Tuesday, November 14, 2017

This is my first painting, based on the 27x27 magic square.  I used the Roman mosaic from the last post as my inspiration (see below).  This is to demonstrate how the magic square can be used as a template for art and architecture.

I first sketeched a grid of 28 vertical lines and 28 horizontal lines creating a 27x27 grid of 729 squares, each measuring one square inch.  Then I outlined the cross within the square which can be identified by the cruiform shapes.  Using a compass and ruler, I then drew my circles and ovals and squares.  Then it was simply a matter of coloring in the shapes that resulted to highlight the circles, ovals, and squares and this is what was created using different colors.

What resulted is the same Pythagorean symbolism that can be seen in early Christian, oriental, and Roman art and architecture:  the cross-in-square pattern (Heaven), the four cardinal directions which intersect at the axis mundi (or center), the four quadrants (earth), the relationship of the square and circle, and the gnomon or carpenter's square (time and space) all are prominent.  This is referred to as sacred geometry using the quincunx model or four objects around a central object.

These are other examples of the quincunx sacred geometry used in art: Vincent Carpaccio, The Ambassadors Depart (1495), Roman mosaic from Caligula's ship (44 AD), and the cover of the Gospels of Theodelinda (seventh century).

Sunday, October 22, 2017

A remnant from Caligula’s ship, a four-by-four piece of mosaic flooring which was pilfered from an Italian museum prior to World War II, has now been confiscated by the Manhattan District Attorney’s Office.  What makes this discovery so fantastic is not just the mosaic’s exceptional beauty, it’s rarity, value, or provenance but the sacred geometry utilized which is consistent with the symbolic designs used in the flooring of some of the most famous churches in Italy, especially Rome and Ravenna, from the sixth century thru the thirteenth century.  The sacred geometry seen here from Caligula’s era is an early example of a mathematical template used in church design, on covers of illuminated manuscripts, and by famous Cosmati artisans of the twelfth and thirteenth centuries. 

The symbolic meaning of the distinctive geometrical and mathematical design will lead us to the understanding of the use of the quincunx, or geometric patterns of mystical mathematical symbolism based on the Chinese magic square and used famously in Christian art and architecture of the Middle Ages (and now thanks to this discovery, used in early Roman times as well).  But first, before we dive into the mathematical model for time and space, let’s go over a little background, or provenance, of this mosaic from Caligula’s ship.

Caligula was a famous Roman emperor who reigned from A.D. 37-41.  His decadent tastes led him to build giant ships, not for travel, but for entertainment and retreat from reality.  These ships were anchored on Lake Nemi – a circular volcanic lake outside Rome.  Following Caligula’s assassination, the ships were sunk. Mussolini, between 1929 and 1932, had the lake drained and the ships hauled ashore.  This mosaic was discovered and placed in a museum especially built for the artifacts recovered in two of Caligula’s ships in 1936.  By the end of World War II, the museum was burned down and many of the artifacts were damaged or destroyed but this mosaic survived undamaged leading some to speculate that the mosaic was removed before the destruction of the museum. 

In a rare sighting of the stolen artifact, the mosaic was spotted in a gallery in Rome and was photographed in the 1960s.  In 2013, an Italian expert on ancient marbles, Dario Del Bufalo, was lecturing a group of art historians and dealers in New York and displayed the photo.  The mosaic was recognized by some of the people at the lecture as being on display at the home of Mr. and Mrs. Fioratti of New York.  Evidently, an Italian aristocratic family who possessed the mosaic sold the piece to the Fiorattis in the late 1960’s thru a broker, an Italian police official known for recovering looted art by the Nazis.

The Manhattan District Attorney’s Office had a judge issue a warrant to seize the mosaic on September 18, 2017 on the basis that the authorities have evidence that the piece was stolen from an Italian museum before World War II.  The mosaic was returned to the Italian government October 19, 2017. Mrs. Fioratti offered no resistance.

The Caligula mosaic demonstrates the typical design used by the early church in art and architecture during the Middle Ages.  The cross-in-square represents the four cardinal directions and intersects in the center, or axis mundi, where Heaven, earth, and humankind co-exist.  The cross-in-square pattern creates the four quadrants where we can see four circular figures.  The cross-in-square represents Heaven and the four circles represent earth.   This is typical of quincunx geometry, or four objects arranged around a central object.  However, the relevance is mystical mathematical symbolism which incorporates Heaven, earth, the center or axis mundi, and humankind.  In addition, when we consider the Chinese magic square, we also incorporate the concepts of math, the right-angle triangle theorem (or Pythagorean Theorem) and the numbers of the calendar. 

