the resultant cruciform of odd numbers:

Table 2: The Luo Shu, formula, and cruciform of odd numbers

 4 9 2 Y-1 X2 Y-X 4 9 2 3 5 7 X Y 2Y-X 3 5 7 8 1 6 X+Y 1 Y+1 8 1 6 3X3 magic square formula cruciform of odd #'s

Magic squares based on the above formula are called magic squares in the Luo Shu format.  X will always be located to left of center and will equal the order of the square; Y will always represent the center number. When magic squares are constructed in this manner, a unique Pythagorean triplet of numbers appears at the heart of the square involving the odd numbers, X and Y.

Table 3: The Luo Shu and Pythagorean Triplets

 cycle of Pythagorean Pythagorean Luo Shu Triplets Theorem X Y-1 Y X2 + (Y - 1)2 = Y2 1 3 4 5 9 + 16 = 25 2 5 12 13 25 + 144 = 169 3 7 24 25 49 + 576 = 625 4 9 40 41 81 + 1600 = 1,681 5 11 60 61 121 + 3600 = 3,721 6 13 84 85 169 + 7,056 = 7,225 7 15 112 113 225 + 12,544 = 12,769

Another feature of magic squares based on this formula is that a cruciform of odd numbers runs through the horizontal and vertical axis of the square.

Table 4: Higher Order Luo Shu magic squares, Pythagorean triplets, and the cruciform of odd numbers

This post will examine two considerations of prime numbers that occur in magic squares in the Luo Shu format:
• The sum total of all the prime numbers (∑ primes) in the first four magic squares in the Luo Shu format, and,
• The sum total of all the prime numbers in the cruciform portion
(cruciform primes) of the first four magic squares in the Luo Shu format.
The sum total will then be broken down to the basic common denominators that compose the number.

Table 5:  Sum of Prime Numbers in Luo Shu Magic Squares

 magic square ∑ primes broken down ∑ cruciform  primes broken down 3x3 17 17 15 5*3 5x5 100 (5*2)2 61 61 7x7 328 41*23 208 13*24 9x9 791 113*7 305 61*5

A center number from each magic square is a component of the number that represents a total sum (with the exception of a sum that is prime and not a centered number).   Whether it is the total sum of all the prime numbers in a magic square or the total sum of the prime numbers that occur just in the cruciform portion of the magic square that is to be considered, one will discover a “centered” number that is a component of the total entity.

The only center number (25) not represented is from the 7x7 magic square (in the Luo Shu format), however; it really is represented as 52.  Therefore, the 3x3, 5x5, 7x7, and 9x9 magic squares all have prime numbers whose total sum and/or cruciform sum have numerical components that correspond to the center numbers of all of the magic squares from order 3 to order 11 and order 15.

Conclusion

There exists a relationship between magic squares in the Luo Shu format and prime numbers.  This relationship has mostly to do with the sum of prime numbers and the center number of magic squares in the Luo Shu format.

Magic squares in the Luo Shu format:
1. Are based on a formula
2. Reveal a Pythagorean Triplet of numbers
3. Reveal a cruciform of odd numbers
4. Reveal a relationship to prime numbers
5. Form a three dimensional torus
6. Are symmetrical, that is, any two numbers equidistant from the center add up to the same number

## Saturday, September 10, 2011

### The Luo Shu and the He-Tu

There exists a connection with the Luo Shu and He Tu concerning the numerical pattern:

1 - 6 - 2 - 7 -3 - 8 - 4 - 9 - 5

This pattern emerges in the 9x9 magic square in the Luo Shu format. There are several hundred trillion ways to arrange the numbers one thru 81 in a 9x9 magic square, but only one arrangement satisfies the definition of a Luo Shu magic square.

The above illustration defines the Luo Shu as a mathematical formula that can be expanded to create larger squares; this will generate an unique Pythagorean triplet of numbers at the heart of any magic square constructed in this manner.  This was a most significant concept to the early Chinese, Daoists, and even the early Christians (who incorporated magic squares onto book covers of illuminated manuscripts).

In fact, parts of the above illustration originate from a drawing that appeared in 1687 in a book called Astronomia Europaea, by Ferdinand Verbiest, a most courageous Jesuit priest who was one of the earliest and most influential Jesuits to penetrate China.  Matteo Ricci was the first famous Jesuit priest to establish a Christian foundation in China in 1582.

