The Magic Square Blog

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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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.

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