Friday, October 4, 2013

The 27x27 Magic Square – The Most Revered of All Magic Squares

The most important magic square of all is the 27x27 magic square.  This is the magic square that the early Chinese came to realize was indeed magical and relevant to the evolution and prosperity of humankind.  

The 27x27 magic square was also significant to Plato and Pythagoras and is referenced in Plato's Republic (Book IX):
     The king lives sevenhundred-twentynine times more pleasantly than the tyrant....

This passage referencing the number 729 is the key to understanding the puzzle to Plato’s famous quote.  However, it is instructive to understand some background philosophy about Plato, Pythagoras, and the early Chinese before illuminating on the impact the magic square had on religions, society, and the great philosophers and mathematicians throughout history. 

The Key to Understanding Plato, Pythagoras, and the early Chinese
Numbers.  Numbers were considered to be a gift from Heaven and a means for humankind to be connected with Heaven and earth.  The key to humankind’s success would be dependent upon man being in perfect harmony with Heaven and earth.  There existed perhaps a handful of instruments or systems that could place humankind in perfect harmony with Heaven and earth: numbers, astronomical instruments, and musical instruments were a few examples. 

Numbers were considered a special category as this was the system, when combined with mathematics, that could describe the mysteries of nature.  Numbers (and math) would organize the cosmos from random chaos to a predictable pattern that would be integral to the process of calendar making.  Numbers would simplify the understanding of musical scales as well as the interpretation of color. 

In summary, Plato, Pythagoras, and the early Chinese believed that numbers were the basis of human evolution and prosperity and, therefore, the key to being in perfect harmony with Heaven and earth. 

The complete quote from Plato’s Republic:

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…..And yet the number is true and appropriate to lives too, if days and nights and months and years are appropriate to them.

The number 729 is mentioned along with a reference to the cycle of time measured in days, nights, months, and years.  Only a calendar can identify these cycles of time.  The number 729 is a reference to 27 squared or the square of 27 and this corresponds to the Chinese 27x27 magic square.  There are at least two very good reasons why Plato is referencing the 27x27 magic square:

a) The center number of the 27x27 magic square is 365, which corresponds to the solar cycle or the number of days, nights, and months that are in a year, and

b) The Pythagorean triplet of 27, 364, 365 resides at the heart of the square.

Plato is attempting to demonstrate that it is best for a king to rule from the perspective of wisdom rather than tyranny.  This follows the tradition of the early Chinese and their development of the calendar, astronomy, and agriculture through the wisdom of several mathematical systems, one being the Yi Jing.  The 3x3 Chinese magic square is also known as the Luo Shu and is part of the Yi Jing.

According to Platonic, Pythagorean, and Chinese philosophy odd numbers are superior to even numbers. Squares and cubes of numbers also have particular interest. Pythagorean triplets of numbers are of most significance. The two odd numbers of the above triplet, 27 and 365 are special numbers indeed. The number 27 can be recognized as the cube of three, three also representing the triad, Mother-Father-Son. Therefore, three to the third power is doubly special. The number 27 squared gives us 729 – the number referenced by Plato. The number 365 not only represents the solar cycle or calendar, the number is also the center of 1 thru 729 and is a member of the Pythagorean triplet mentioned above.

The principles of the Pythagorean Theorem or Right Angle Triangle Theorem had a very practical use to the early Chinese as well as to the people of Platos’ and Pythagoras’ time because the mathematical relationship was a part of the calendrical system known as the gnomon or gnomonics.  This connection of math, time, and space with Heaven and earth rendered these concepts sacred and relevant to the use of symbols that would become integrated with objects used in religious ceremonies, religious clothing, temple building, city design, art, coinage as well as many other facets of society.

The Pythagorean Theorem and the Calendar

Traditional observation of the gnomon and use of the right angle triangle led to an ordering of time and space.

From the above illustration, one can see how the shadow generated from the sun exhibits a right angle as well as a calendrical system. (Please refer to the brilliant Joseph Needham whose work on the subject can be reviewed in perhaps the greatest of all publications, Science and Civilisation in China, Vol. III, pages 284-461).

