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 registration number TXu 2-483-178
effective date of registration: January 30, 2025
Registration decision date: April 25, 2025
Author: Robert Dickter

United States Copyright Office

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
$7.00

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

MUSIC AND THE MAGIC SQUARE

I have always wondered why this 3x3 pattern appears on bronze bells from over 2,000 years ago.  These bells were known to be used for ceremonial purposes and were played in the royal court. Examining the relationship of the Luo Shu, He Tu, and number manipulation may lead to some interesting results pertaining to music.
Upon close examination of the 9x9 magic square in the Luo Shu format, it can be demonstrated that the He Tu pattern presents itself. 

First, a review of the 3x3 magic square also known as the Luo Shu. It is important to realize that the 3x3 magic square has a formula:

3x3 Magic Square				FORMULA
4	9	1		Y + 1	X2	Y - X
3	5	7		X	Y	2Y - X
8	1	6		Y + X	1	Y - 1

A) Formula for 3x3 magic square which can be expanded to larger magic squares.¹

This relationship can be expanded into larger order magic squares that all feature a Pythagorean triplet of numbers at the heart of every magic square. In addition, the numbers of the calendar are featured in the 13x13 and 27x27 magic squares. For these reasons, the Luo Shu was considered an ideal model of time and space. 

[For convenience, Y = (X² + 1) ÷ 2].

(Note: The Chinese translation of Luo Shu and the He Tu deserves a moment of reflection. He Tu translates as Yellow River Chart indicating a singular image. Luo Shu translates to Luo River Document or Luo River Writing possibly indicating multiple images. It is best to think of the Luo Shu as a set of magic squares based on a formula that describes the simplest magic square, the 3x3, but can be expanded to include larger order magic squares that have relevance to the Chinese numerology system. 

The Luo Shu is from Heaven

In a perfect world, the music of the royal court would be composed by Heaven and not by humankind. Music was considered to be a gift from Heaven, as are numbers and the math that follows.

The early Chinese placed a sacred bestowment on all instruments that connected civilization with Heaven. Astronomical instruments, musical instruments, and mathematical tools such as the carpenters’ square and numbers were entities that allowed humankind to connect with Heaven. Some of these astronomical instruments (the bi disc and cong tube) became relics used in ceremonies, other tools such as the carpenters’ square became iconic symbols.  These religious relics and iconic symbols could find their way into the royal tomb to assist in the emperor’s journey thru the underworld.

The early Chinese believed that to evolve and prosper, the sacred rules of Heaven must be followed. Within the math of numbers (i.e. the language of Heaven) was a plan; society could apply mathematics and succeed. This so called “plan from Heaven” comes from the Hong Fan, a legendary Chinese belief system that claims the Luo Shu Magic Square contains the “plan for humankind”.  Regardless of Chinese mythology, it is clear that the “set” of magic squares in the Luo Shu format are not man-made, these mathematical relationships must be discovered by man and therefore are constructed by “nature”.

Hence, magic squares were part of an armamentarium of important tools (or instruments) that helped guide the early Chinese as magic squares (in the Luo Shu format) were a major influence on:  temple design, pictograms of the Chinese language, art, and even tomb and city design.²  This is consistent with early Chinese classification systems that depend on centrality, order, and symmetry.  As the magic square was the perfect model for these requirements, the early Chinese, especially the royal court, invested heavily (labor, materials, money) by incorporating the esoteric, yet mathematical, philosophy of the Luo Shu into society over a period of 2,000 years (1,100BC to 1,000AD). Although the Luo Shu was not included in the Yi Jing until about 1,000AD, there can be no doubt that the legacy of the Luo Shu had begun over 2,000 years prior to this inclusion!

THE 9X9 MAGIC SQUARE AND THE HE TU

9x9 Magic Square 9x9 reduced to Pythagorean root number 37 78 29 70 21 62 13 54  5 1 6 2 7 3 8 4 9 5  6 38 79 30 71 22 63 14 46 6 2 7 3 8 4 9 5 1 47  7 39 80 31 72 23 55 15 2 7 3 8 4 9 5 1 6 16 48  8 40 81 32 64 24 56 7 3 8 4 9 5 1 6 2 57 17 49   9 41 73 33 65 25 3 8 4 9 5 1 6 2 7 26 58 18 50  1 42 74 34 66 8 4 9 5 1 6 2 7 3 67 27 59 10 51  2 43 75 35 4 9 5 1 6 2 7 3 8 36 68 19 60 11 52  3 44 76 9 5 1 6 2 7 3 8 4 77 28 69 20 61 12 53  4 45 5 1 6 2 7 3 8 4 9

B) Here one one can see the He Tu pattern emerge when the numbers are reduced to their Pythagorean root number.

