The Magic Square Blog

The 5x5 magic square and Christian Symbolism

2012-01-01T01:55:00.000-08:00

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

The incorporation of mathematical symbolism in art was used by the early Christian church to stress the importance of math with the evolution and prosperity of humankind. The early church was greatly influenced by Pythagorean mathematics and the mystique of magic squares. Numbers (and math) would help humankind be in harmony with the earth and Heaven.

In the above 5x5 magic square, the numbers can be separated into two groups: the bronze colored group of numbers sum to 221 and the light blue colored group of numbers sum to 104.

The number 221 is significant as it is a part of the 21-220-221 Pythagorean Theorem, as demonstrated in the below 21x21 magic square in the Luo Shu format.

This magic square is based on the numbers 10 and 11:
X = 11²- 10² = 121 – 100 = 21
Y = 11²+ 10² = 121 + 100 = 221
magic constant = X*Y = 21*221 = 4,641
∑ 21 = 231

The numbers of the central 3x3 grid in the above 21x21 magic square can be reduced to their Pythagorean root numbers to generate the classic 3x3 Luo Shu magic square. (There are some interesting correspondences with this 21x21 magic square and the 5x5 magic square: a) the center number 221 reduces to five, and b) the numbers 5, 25, and 65 - the magic constant for the 5x5 magic square - all line up in the same row.)

But let us return to the 5x5 magic square. In alchemy, the 5x5 magic square corresponds to the planet Mars = Ares, the principle of life-prolonging medicine. Mars rules the month of Aries, the season of spring, birth, and regeneration. The opus alchymicum - the work of transformation - should also begin in the spring (C.G. Jung, Alchemical Studies).

From Joseph Henderson and Donna Sherwood, Transformation of the Psyche:

In medieval astrology, each known planet had a ‘ruler’, personified as a god or goddess, and in medieval alchemy, each of the seven planets was associated with a stage of the alchemical opus in the following order: Saturn, Jupiter, Mars, Sol (Sun), Venus, Mercury, and Luna.

Other correspondences with Mars are the color red, the metal iron, maleness or yang energy, the stone ruby, and the element fire.

The astrological symbol for Mars, the alchemical symbol for iron, the gender symbol for male, and the symbol for the Greek god Ares and the Roman god Mars.

Evidently, the early Christian church believed in these concepts as demonstrated in church art. The 5x5 magic square of Mars forms part of the iconography of church mosaics that help to identify the “doctors” or “fathers” of the church: St. Athanasius, John Chrysostom, Gregory of Nazianzus, and Basil the Great of the East and St. Ambrose, Pope Gregory, St. Augustine, and St. Jerome of the West.

Doctors of the church: St. Athanasius, Gregory of Nazianzus, and Basil the Great with John Chrysostom and Gregory the Theologian. (source: Wiki)

Magic Squares and Prime Numbers, Part Three

2011-12-06T14:54:00.000-08:00

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 e-book, The Language of Numbers Dymystified. Scroll to top of page, click on "a book for the library", and order for only $15.

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.

Magic Squares and Prime Numbers, Part Two

2011-10-01T14:25:00.000-07:00

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
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

					9x9 magic square
					37	78	29	70	21	62	13	54	5
					6	38	79	30	71	22	63	14	46
3x3 magic square					47	7	39	80	31	72	23	55	15
The Luo Shu					16	48	8	40	81	32	64	24	56
					57	17	49	9	41	73	33	65	25
	4	9	2		26	58	18	50	1	42	74	34	66
	3	5	7		67	27	59	10	51	2	43	75	35
	8	1	6		36	68	19	60	11	52	3	44	76
					77	28	69	20	61	12	53	4	45

5x5 magic square						7x7 magic square
						22	47	16	41	10	35	4
11	24	7	20	3		5	23	48	17	42	11	29
4	12	25	8	16		30	6	24	49	18	36	12
17	5	13	21	9		13	31	7	25	43	19	37
10	18	1	14	22		38	14	32	1	26	44	20
23	6	19	2	15		21	39	8	33	2	27	45
						46	15	40	9	34	3	28

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
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:

Are based on a formula
Reveal a Pythagorean Triplet of numbers
Reveal a cruciform of odd numbers
Reveal a relationship to prime numbers
Form a three dimensional torus
Are symmetrical, that is, any two numbers equidistant from the center add up to the same number

The Luo Shu and the He-Tu

2011-09-10T08:27:00.000-07:00

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

2011-05-07T08:23:00.000-07:00

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

2011-03-07T14:59:00.000-08:00

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

2011-02-19T16:58:00.000-08:00

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

2011-02-15T07:13:00.000-08:00

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

2011-02-04T20:16:00.000-08:00

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

2011-01-28T08:31:00.000-08:00

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:

Odd numbers are superior to even numbers
Powers of numbers play a significant role
Pythagorean triplets exist in any order magic square in the Luo Shu format
Both magic squares and Pythagorean numerology are great teachers of mathematics
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?

ART AND THE MAGIC SQUARE, PART ONE

2011-01-22T11:18:00.000-08:00

ALBRECHT DURER, MELENCOLIA I

The first known use of magic squares in Western art was the 4x4 magic square in Albrecht Durer’s Melencolia I (1514).

This was during the embryonic phase of the Italian Renaissance with legends like Brammante, Da Vinci, and Pacioli introducing math into art and architecture – an ancient concept which was enjoying a rejuvenation of sorts.*

Plato was an important influence to the philosophies of the great mathematicians (artists and architects) during the Italian Renaissance as his books were just coming onto the market after being out of print for about a thousand years (the "Dark Ages").

