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

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.

This book,  The Language of Numbers Demystified, is now available as an e-book. Deeper explanations than the present blog is able to elaborate upon are provided in this unique book.

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.

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!