Taoist Cultural Secrets (Part 2): Taoist Mathematics
Taoist culture places great emphasis on mathematical divination (Shu Shu), which is composed of two branches: arithmetic and mathematics. Arithmetic uses a series of symbols to simulate the natural laws of change between heaven and earth, summarizing a set of models. The greatest utility of this model lies in its ability to simulate reality, thereby deducing the past and future. Mathematics, on the other hand, is the study of numbers used to analyze everything in the world. Many people are familiar with Taoist arithmetic but less so with Taoist mathematics; here we discuss mathematics.
This mathematics differs from mathematics in the modern sense. Here, mathematics refers specifically to the study of numbers themselves, including Taoist reflections and summaries on the nature of numbers. If one understands the Pythagorean school's philosophy of 'all is number' in ancient Greece, one will have a deeper comprehension of the concept of Taoist mathematics.
Taoism most strongly opposes making bizarre guesses without practice or reason and then treating those guesses as conclusions. Therefore, here we unveil just the tip of the iceberg of Taoist mathematics, hoping to give those unfamiliar with it a preliminary understanding.
In mathematics, numbers are supreme; all things in the world can be categorized into numbers. Due to certain constraints, we will only introduce some mathematical theories from the Luoshu model here.
[Luoshu Nine Palace Numbers]
In Qimen Dunjia, it is said, 'First arrange the Nine Palms on the palm, fifteen runs through the vertical and horizontal.' This is the Taoist Nine Palace number, which corresponds to the magic square of order three in modern mathematics. As is widely known, the characteristic of a magic square of order three is that the sum of any row, column, or diagonal is always fifteen, which is the most basic expression of the Nine Palace numbers.
Taoism believes that the Nine Palace number represents a model of perfect balance. The existence of our world is necessarily determined by a series of rules, and these rules must form a balance. No matter how the world changes, it evolves around this balance, giving rise to our world.
If we wish to study this world, we must first find the initial state of all things. In terms of numbers, this is expressed by the Nine Palace numbers.
So, what characteristics indicate that the Nine Palace number is the most stable balance? And does it possess properties that higher-order magic squares do not have? If it were merely that the sums of rows, columns, and diagonals are equal, that would be easy; it is easy to calculate magic squares of order four, five, six, and higher.
However, the Tao Te Ching says, 'The Tao produced One; One produced Two; Two produced Three; Three produced All Things.' Why did 'All Things' begin to be generated from Three, and not Four or Five? Why only from Three?
Is it because the wisdom of the ancients was limited and they could not deduce higher-order magic squares? No, absolutely not.
The shape of the Nine Palace numbers is as follows:
4 9 2
3 5 7
8 1 6
Characteristic One: The Nine Palace number has only one unique expression; other forms are merely rotations of it. The number 5 always remains in the center, which is unchangeable. Higher-order magic squares that satisfy the condition of equal sums for rows, columns, and diagonals often have more than one solution.
Characteristic Two: In the Nine Palace numbers, the sum of opposite pairs forming two-digit numbers is consistent. This implies the interdependence of the Five Elements. For example: 27+61+83+49 = 92+76+18+34+92=220.
Characteristic Three: In the Nine Palace numbers, the sum of three-digit numbers formed by rows, columns, and diagonals is consistent. This implies the changes of the Three Powers (Heaven, Earth, Man) unified by one Qi. For example: 456+951+258 = 618+753+294 = 1665.
The above three characteristics are written extensively in general modern mathematics books, but these three characteristics are actually insufficient to describe the mystery of the Nine Palace numbers. Note that the following characteristics are not mentioned in modern books studying magic squares of order three.
[Rotation Category: Taking numbers by clockwise or counter-clockwise rotation]
Characteristic Four: In the Nine Palace numbers, the sum of four-digit numbers formed by rotation is consistent. This implies the balance of Yin and Yang in the division of Taiji. For example: 1834+9276=4927+6183.