At the same time, books on alchemy and magic were introducing magic squares to the public. Luca Pacioli and Cornelius Agrippa would publish books with a set of magic squares and their magical correspondences.

The symbolic significance of the 4x4 magic square corresponded to related Pythagorean themes:

Plato’s mantra that “number” was essential for the evolution or destiny of humankind as Number could provide the way to measure.
An alchemical or "magical" correspondence to the planet Jupiter; that is, part of a complicated system where powers of numbers related to magic squares and a mathematical language that could explain the universe.

The 4x4 magic square corresponded to Jupiter and could be used to invoke the spirits of Jupiter to counter the effects of Saturn, that is whatever affect Saturn has on the planet earth and its inhabitants - which in this case could be the psychological affects of the mind and its physical impact on the body. What was referred to as "Melancholia" in the sixteenth century may be commonly known today as Major Depression, which can be associated with chronic widespread pain and Fibromyalgia.

Plato, from the Republic

“And haven’t measuring, counting, and weighing come to light as the most charming helpers in these cases? As a result of them, we are not ruled by a thing’s looking bigger or smaller or more or heavier; rather we are ruled by that which was calculated, measured, or, if you please, weighed.”

Other Platonic (measuring) symbols in Durer’s Melencolia I include:

The carpenter’s T-square
The compass
The ruler
The hourglass
The scales

Additional symbols include a bell, a pot of gold, keys, a purse, a ladder with seven rungs, a polyhedron, a brooding Goddess with her muse, a brooding dog, carpenter’s tools and materials, and more.

There are 880 different ways to arrange the numbers 1 thru 16 in a 4x4 magic square, Durer chose the one square that appeared in Fra. Luca Pacioli's book De Viribus Quantitatis (1509) with these particulars:.

The numbers fourteen and fifteen are arranged to identify the date of this masterpiece: 1514.
Each row, square, and major diagonal add up to 34 (the conventional definition of a magic square)
Each quadrant of four numbers of the square add up to 34
The middle four numbers add up to 34
The numbers in the four corners add up to 34
The middle two numbers of the first and last row add up to 34
The middle two numbers of the first and last column add up to 34

The use of magic squares in art was a reference to a Platonic or Pythagorean world view that emphasized number as the most integral part of human existence.**

This paralleled the ancient Chinese reverence for number (the Luo Shu) as represented in their art (the TLV Bronze Mirror, Jade Bi disc) and architecture and should be considered as an influence on Platonic, Neoplatonic, Pythagorean, and early Christian philosophy and numerology.

* The magic square was part of an elaborate "sacred geometry" that was incorporated into the design of metal and jeweled covers of Illuminated Manuscripts during the Carolingian era (example: upper cover of the Lindau Gospels, c. 975 AD). Monks from monastaries (such as Hrabanus Maurus) incorporated math (numerology and magic squares) into poetry, art, and the design of several bookcovers during this period.

** The use of magic squares in art can also make alchemical correspondences to the planetary spheres that was part of a complicated system connected to Renaissance magic.

SOME GOOD WORDS ON THE NUMBER 13

2010-12-02T17:52:00.001-08:00

THE 13X13 MAGIC SQUARE

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 NUMBER 13

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.

THE TRIABOLO

Conclusion

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

THE TIBETAN MANDALA AND THE 3X3 MAGIC SQUARE, aka the LUO SHU (or the Philosophy of Architecture)

2010-11-22T19:58:00.000-08:00

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:

a unique Pythagorean triplet occurs in the heart of every Luo Shu magic square,
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

2010-09-09T12:13:00.000-07:00

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

2010-07-04T15:50:00.000-07:00

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

2010-06-14T15:19:00.000-07:00

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

2010-06-13T12:23:00.000-07:00

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

RETENTION MAGIC SQUARE CONTEST

2010-06-11T12:39:00.000-07:00

PEOPLE DROPPING OUT

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

UPDATE

HERMANN JURKSCH HAS WON, CONGRADULATIONS HERMANN!!!!

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

ENIGMA MAGIC SQUARE CONTEST

2010-04-30T06:31:00.000-07:00

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.

WELL DONE TOSHIHIRO!!!!

RETENTION MAGIC SQUARE CONTEST

2010-04-29T15:36:00.000-07:00

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!!

MAGIC SQUARES AND PRIME NUMBERS

2010-04-22T20:44:00.000-07:00

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

ANOTHER MAGIC SQUARE CONTEST

2010-04-19T19:38:00.000-07:00

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!!!!

CASH REWARDS AND CHAMPAGNE!!!!!

Did you know that Lady GaGa is into magic squares?

CONTEST DETAILS : http://www.multimagie.com/English/MagicSquaresEnigmasE.pdf

Leonardo Da Vinci and the Magic Square

2010-04-15T09:58:00.000-07:00

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.

The Magic Square and Chinese Talismans

2010-04-13T14:07:00.000-07:00

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.

Magic Square Contest

2010-04-12T08:09:00.000-07:00

ANNOUNCEMENT

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.

MAGIC SQUARE CONTEST

2010-04-11T12:59:00.000-07:00

Announcement:
Water Retention Magic Square contest at:

http://www.azspcs.net/Contest/MagicWater

What is a magic square:

http://luo-shu.com/book/part_two/3X3_magic_square_lo_shu_format_examples_luo_shu_formula_pythagorean_triad

What is a water retention magic square:

http://luo-shu.com/book/part_three/outside_contributions

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++
C #
Brainfuck: no kidding, that is the name: reputed to be one of the simplest languages.
PowerShell
zetagrid algorithm

Languages/environments lacking:
Java