In the Chinese scheme of systems that helped to align humankind in a harmonious relationship with Heaven and earth, the gnomon and the magic square became intimately inter-related to one another based on calendrical and Pythagorean Theorem correspondences. The mathematical correspondences with the calendar and the right angle triangle theorem represented by the Chinese magic square known as the Luo Shu became the basis for a system of prognostication that was integral to the success of agricultural and as a byproduct, humankind. The sacred geometry that corresponds to the mathematical arrangement of numbers in a 3x3 magic square is known as the quincunx; this would become the model of symbolic representation of the highest form of being one with Heaven. 
The quincunx is a symbol in Christianity, here it is seen in the ciborium in the nave of Santa Maria Cosmedin church in Rome, Italy

The Chinese magic square had a major influence on early Chinese society.  Excellent books by Albert Schinz (The Magic Square, Cities in Ancient China) and  Paul Wheatley (Pivot of the Four Quarters) make clear that the whole of Chinese society used the gnomon and magic square as a model for agricultural and city planning, temple building, and ceremonial objects.

The Luo Shu

The 27x27 magic square is the key to the Luo Shu, which is a progression of magic squares not limited to only the 3x3 magic square. My book, The Language of Number Demystified, is the only publication that contains the full set of magic squares 3 thru 27. A thorough analysis explains the significance of the Chinese magic square and why it is was so revered in Chinese civilization. The illuminating information contained within the set of magic squares in the Luo Shu format is too detailed for this blog but can be accessed in The Language of Number Demystified.

Most interesting to me was the discovery of the early Church integration of magic square symbolism on the covers of elaborate illuminated manuscripts. The magic square also enjoyed a resurgence during the Italian Renaissance and appeared in some classic works of art. The story of the magic square is endless, please enjoy my book.

Sunday, February 3, 2013

Hans Holbein, The Ambassadors

This is one of my favorite paintings.  Much has been written about this masterpiece by Hans Holbein, one of the greatest artists of the Renaissance era.  Rather than rehash what has already been reported by some excellent researchers, the purpose of this post is to bring to light certain aspects that have never been discussed in published form.

The Significance of the Axis Mundi and the Pole Star

Let us start by examining comments made by Paloma Pajares-Ayuela from her outstanding book Cosmatesque Ornament:

The center (or axis mundi) 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.  

Next, from the writings of the great Joseph Needham, Science and Civilisation in China, Volume 3, page 230:

The pole was thus the fundamental basis of Chinese astronomy.  It was connected therein with a background of microcosmic-macrocosmic thinking.  The celestial pole corresponded to the position of the emperor on earth, around whom the vast system of the bureaucratic agrarian state naturally and spontaneously revolved.

The pole refers to the Pole Star, also known as the North Star, and is equated with the celestial north pole, which is identified in The Ambassadors painting on the celestial globe.  

The Chinese believed the sun is attached to the heavens and shifts its position between the seasons of winter and summer.  The measurements of distances of the pole, and of the motions, are all obtained from the use of the gnomon and the right-angled triangle which it forms.  The carpenters’ square was the Chinese symbol for these concepts.   

Needham adds:

The inclination of the polar axis must have been among the earliest of astronomical observations.

Lastly, from Paul Wheatley, Pivot of the Four Quarters, page 462:

The principle of symbolic centripetality was also clearly manifested in the traditional Chinese city…whose primary concern was with the ordering of society.  In the imperial capitals the symbolism of the center was more strongly developed, for it was at this quintessentially sacred spot that was raised the royal palace, which corresponded to the Pole Star.  It is explained in the Chou Li how the precise position of the axis mundi was calculated, which is characterized as ‘the place where earth and Heaven meet, where the four seasons merge, where wind and rain are gathered in , and where ying and yang are in harmony.’  A gnomon erected there was held to cast no shadow at the summer solstice. The axis of the world was the point where earth most nearly approached Heaven…The axis mundi was also held to extend below the earth to establish contact with the underworld.

So where is the axis mundi in Holbein’s The Ambassadors?  And what would happen if Holbein’s axis mundi is connected with the celestial north pole, or Pole Star?  