Characteristic Five: In the Nine Palace numbers, the sum of five-digit numbers formed by rotation is consistent. If the first and last digits of the second five-digit number are the last digit of the previous five-digit number, the sum of any selected combination is always equal. This implies that by following the principle of connected Qi, a balance can be maintained. For example: 49276+61834=34927+76183=111110; furthermore, if we continue to deduce, we will find that for six-digit, seven-digit, eight-digit, nine-digit, and even infinite-digit numbers, as long as the principle of connected Qi is followed (taking two connected Qi in six-digit numbers, three connected Qi in seven-digit numbers, and so on), the sums will always be equal. This implies a simple principle of connected Qi, which is a secret in Feng Shui and divination.
Characteristic Six: Taking numbers at intervals, any two added together yield an equal sum. This means the sum of jumping palaces is also equal. For example: 24+86=62+48=110. 17+93= 31+79=110
[Flying Step Category: Taking numbers by flying steps according to the Nine Palace]
Characteristic Seven: The sum of taking four palaces in reverse and four palaces in the forward direction is always equal. 9876+1234 = 8765+2345 = 11110.
Similarly, taking three palaces in reverse and three palaces in the forward direction: 987+123 = 1110, 876+234 = 1110
Taking two palaces in reverse is also the same.
There is a connection between rotation and flying steps. What is this connection? It becomes clear from the table below.
| Fly | Rotate |
| 9+1=10 | 9+1=10 |
| 98+12=110 | 92+18=110 |
| 987+123=1110 | 927+183=1110 |
| 9876+1234=11110 | 9276+1834=11110 |
| 98765+12345=111110 | 92761+18349=111110 |
Characteristic Eight: The Connection Between Four Dimensions and Opposite Palaces
| Four Dimensions | Opposite Palace |
| 3+9+4 = 16 | 6 |
| 1+7+6 = 14 | 4 |
| 7+9+2 = 18 | 8 |
| 1+3+8 = 12 | 2 |
There are many other characteristics, which are not listed one by one. From the above characteristics, it can be seen that the distribution characteristics of the Nine Palace numbers are extremely precise, and higher-order magic squares cannot satisfy these features.
[Applications of Taoist Numbers]
So, what are the practical applications of these Taoist numbers?
In the Nine Palaces, except for the central five, every number corresponds to a Hexagram (Gua). By borrowing the Hexagrams, many wondrous things can be discovered.
For example, the Youngest Son corresponds to the Gen Hexagram, and the Youngest Daughter corresponds to the Dui Hexagram. The Gen Hexagram corresponds to the number 8, and the Dui Hexagram corresponds to the number 7. Using these two numbers as examples, let's see how the Huangdi Neijing (Yellow Emperor's Inner Canon) describes it:
Huangdi Neijing: Treatise on the Pure Yang of Antiquity
The Emperor said: When people grow old and have no children, is it because their strength and vitality are exhausted? Or is it the natural course of heaven?
Qibo replied: When a girl is seven years old, her kidney qi flourishes, her teeth change, and her hair grows.
At fourteen, her Tian Gui arrives, the Ren vessel is open, the Taichong vessel is flourishing, her menstruation comes on time, so she can have children.
At twenty-one, her kidney qi is balanced, so her wisdom teeth grow and her body reaches its full development.
At twenty-eight, her sinews and bones are firm, her hair reaches its full length, and her body is in its prime.
At thirty-five, the Yangming vessel declines, her face begins to wither, and her hair begins to fall.
At forty-two, the three Yang vessels decline above, her face is completely withered, and her hair begins to turn white.
At forty-nine, the Ren vessel is deficient, the Taichong vessel declines, Tian Gui is exhausted, the earthy way is blocked, so her form deteriorates and she cannot have children.
A man at eight years old, his kidney qi is solid, his hair grows and his teeth change.
At sixteen, his kidney qi flourishes, Tian Gui arrives, his essence overflows and is discharged, Yin and Yang harmonize, so he can have children.
At twenty-four, his kidney qi is balanced, his sinews and bones are strong, so his wisdom teeth grow and his body reaches its full development.
At thirty-two, his sinews and bones are robust, his muscles are full and strong.
At forty, his kidney qi declines, his hair falls and his teeth wither.
At forty-eight, his yang qi衰竭 (declines) above, his face withers, and his sideburns turn white.
At fifty-six, his liver qi declines, and his sinews cannot move.
