Introduction to The Secret of the Luo Shu, Numerology in Art and Architecture

If numbers can be considered the letters of a universal alphabet, then the perfect arrangement of numbers that can describe the formulas of mathematics correspond to heavenly statements of mathematical fact.  (This represents the only pure truth of our universe).

When humankind discovers these mathematical equations and can apply them to daily life, society reaps the benefits of evolution and prosperity. It would be a natural progression for symbols to be born, symbols that are directly related to the advancement of humankind.  The integration of these math symbols with art and architecture can then tell the story of civilization.  The Chinese magic square is one such symbol.

This paper is dedicated to the basic elements of numerology, i.e. the symbolism of numbers based on the Chinese magic square of three, otherwise known as the Luo Shu.

The Luo Shu represents the ideal model of time and space: time as in the numbers of the calendar and space as in the Pythagorean Theorem, the most important mathematical relationship to ancient civilization.  Hence, the Luo Shu was the basic model the early Chinese determined as their concept of the world and was incorporated into art, architecture, city design, royal tomb design, and served as an important correspondence to a select group of Chinese pictograms.

As time unfolded, different cultures would adopt the model of the Luo Shu with usage of its perfect symmetrical template to be found in Roman mosaics, early Christian art and architecture, Islamic pottery, tiles and textiles, and Italian Renaissance art. 

The magic square is represented by the quincunx design, or quincuncial composition, which is based on the arrangements of numbers within the magic square and the powerful math that lies within its manifestation.  The model of the magic square became a motif repetitive in art and architecture which represented the implementation of math to establish order on earth.

Christianity incorporated magic square symbolism in its early church design, illuminated manuscripts, textiles and more.  The walls, windows, pavements, and mosaics of the church were adorned with this motif.  In fact, the cross and square ground plan, also known as the nine-bay church ground plan also known as the quincunx ground plan, was the most popular church design in Medieval times.

Why the Magic Square?

Table 1.  The first magic square and the corresponding formula; in the example of an order 3 magic square, X = 3 and Y = 5.  A magic square is defined as an arrangement of numbers in a square such that the sum of any row, column, or major diagonal will yield a singular result, in this case 15.  Pythagorean triads highlighted in green.  (Note: it is necessary to use Y as a means to express the fraction
(X² + 1) ÷ 2 for an easier visual processing of the formula).

The formula above allows the generation of higher order magic squares in the Luo Shu format as long as X is odd and greater than one. 

To understand the wisdom that the Luo Shu offers, one must consider that the Luo Shu represents a “set of magic squares” based on a common formula that can be expanded to include higher order magic squares; the Luo Shu is not limited to just the 3x3 magic square.  A “set” of magic squares traditionally used in alchemy would include the odd and even magic squares up to order nine; this author considers the Luo Shu set of magic squares to include only odd magic squares up to the order twenty-seven.  Examination of the math of the higher order magic squares will give insight to the various mathematical properties associated with magic squares in the Luo Shu format.

The fraction (X² + 1) ÷ 2 will be represented by Y and will always equal the center number of the corresponding magic square.  X and Y will always be the odd components of a Pythagorean triad, (X)(Y) will equal the magic constant, and X²Y will equal the sum total of all the numbers in the square, or ∑X².

Table 2.  The first three magic squares of the Luo Shu demonstrating the Pythagorean Theorem in the form of Pythagorean triplets of numbers (in green) in the shape of a gnomon or right angle.

The first significant feature of the Luo Shu is the Pythagorean triad of numbers in the shape of the carpenter’s square or gnomon, an important symbol that is closely associated with the Luo Shu as it also represents time and space as well as applied mathematics (see table 2).

The second outstanding feature of magic squares in the Luo Shu format is the cross of odd numbers that run through the horizontal and vertical axis with a center point of reference.