The axis mundi can be clearly identified by the pattern on the floor of the painting. 
The Pavements of Westminster Abbey and The Ambassadors, and the Quincunx Symbol

Holbein was the King's painter in England for Henry VII from 1535 until Holbein's death in 1543.  He most likely had access to Westminster Abbey and its pavement.  In "The Ambassadors”, Holbein uses the quincunx symbolism on the pavement and carpet. The quincunx is a universal geometric symbol that represents the structure of the universe.  This would involve the four cardinal points, the four seasons, the four elements, axial correspondence relating the north pole of Heaven with earth, the gnomon and its right angle shadow, Pythagorean math and its tools: the carpenters’ square and compass.    

These concepts are also closely related to music according to Pythagorean and Chinese tradition.  (In fact, the Chinese words for gong,  工, and for carpenters’ square qu, old style: 曲, are both associated with music and the astronomical instrument, the carpenters’ square.)  Equipped with these math tools, humankind could establish an ordering of the cosmos with an amazing accuracy considering the crudeness of their instrumentation utilizing only observation with no magnification.  

All of these concepts are demonstrated in Holbein’s The Ambassadors.  A close examination of the carpet will reveal the quincunx pattern and the swastika.  The swastika or wan, 卍,in Chinese philosophy is a math symbol meaning the ten thousand things and could very likely be based on the Chinese 3x3 magic square (see Needham, ibid, page 58 and 308).   The quincunx (the Chinese 3x3 magic square and the swastika) were cosmo-magical symbols that emphasized math and numbers and all the previously mentioned concepts.  The quincunx can be represented as four around one: 

Holbein is not only referencing the pavement at Westminster Abbey (see John North’s The Ambassadors Secret and the exceptional Patterns of Thought by Richard Foster), he is also identifying the axis mundi of his masterpiece.  Here is Holbein’s The Ambassadors with the vertical axis that connects Heaven with earth and also the earth’s axis that connects with Jesus being crucified:

The two lines intersect between the lute and the carpenter’s square (at a 27 degree angle?) with the vertical line from the Pole Star connecting with the underworld via the eyes of the skull.  The gentleman’s fingers are pointing to the earth and a math book, with the carpenter’ square holding the book open to this page: 

The numbers 1,890,000 and 81,648 are related to the number 27 (the triad cubed), a theme persistent throughout Holbein’s masterpiece The Ambassadors.  The large numbers relate to the 27x27 magic square with 365 in the middle.  Also, an ingenious Pythagorean exercise involves these large numbers.  But you will have to purchase my ebook for the answer of this Pythagorean riddle.   My book will make the case that the ancient Chinese, the Pythagoreans, and Plato all knew about these magic squares and their relevance.

The book has been identified as a mathematical treatise by Peter Apianus.  The numbers 81648 and 1,890,000 are related to the number 27, a theme persistent thru out Holbein’s masterpiece The Ambassadors.  Apianus had written an important book on the mapping of the cosmos in 1524 and was highly praised in Europe as a mathematician and astronomer.  In the book is featured this image of the “nocturnal”:

This image is from Peter Apianus’ 1524 book and also from Joseph Needham’s book, Science and Civilisation in China, Volume 3.


Joseph Needham, Science and Civilisation in China, Volume 3, page 338:

The nocturnal was a device for telling the time at night by means of the fixed stars, and consisted of a graduated disc or dial with a hole In the centre, and a long projecting arm capable of rotations,  The centre being placed in a direct line  between the observer’s eye and the pole-star….

The Circumpolar Constellation Template

According to Needham, the nocturnal had evolved from the circumpolar constellation template, an ancient Chinese disc that was used in conjunction with the sighting tube and would identify the circumpolar  rotation of β Ursae minoris.  

Wikipiedia:   Ursa Minor is notable as the location of the north celestial pole, although this will change after some centuries due to the precession of the equinoxes.

It may be difficult to accept that so many mathematical and cosmological philosophies were common to the Pythagoreans, the ancient Chinese, and the Renaissance artists, mathematicians, and astronomers.  But there can be no denying the roles of the gnomon and the resultant shadow that follows the suns path, the right-angle triangle theorem, and the carpenters’ square in combination with mathematics sheds illumination for humankind's understanding of the stars, Heaven, and earth.  And according to the Chinese Yi Jing, this would include the magic square as well.