In order to apply musical notes to this arrangement of numbers it is useful to know that in the ancient Chinese musical gamut, 5 notes made up the pentatonic scale. The Pythagorean root numbers will be re-written in Base 5 notation to match the pentatonic scale:

Translating to Base 5 notation
1	11	2	12	3	13	4	14	5
11	2	12	3	13	4	14	5	1
2	12	3	13	4	14	5	1	11
12	3	13	4	14	5	1	11	2
3	13	4	14	5	1	11	2	12
13	4	14	5	1	11	2	12	3
4	14	5	1	11	2	12	3	13
14	5	1	11	2	12	3	13	4
5	1	11	2	12	3	13	4	14

C) The Pythagorean root numbers translated to Base 5 notation

Note C is selected as the middle note of the pentatonic scale, note C is assigned to the number five, and with that the rest of the notes (A, B, E, and G) fall into place. The six is translated to eleven in base five notation and refers to the A note of the second ascending scale and is noted as A(2).

NUMBERS IN	NUMBERS IN	NOTE
BASE 10 NOTATION	BASE 5 NOTATION
1, 6	1, 11	A, A(2)
2, 7	2, 12	B, B(2)
3, 8	3, 13	E, E(2)
4, 9	4, 14	G, G(2)
5	5	C

D) The numbers have been translated into Base 5 notation and assigned musical notes in the Pentatonic scale.

E) The He Tu combined with the Luo Shu and the musical notes assigned to the numbers.

Sequence of notes

FIRST STANZA

5, 1, 6  =  C, A, A(2)

5, 7, 2  =  C, B(2), B

5, 3, 8  =  C, E, E(2)

5, 9, 4  =  C, G(2), G

F) Sequence of notes determined by numbers of the He Tu combined with the Luo Shu

The wan character, 卍,has been associated with the He Tu and the Luo Shu.  Berglund  maintains that the symbol 卍 was based on pairs of numbers of the Luo Shu Magic Square representing the five elements and, by extension, the universe.3  (Aihe Wang, A.C. Graham, and Sarah Allan all support the Si Fang/Wuxing philosophy involving the Luo Shu and He Tu).4  If this is true, then the 卍 comes from the He Tu numerical pairs being emphasized on the Luo Shu 3x3 grid of numbers, creating the pairs of numbers associated with the middle number, 5. This gives the symbol  卍 mathematical relevance and corresponds to several important Chinese philosophical concepts: the four cardinal directions, Heaven and earth, the five elements, the axis mundi or cosmic center, the carpenters’ square, the right angle triangle theorem, the gnomon and time keeping.  As this pattern (1,6; 2,7; 3,8; 4,9; 5) appears in the 9x9 M.S., then there can be no doubt that the He Tu pattern is related to, and even comes from, the 9x9 M.S. (see table B above).  The Luo Shu and He Tu were the two main mathematical philosophies by which everything in the YinYang/Wuxing compendium was connected.

Therefore, are the He Tu and Luo Shu the basis of early Chinese ceremonial music? If a musical composition were to be based on the mathematical patterns of the Luo Shu and He Tu, then this music could be hypothesized to be “from Heaven” and be consistent with YinYang/Wuxing philosophies. In other words, music derived from the mathematics of the Luo Shu would be ideal for sacred and ceremonial music as well as music for political purposes.

Footnotes and References

1.       1. Dickter, Robert, Number Time Archetype, California (2014)

2.       2. Schinz, Alfred, The Magic Square, Great Britain (1996)

3.       3. Berglund, Lars, The Secret of the Luo Shu, Sweden (1990)   
          4. Wang, A.E., Cosmology and Political Culture in Early China, Cambridge (2000)

Symbolic images and numerology belong to the very essence of the alchemists’ mentality. The use of symbols and numbers convey a much deeper communication, or language, and therefore, have a much more profound affect on the subconscious than the mere words of philosophy or mythology.  The most profound symbols are archetypes, symbols that live as ideas in our subconscious but link us to a collective society, as if archetypes are an intellectual vestige of an earlier civilization.

Magic squares play a vital role in alchemy and magic.  Magic squares have a long history as talismans that date back 2,000 years. Magic squares possessed magical qualities that could ward off bad spirits or bring good luck. Magic squares in the Luo Shu format form a “set” of magic squares that are closely related to the Pythagorean Theorem (see last post).

3x3				5x5						7x7
Magic Square				Magic Square						Magic Square
										22	47	16	41	10	35	4
				11	24	7	20	3		5	23	48	17	42	11	29
4	9	2		4	12	25	8	16		30	6	24	48	18	36	12
3	5	7		17	5	13	21	9		13	31	7	25	43	19	37
8	1	6		10	18	1	14	22		38	14	32	1	26	44	20
				23	6	19	2	15		21	39	8	33	2	27	45
										46	15	40	9	34	3	28
SATURN				MARS						VENUS
LEAD				IRON						COPPER
EARTH				FIRE						METAL
BLACK				RED						GREEN
ONYX				RUBY						EMERLAD
CROCO				HORSE						DOVE

Table One:  The first three magic squares in the Luo Shu format with each demonstrating a unique Pythagorean triplet of numbers.  Each magic square has an alchemical correspondence with a specific planet, metal, element, color, stone, and animal. 

Magic Squares and the Kabbala

We are all familiar with how the vast majority of people, for thousands of years as well as presently, believe in the influences of the planets on the character of humankind. The early Chinese, the Kabbalists, the shamans and magis all developed systems to obtain the most auspicious of these influences that the planets may impart.

