Plato and the Magic Square

"The king lives 729 times more pleasantly..." is a reference to the 27x27 Magic Square, the calendar, and an order of society through wisdom rather than tyranny.

Passages from The Republic of Plato (Book IX, paragraphs 571 – 592)

This book is the defense of Socrates who is accused of unjust behavior, corrupting the youth, and not believing in the gods of the city.  It is the apology of Socrates and his defense of philosophy, the love of wisdom.  Socrates must defend himself as well as the philosophical pursuit, for such a belief calls into question one’s allegiance to the city’s gods, who are the protectors of its laws.   

In the beginning of Book IX, Socrates and Glaucon compare and contrast the attributes of the democratic man and the tyrant.  Once the pursuit of wisdom is absent, the mind has no pleasures of its own and the desires of the body overcome the moral compass of the mind.  

The three primary classes of human beings are the wisdom loving, the victory loving, and the gain loving. 

Socrates  proclaims, “Of these three pleasures, the most pleasant would belong to that part of the soul with which we learn, and the man among us in whom this part rules has the most pleasant life.” 

A tyrant is a man who is consumed  of his desires, be it food, drink, sex, or power.  He pushes away sleep to pursue the unnecessary pleasures and desires that are in violation of the moral standards and law.  Such a man becomes mad with power and expects to rule not only over men but gods as well.  Such a man never experiences friendship or freedom.  Such a man is faithless. 

A kingly man is just the opposite.  He may have such immoral thoughts, in his dreams, but does not act on such thoughts.  A democratic man can use discernment to rule over evil desires. 

There is no city more wretched than one under a tyranny and none happier than one under a kingship, and such a city ruled by a tyrant is a slave city, without honor, and full of fear. 

Socrates:  “Do you know how much more umpleasant the tyrant’s life is than the king’s?”
Glaucon:  “I will if you tell me.”
Socrates:  “There are , as it seems, three pleasures – one genuine, and two bastard.  The tyrant goes beyond the bastard ones, flees law and argument, and dwells with a bodyguard or certain slave pleasures, and the extent of his inferiority isn’t at all easy to tell, except perhaps as follows.”
Glaucon:  “How?”
Socrates:  “How far the king is removed from the tyrant in truth of pleasure, one will find at the end of the multiplication that the king lives 729 times more pleasantly, while the tyrant lives more disagreeably by the same distance.”
Glaucon:  “You’ve poured forth a prodigious calculation of the difference between the two men – the just and the unjust – in pleasure and pain.”
Socrates:  “And yet the number is true and appropriate to lives too, if days and nights and months and years are appropriate to them.”

This is a reference to the 27x27 Magic Square, in the Luo Shu format, and the calendar.  729 is equal to twenty-seven squared, code for the “square” of twenty-seven.  The magic square of 27 has at its center the number 365.   This is a magical number because it also represents the number of days in a solar year.  To keep the information coded, the number 729 could also be identified as the number of the sun and was referred to as such in Plutarch’s system of numerology. 

The compendium of literature concerning the Luo Shu generally attributes the root origin to be Chinese.  Knowledge and reverence for magic squares seemed to have spread through India, the Islamic world, and perhaps Alexandria following China but no one can say for certain.  The use of magic squares in Plato’s time is unknown but this association with the number 729 and the Chinese reverence for this particular square deserves to be examined.  

Suffice it to say that magic squares were tremendously influential in early Chinese culture, perhaps as much as 2,000 years before Christ, and became revered in other cultures as well.  The Luo Shu in fact, is an integral part of the Yi Jing, one of the five classics that make up the canon of ideology of the early Chinese.  Mandalas from Tibet and India, amulets from Babylonia and Assyria, and bronze mirrors from China are examples of how different cultures held these squares to be sacred.  Symbols of magic squares and the use of magic squares apparently occurred in early Christian art.  The mosaics in Ravenna, Italy and book binding covers such as the Lindau Gospels are clear examples of the use of Chinese iconography (see Chapters eight and nine) in early Christian art.

It is likely that the reverence for the number 729 is due to the harmony of the arrangement of the first seven hundred twenty nine numbers in a square.  The 27x27 magic square is a great teacher of algebra.  There are at least four triads of Pythagorean numbers in the square, the 3x3 Magic Square and the 5x5 magic square are also housed within the square.  And the number seven plays a crucial role in the construction of this square. 

But more importantly, the numbers of the calendar are present in the form of a triabolo, a geometric figure composed of three right angle triangles. 

The five numbers (7, 13, 85, 91, and 169) that form this triabolo add up to 365.  The numerical significance of these numbers is discussed in chapter eight, however one of the most important numbers is 91 as it identifies the number of days in a season.  All these numbers are essential in “calendar making”, integral to an agriculturural based society. 

The 27x27 Magic Square is derived from the 3x3 Magic Square, also known as the Luo Shu River Document.  The Chinese believed that these numbers, in this arrangement, were presented to man from Heaven.  The Pythagoreans also believed that numbers were presented to humankind from God and that numbers could explain all matters of importance.  Numbers, to some, were a religion and magic squares were a basis for this ideology. 

There are many interesting mathematical relationships with these five numbers. Here are some:

7 is the seventh number, and represents the number of days in a week.
13 is the seventh odd number and its magic square has 85 as the center cell.
The product of 7 and 13 equals 91, the number of days in a season.
91 is the sum of the first 13 integers.
169 is equal to the square of 13, part of the Pythagorean triad of 13, 84, and 85.

The product of 169 and 85 equals 14,365, the sum total of all the numbers in the 13x13 Magic Square.  The numbers can be separated by the comma to yield the number 14, which, together with 13 are the basis of the 27x27 Magic Square, and the number 365 is the center cell of the 27x 27 Magic Square. 