Table 3.  Example of the cross of odd numbers that will appear in every magic square in the Luo Shu format. Here, the 9x9 magic square. 9, 40, and 41 satisfy the Pythagorean or Right-Angle Triangle Theorem. (X)(Y) = 9 x 41 = 369, the magic constant.  In alchemy, this square corresponds to the moon.

The third important feature of the 3x3 magic square are the quincuncial relationships of even and odd numbers around the center number.  A quincunx is defined as an arrangement of five things with one at each corner and one at the center of a square. The even numbers of the first nine numbers will occupy the corners (or four quadrants) and the odd numbers will form a cruciform with the number five at the center; this is known as a double quincunx.

Table 4.  All magic squares in the Luo Shu format will have a cross of odd numbers that run through the vertical and horizontal axis.  The ya cartouche 亞 – cosmic temple – reflects this important feature of the Luo Shu.  The odd numbers equate to the heavenly yang force, the four cardinal directions and the center; the even numbers represent the earthly yin force.

The early Chinese believed these squares were a communication from Heaven because of the wisdom contained within.  As these squares exhibit the Pythagorean Theorem, a center, and a cross of odd numbers that separates the square into four quadrants, the “luo shu” squares became a perfect model to represent Heaven and earth in microcosm, to be used in art and architecture while at the same time communicate a wisdom of mathematics that would remain hidden, known only to the intelligentsia.  This explanation can help to understand why this pattern shows up across a diversity of cultures, beginning with the early Chinese and continuing thru the Renaissance era.  This model, or symbol, of numerology would therefore be transformed into an artistic composition of motifs beautifying tiles, pavements, mosaics, books, and textiles to mark structures and objects of religious or political import. 

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PORTABLE SUNDIAL, TYPE B

That the Luo Shu appears on Chinese sundials from the seventeenth century has never been documented until now.

This type of sundial did not appear much before 1600 AD.  Many replications of this instrument can be found at several museums.  The Type B sundial, or equatorial dial, became popular in Europe during the seventeenth century.  

One can clearly see the numbers of the Luo Shu in the second ring adjacent to the ring with the eight trigrams.  Even numbers are in black, odd numbers are in red.  It is perfectly appropriate for the Luo Shu to be used as a motif to mark astronomical instruments that are related to time measurement and space directionality.  

Instruments such as the portable sundial help humankind to connect with Heaven.  The Luo Shu has a legacy of being related to the calendar, the Pythagorean Theorem, the carpenter's square and the gnomon.  

From Science and Civilisation in China, p. i of every volume:

The carpenter's square is no ordinary tool, but the gnomon for measuring the lengths of the sun's solstitial shadows. 

The above pictures were taken from Time, Science, and Society in China and the West edited by J.T. Fraser, et al (1986).  Another example can be found in Science and Civilisation in China, Vol. 3, p. 308, Figure 134.  Please note the editor offers explanations of the markings in each of the rings around the compass but makes no mention of the Luo Shu being present in the second ring.  This is the first time this observation has been noted.

Symbols such as the Luo Shu and carpenter's square are frequently used in art and architecture to identify things of political or religious significance such as temples, churches, art, architecture, and astronomical instruments.  

My paper, The Secret of the Luo Shu, Numerology in Art and Architecture explains in detail the use of the Luo Shu and related symbols as an artistic motif used by many cultures for over two thousand years.  

Order you PDF today! (Sale, PDF is $3 until Dec. 31, 2024).

9X9 MAGIC SQUARE

The feature that makes this magic square most unusual is another magic square appears within its framework.  That demands further inspection.

I found this square on Jollymaths.com

Reducing the numbers of this 9x9 magic square to their Pythagorean root number generates a square of repetitive Luo Shu magic squares.  The entire square becomes nine individual 3x3 magic squares.

Notice there are nine consecutive numbers in each grouping with the first nine numbers located in the first row middle column grouping.  

The second grouping of consecutive nine numbers, ten thru 18 can be found in the lower right grouping of numbers.  The third set of consecutive nine numbers, nineteen thru twenty-seven, can be discovered in the first column middle row group.  If each group of nine numbers is assigned its position in a 3x3 grid according to numerical order, then the Luo Shu magic square shows up once again.