The Kabbalists assigned names and symbols to each planet.  These signs were followed by magic squares.  The signs and magic squares for the planets have been published in several books, one of the first by Cornelius Agrippa, who completed De occulta philosophia in 1510.  The set of magic squares also appeared in books by Luca Pacioli, De viribus quantitates (1498), and Athanasius Kircher, Arithmologia (1655).  Magic squares were popular during the Italian Renaissance and were known to the famous artists, mathematicians, and astronomers of the time. 

Magic Squares and Symbols

Time and space have always been important concepts to philosophers, mathematicians, and scientists.  The carpenter’s square and the magic square are ideal symbols for time and space with the history being Chinese in origin. 

A prosperous, evolving civilization begins with the gnomon, an astronomical instrument which served as the keeper of time for thousands of years.  The light of the sun would cause a shadow from the gnomon to appear on the ground and measuring this shadow would determine the time of year.

 It was because of the gnomon and  the ordering of sequential time that allows for the necessity of Numbers and the preservation of the “Numbers of Time.” 


Understandably, this took years of observing and record keeping but this was the method to determine summer and winter solstice. 

The definition of gnomon, in Webster’s Dictionary, indicates that the word is inherently related to the carpenter’s square.  In Joseph Needham’s Science and Civilisation in China (page i of any volume), the words carpenter’s square and gnomon are used interchangeably:

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

Therefore, the carpenter’s square measures time via the role of the gnomon and measures space via it’s graduated numbers of measurement in combination with the right-angle triangle theorem (the Pythagorean Theorem). 

                                                        
The gnomon, the most important astronomical instrument known to civilization for thousands of years.

The Chinese Connection to the Carpenter’s Square

A very old Chinese pictogram, 工 (gong), refers to an artisan who carries his carpenter’s square; the glyph is used in many words and also has a relationship with music and astronomy.  Another very old pictogram, 曲 (chu, ju, or jue) also means carpenter’s square.  One can see that, according to Mathew’s Chinese – English Dictionary (1956), the glyph is related to many words, but like 工 (gong), is also related to music.  Alfred Schinz, in his book The Magic Square (1996), reports that the pictogram 曲 is based on the 3x3 grid of the magic square and appeared on oracle bones dating to the Shang dynasty (1100 BC).

If we re-examine table one above, we can observe Pythagorean triplets of numbers in the shape of a right angle or gnomon residing at the heart of each magic square.  A gnomon shape of Pythagorean numbers in a 3x3 grid can explain why a pictogram such as 曲 (chu) means carpenter’s square. 

Another interesting link exists with the English definition of gnomon with that of the Chinese.  Referring to Webster’s Dictionary once again, the fourth definition of gnomon is canon or tenet. 

This relates to the Chinese phrase 規矩 (gui chu) which combines the pictograms for the compass and carpenter’s square and means:  following the rule of tradition and moral standards for establishing order; the way things should be.  Measuring the shadow of the gnomon involved following the rules of tradition thereby creating a moral standard.  Therefore, the ancient method to establish order involved using mathematical tools and traditional ways of observing and documenting the numbers of the shadow length.  Thus, the English definition of gnomon referring to canon or tenet, that is, the way things should be may lead one to believe that the etymology of the Greek derived word gnomon may have a Chinese connection.

The importance of the gnomon to civilization was documented by the early Chinese in one of the oldest math books known, the Zhou bi suan jing, The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, wherein it states, “the combination of the right angle with numbers (the carpenter's square/gnomon) is what guides and rules the universe.” 

These Chinese pictograms from thousands of years ago represent one of the earliest written languages known to linguists.  The use of mathematical symbols with the incorporation of traditions were the basis of some important words (pictograms) in the Chinese language and demonstrates the heavenly role the magic square played in early Chinese civilization.  Diagrams such as the Chinese magic square contain vestiges of knowledge of the most ancient human beings, reveal a mystical mathematical vision, and may even be a link to a primitive, universal language.

Numbers, the right-angle triangle theorem, the carpenter’s square, and the gnomon are inter-related with the magic square, alchemy and magic; all of which make correspondences to math, time, and space.  Symbols and archetypes are a natural result from the traditional and repetitive use of these functional tools that help advance the prospects of human evolution and prosperity. 

       

Mosaics from Ravenna, Italy represent some of the finest examples of early Christian art from the fifth to tenth century.  It may be a far-fetched theory, but the symbols on the religious garments simulate the carpenter’s square and the Chinese 工 (gong) pictogram.  Art historians have no explanation for these markings.  However, it is known that in the middle ages, the Christian hierarchy was obsessed with obelisks, gnomons, astronomy instruments, Pythagorean symbolism, numerology, and magic squares, in other words, instruments of time and space.  Was there a Chinese influence on early Christian art and architecture?

This blog is dedicated to mathematics and the magic square – to learn more about how the magic square influenced Chinese civilization as well as early Christian art and architecture, please order my book, Number Time Archetype.

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