As described many times before in this blog and in my book Number Time Archetype, there exists a formula to apply to the Chinese magic square that will lead to a set of magic squares called “magic squares in the Luo Shu format”.  These magic squares reveal a Pythagorean triplet of numbers at the heart of each square – a reference to the Pythagorean Theorem – which represents the describing of space in mathematical terms.  In addition, these magic squares will always have a cross of odd numbers thru the vertical and horizontal axis which creates the four quadrants.  Upon closer examination of the mosaic, one can see within the cross-in-square pattern a grid and within this grid exists a second quincuncial arrangement of four gnomon shapes or right angles around an identifiable center, a clear reference to the gnomon, a keeper of time, and the right angle or carpenter’s square, a symbol of time and space.  And the number 365, the number associated with the solar cycle, appears at the center of the 27x27 magic square, which represents time.  These squares, a gift from Heaven, had special reverence to the ancient Chinese a thousand years prior to their adoption by the Romans and Christians.

Therefore, the symbolic meaning of the quincunx incorporates all these concepts: Heaven, earth, humankind, the axis mundi, the four cardinal directions as well as time and space.  This is why the quincunx geometry was used as the ground plan for church design during the Middle Ages (known as the quincunx or cross-in-square ground plan) and why this same geometry appears on several covers of illuminated manuscripts during the Carolingian period, as well as on church art such as altar cloths and tapestries. 

However, the provenance of the magic square is Chinese and is known as the Luo Shu.  The Chinese used the magic square as their template for temple design as well as for city layouts and royal tomb design (see Alfred Schinz, The Magic Square: Cities in Ancient China and Lars Berglund, The Secret of the Luo Shu: Numerology in Chinese Art and Architecture).

The Quincunx as a Cosmic Symbol

Paloma Pajares-Ayuela sums it up best in her book, Cosmatesque Ornament:

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 circlein space, the sphererepresents the infinite, transcendent, and complete, in sum, the divine, God. The squarein space, the cubeis the symbol of the material, of the finite, non-transcendent, limited, solid; it 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 (solstices and equinoxes; change of seasons).  The cross is the mediating symbol that connects heaven and earth.

Tuesday, March 7, 2017


The circle and the square have a sacred place in human evolution.  Examples of the two icons are replete in art and architecture from the earliest of times.  The circle represents Heaven, the square represents the earth.  Adding to this, the male energy is represented by Heaven as the female energy is represented by earth.  The early Chinese refer to these energies, or qi, as yin (earthly and female) and yang (heavenly and male). 

My expertise is the study of magic squares in the Luo Shu format.  These squares are of an odd order and have peculiar characteristics that gave this “set of squares” a revered status.  The Luo Shu magic square was used in religious ceremonies, was the basis for religious art, and served as the ground plan for temple design and city layouts as this arrangement of numbers had a mystical connection to Heaven. 

These observations have been well documented by some of the greatest Sinologists such as Schuyler Cammaan, Alfred Schinz, and Paul Wheatley.  However, it was not until my discovery of the formula that describes the 3x3 magic square that it became evident that the Chinese magic square, referred to as the Luo Shu, actually refers to all orders of magic squares that can be derived from this formula.  Also, constructing magic squares using the Persian or Hindu method will generate squares of this type.

The study of these magic squares will demonstrate a unique Pythagorean triad of numbers associated with each order of square.  A cross of odd numbers run thru the vertical and horizontal axis of every square, regardless of size, and generates the cross-in-square template used in temple and tomb design (referred to as si fang).  Therefore math, and in particular, the Right-Angle Triangle theorem, is being highlighted in these squares.  Because the Pythagorean Theorem is the most important mathematical relationship to the evolution of humankind, these squares had special status to many cultures.

There has now been a new discovery with these squares that I refer to as magic squares in the Luo Shu format.

Tasha Lindsay has created spherical images based on these squares that are perfectly symmetrical generating as beautiful a mandala as can be since these images are based on a perfect kind of math.  The early Chinese believed the Luo Shu was a gift (or mandate) from Heaven and if used ceremoniously as well as functionally, humankind could improve their prospects to evolve and prosper. 

If it is true that these magic squares are not human constructs but the stuff bestowed upon us from Heaven, then Ms. Lindsay has made the visual and artistic* connection with Heaven.  Or at least with the magic square and circle. 

Tasha Lindsay