THE 21X21 MAGIC SQUARE and HE TU

If all the numbers in the 21x21 magic square are reduced to their Pythagorean root number, the He Tu pairing emerges.  In addition, the 3x3 magic square appears along the major diagonal that runs from the upper right to the lower left corner cells.  The He Tu pairing can be found in the 3x3, the 9x9, the 15x15, the 21x21 and the 27x27 magic squares.

The magic constant of this square is (21)(221) = 4,641; every row, column, and major diagonal sum to this number.  Note the cross of odd numbers that run through the horizontal and vertical axis of the square.

Every number one is paired with every number six and so on for two-six, three-seven and four-nine.  The pattern that emerges is different for each square the pairing occurs in.

This post demonstrates that the He Tu pairing is to be found in the Luo Shu magic squares.

Note:  There are some unpaired numbers along the borders of the magic square.  Keeping in mind the magic square is the two dimensional image of a torus, the numbers that are unpaired actually have their paired mate located at the opposite border in the same row or column so in three dimensions the pairing is complete.  Five is unpaired as it represents the central pivot.  

As stated above, the pairing can be found in several squares, beginning with the 3x3 M.S. and skipping every six orders so the next magic square that the He Tu will be discovered is the 9x9 M.S.

This proves that the He Tu pairing originates from the Luo Shu format of magic squares.  This assertion has been made previously by Lars Berglund in his book, The Secret of the Luo Shu, Numerology in Chinese Art and Architecture (1990).  Berglund puts forward his theory but offers no evidence, now we have the evidence.

La Sagrada Familia

FROM THE PAPER  "THE SECRET OF THE MAGIC SQUARE, NUMEROLOGY IN ART AND ARCHITECTURE
(copyright pending 2024)

La Sagrada Familia, Barcelona, Spain

The magnificent church, La Sagrada Familia, was started in 1892 by architect Antoni Gaudi and is presently still under construction although a completion date of 2026 has recently been announced.  Two significant features relevant to this paper are the appearance of a magic square albeit an even magic square of order 4, and the coded use of numbers reflecting the Pythagorean Theorem. 

Image 19.  A 4x4 magic square at the entrance of La Sagrada Familia with a magic constant of 33.

This is not the traditional magic square of order 4 as that square would have 34 as the magic constant.  This square uses two 10s and two 14s in order to have the magic constant equal 33, a reference to the age of Christ when he died.  A traditional 4x4 magic square would have the numbers one thru sixteen and there are 880 different arrangements of numbers to achieve this end.                        

Stained glass image of quincuncial composition in La Sagrada Familia using the classic quincunx and comparing to jewelry from the Lazaro Galdiano Museum, Madrid.

Stained glass window exhibiting the Pythagorean Theorem in code, La Sagrada Familia.

Counting the number of petals in the large circle, one comes up with 16 or 4².

Counting the number of circles including the large circle (but not the real small circles) one comes up with 25 or 5².

Counting the tomb shaped figures on one half of the window, one comes up with 9 or 3².

3² + 4² = 5²

Could it be a coincidence that Gaudi uses a magic square, makes a coded reference to the Pythagorean Theorem, and uses quincuncial composition?  The magic square represents the establishment of order on earth by connecting with Heaven thru the use of mathematics; repetitive concepts familiar to early Christian art and architecture. 

La Sagrada Familia, outside of church.

Another example of Gaudi's use of square numbers:  there are 81 (9²) circular windows in the above picture, the last grouping has 9 windows and both 81 and 9 are powers of three (the triad). 

for PDF of the complete paper (36 pages), 
email:  robert.luoshu@gmail.com
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The Secret of the Magic Square, Numerology in Art and Architecture

25 images and 15 tables, discusses in detail the magic square and its relevance in early China and the West as a motif in art and architecture.

©2025 pending