These relationships did not go unnoticed by the early Chinese.  Furthermore, the constructing and deciphering of magic squares was a fantastic teaching tool to students of mathematics.  Practical information, such as algebra and the right angle triangle theorem could be gained from the study of magic squares as every odd magic square in the Luo Shu format generates a Pythagorean triad in the center of the square.  Practical use of the right angle triangle theorem could lead to the ability to measure distances and survey land as well as measure celestial distances, which is the foundation of astronomy.

Working with magic squares could also lead to philosophical uses as well.  Magic squares are an excellent tool to meditate upon and can push out the thoughts that preoccupy the mind.  One could spend literally hours on end discovering the mathematical applications of these squares and reach a state of nirvana.  This alone could be the objective for some who enjoy working with magic squares. 

Now, lets examine more closely the phrase:

“…the interval which seperates the king from the tyrant in truth of pleasure is 729.”

The king is a reference to the sage who rules through the wisdom of mathematics or number, whereas the tyrant is one who rules with ignorance, brute force, and selfish desires. 

The root origin of the Luo Shu is Chinese.  The mythology of the discovery of the Luo Shu is credited to the great sage Fu Hsi, one of the four great mythological leaders of China, who was presented the Luo Shu by Heaven in the form of a turtle.   The great emperors of China were considered to be sons of Heaven and great sages.  They were considered to be the wisest of men and were referred to as sage-kings, priest-kings or sky-priests. 

Knowledge of numbers, mathematics, and its practical application was crucial in the evolution and prosperity of humankind.  Magic squares were an integral part of this wisdom, hence the reverence for the wise men who possessed this knowledge and who were designated the leaders of their communities. 

Chapter X paragraph 602

“And haven’t measuring, counting, and weighing (authors italics) 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 has calculated, measured, or, if you please, weighed.”

“But this surely must be the work of the calculating part in a soul.”

“And to it, when it has measured and indicates that some things are bigger or smaller than others, or equal, often contrary appearances are presented at the same time about the same things.”

“Did we not say that it is impossible for the same thing to opine contraries at the same time about the same things?”

“Therefore, the part opposed to it would be one of the ordinary things in us.”

“Well, then, it was this I wanted agreed to when I said that painting and imitation as a whole are far from the truth when  they produce their work; and that, moreover, imitation keeps company with the part in us that is far from prudence, and is not comrade and friend for any healthy or true purpose.”

“Therefore, imitation, an ordinary thing having intercourse with what is ordinary, produces ordinary offspring.” 

“Does this apply only to the imitation connected with the sight or also to that connected with the hearing, which we name poetry?’

“It is likely that it applies also so this.”

In this last passage, Plato is emphasizing rational truth can only be explained by numbers and math.  Using measurements of weight, counting, and measuring time with a calendar brings order to the platonic world.  The artist is an imitator who can not explain nature through artistic creation. 

Perhaps Plato is being too harsh on the creative, non-mathematical mind who can also be a genius and, more importantly, a record keeper of history with art.

The survival of art through the past several thousand years can sometimes be an excellent portal into the thoughts of humankind at that point in time.  In this book, the art that is used to exemplify the use of magic squares will include:  pottery plates from Ur, Jade bi discs and TLV Mirrors from China, book covers prior to the ninth century, beautiful church mosaics pre-sixth century, Tibetan mandalas, and works by Leonardo da Vinci and Albrect Durer. 

This art has provided great insight into the role of Magic Squares for the past five thousand years.  The magic square, (which symbolizes time, space, and order over chaos via mathematics) was a very important concept to our forebearers and has survived political and religious suppression for thousands of years partly because of the survival of art, but mostly due to mathematicians who feel that numbers reveal the only truth.  Plato and Pythagoras certainly subscribed to these philosophies. 

Conclusion

Pythagoras, Pythagoreans, and Plato were familiar with Magic Squares and realized the connection with the right angle triangle theorem (also known as the Pythagorean Theorem), the calendar, and philosophical meditation.  It was  necessary to keep this information available only to trusted members of their secret societies to avoid persecution.  Hence the reason of no written records of magic squares for over a thousand years.  The first published set of magic squares in the western world appeared in the encyclopedia Rasa’ il ( fourteenth century) and is credited to a Muslim sect in Basra, known as “The Brethren of Purity”.   The library of Alexandria was burned down several times and the western world was thrust into the Dark Ages as books and information were seen as threats to the authority of the rulers during those times, therefore, it is unknown how early Magic Squares gained popularity in the western world.

The early Chinese of the Shang Dynasty (1800 - 1100 BC), the western (1028 - 771 BC) and eastern (771 - 257 BC) Zhou Dynasty, and the Han dynasty (206 - 220 BC) used magic squares as a magical tool to impart balance and harmony to man in the ever changing universe.  Magic squares were essential tools for establishing order over chaos and imparting an ideal yin yang balance.  The practical use of applying the math of the Luo Shu led to a tradition of moral standards.

Plato wrote in these terms, that the pursuit of knowledge and love of wisdom would free man of tyranny and corruption ultimately leading to an orderly society.  Temple design, city planning, and ritual ceremonies are examples of how the magic square were used in places of religious and social importance to help establish social order. 

Numbers are a language, perhaps the very first language on this planet, waiting to be discovered by humankind.  Numbers, when arranged in squares can reveal practical information.  This numerical language was re-discovered time and again throughout the history of civilization.  For some societies and people, this discovery served as a foundation to an ideology and would be incorporated into the management of society to insure the evolution and prosperity of its people.