The Jesuits were no doubt interested in older Chinese traditions such as the Luo Shu.  The page in Verbiest's book which demonstrated the Luo Shu as the model of the Universe was the first indication that the Western world (specifically the Jesuits) knew the connection of the Luo Shu with the Pythagorean Theorem. This was in 1687.  Athanasius Kircher's frontpiece of Arithmologia (1666) strongly suggests that he was aware of this connection as well.

The two Jesuits, Verbiest and Kircher, shared many interests and did know of each other.  Verbiest stayed at the College of Rome in 1652 and 1653 while Kircher was living there and teaching mathematics as well as  overseeing the Museum Kircherian.  The two also exchanged letters.

Kircher was influenced by Pythagorean and Platonic philosophies as well as magic squares as these concepts were expressed as symbols in his art.  Mathematics and astronomy were very important concepts to the Jesuits who were a part of the China mission over four hundred years ago.  The Christian expertise in math and astronomy would help to establish credibility with the Chinese government.  Soon the Jesuits were running the Bureau of Astronomy and correctly adjusting the Chinese calendar.

Was the Luo Shu of interest to the Christian religion?

Yes, the Luo Shu was a vehicle to incorporate mystical mathematical symbolism into art and architecture and was used by the early Christians as early as the eighth century.
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## Saturday, May 7, 2011

### Christianity and the Magic Square

This 7x7 magic square recently appeared on the side of a religious building  in a very small town called Zurgena in Almeria, Spain.

Why would church authorities put such an odd matrix of numbers on a building and what does it mean?

Numerical Symbolism

Mystical mathematical symbolism with Magic Squares has been used by Christianity since medieval times (eighth century) as demonstrated by several book covers of illuminated manuscripts and the works of Hrabanus Maurus. Incorporating mathematical symbolism into art and architecture would identify objects and places of religious significance.

The reason for using math and squares of numbers is to acknowledge the revered Luo Shu, which appears in the central 3x3 square as the last digit of each number.
The Luo Shu

 28 21 26 8 1 6 4 9 2 23 25 27 → 3 5 7 → 3 5 7 24 29 22 4 9 2 8 1 6

The gnomon and the Luo Shu would provide humankind with a calendar (Time) and the right angle triangle theorem (a formula for Space).  In other words, the gnomon and numbers corresponded to the concepts of Time and Space and were integral to evolution and prosperity.  The early Chinese, the Buddhists, the Daoists, and early Christianity all recognized this cosmology.  This is why this magic square appears on this building.

If we examine the square closely we should observe the basic Pythagorean, Platonic, and Chinese characteristics:
• The axis mundi
• references to space
• references to time
The above magic square meets the basic requirements for a magic square:
All the columns, rows, and major diagonals sum to 175.
The total sum of all the numbers in the square is 1,225 or 352  = (7x5)2
In addition, the magic square is symmetrical, that is, any four numbers equidistant from the center number, 25, will sum to 4 x 25 or 100, which is ten squared.

Let us now examine the details.

A.  The inner shaded square is a 3x3 grid of the numbers 21 through 29, these numbers sum to 225 or 152.  The numbers 21 through 29 are the nine centered numbers between 1 and 49.

B.  The numbers of the light blue tiles that surround the central 3x3 magic square sum to 400, or 202.  These are numbers 13-20 and 30-37, inclusive.

C.  The numbers of just the outer perimeter sum to 600 which is not a square number but if you add the central number 25 to 600 then you get 625 or
252.
 Square of Numbers Numbers Sum Sum ÷ 52 Inner 3x3 shaded square 21 - 29 225 or 152 32 Middle shaded square 13 - 20, 30 - 37 400 or 202 42 Inner + Middle squares 13 - 37 625 or 252 52 Perimeter shaded square 1 - 12, 38 - 49 600 (+25) = 625 52
As one can see from the table above, the numerical sums from each shaded area of the magic square reduce to the 3 - 4 - 5 Pythagorean triplet

The
axis mundi

The center number or axis mundi is 25 or (5)2.   The center 3x3 square with the numbers 21 through 29 are also the nine centered numbers with in the numbers one through 49.  These numbers also add up to a square number, 225 or152.   Both of these numbers 25 and 225 symbolize the axis mundi, and 225 ÷ 25 = 9, another square number.

The Knights Tour and the Star of David

The Knight's Tour - draw a line sequentially from number one to forty-nine.
One of the more interesting features of the above magic square is that it features a Star of David around the axis mundi of number 25. This is a similar characteristic that was featured in a magic square in the last blog post, just below.