At sixty-four, Tian Gui is exhausted, his essence is scarce, his kidney organs decline, his form is exhausted, and his teeth and hair fall out.
The kidney governs water, receiving the essence of the five zang and six fu organs to store it. Thus, when the five zang are flourishing, they can discharge. Now, if all five zang decline, sinews and bones loosen and fall, Tian Gui is exhausted, so the sideburns turn white, the body becomes heavy, walking is unsteady, and he cannot have children.
It can be seen that using seven for the young girl, the physiological changes starting from the young girl are calculated in seven-year cycles. Using eight for the young boy, the physiological changes with age are calculated in eight-year cycles. Finally, when the female seven numbers are exhausted and the male eight numbers are exhausted, the natural ability for biological reproduction is lost.
At the same time, since the Nine Palaces start from one, and one corresponds to Kan and the kidney, one-seven or one-eight both indicate solid kidney qi. For example, three-seven or three-eight, because three belongs to Wood, representing the generating force of Wood, it governs strong sinews and bones. Similarly, four-seven or four-eight, four corresponds to the Xun Hexagram, and Xun represents muscles, so it governs strong muscles. For women, this means firm sinews and bones; for men, it means robust pelvis and bones.
This applies to humans. Is this number theory useful for animals? For animals, looking at the most common livestock, when do they produce their reproductive ability:
Taking dogs as an example, dogs correspond to the Qian Hexagram, which is the number six. The average sexual maturity period for dogs is 6 months, exactly corresponding to the number 6.
Taking cattle as an example: Here specifically referring to domestic yellow cattle, which correspond to the Gen Hexagram, number eight. The estrus period for female cows is at about 8 months, exactly corresponding to the number 8.
Taking pigs as an example: Pigs also correspond to the Qian Hexagram, number six. The estrus cycle for sows is generally 5-7 months, averaging 6 months, exactly corresponding to the number 6.
The above are all mammals. Is it useful for non-mammals?
Taking chickens as an example, chickens correspond to the Dui Hexagram, which is the number seven. The normal growth cycle for domestic chickens is 14 months, which is exactly a multiple of seven. Incubation takes 21 days, which is also a multiple of seven. After another six months, they can lay eggs. Adding the 21 days of incubation, they can roughly start laying eggs at around 7 months.
Similarly, ducks are the same. The hatching time is 28 days, which is also a multiple of seven. After another six months, they lay eggs, totaling 7 months, still corresponding to the number 7.
These numerical principles permeate all things. The Luoshu Nine Palace numbers are applied in the Five Taoist Arts roughly as follows:
In Divination Arts, it is mainly applied in Qimen Dunjia and related Nine Palace calculation series, used for predicting human affairs.
In Destiny Arts, it is mainly applied in Nine Star Destiny Calculation and other methods.
In Physiognomy Arts, it is mainly applied in Eight Mansions Feng Shui and Xuan Kong Feng Shui, used for adjusting luck.
In Mountain Arts, it is mainly applied in stepping the Dipper and selecting time and space; no explanation needed.
In Medical Arts, it is mainly applied in image-number theory. For example, treating stomach diseases is mostly related to the numbers 2, 5, and 10, and strengthening the dosage is often around 20g, etc.
Next, let's discuss another manifestation of the application of Taoist numbers in reality: Coordination/Overall Planning.
The Pythagorean school emphasizes a proportional relationship among all things, which is a relatively clear expression. In Taoist mathematical concepts, there is also a similar saying about making numbers proportional. However, Taoist numbers emphasize coordinating real-world things more. Facts prove that this coordination has achieved excellent results.
Among Chinese musical instruments, the one that best reflects cultural heritage and has the most complex and exquisite craftsmanship is the Guqin. Making the Guqin to achieve an approximately perfect musical effect is difficult because it involves complex physical principles. The ancient Chinese were very smart, but their grasp of physical knowledge was not as detailed as modern theory. However, this does not mean they lacked the method to make a good Guqin.
The making of the Guqin directly uses the method of imitating heaven and earth. The length of the qin is 3 chi, 6 cun, and 5 fen, representing the 365 days of a year. The qin surface is curved, representing heaven; the qin bottom is square, representing earth; this is the theory of 'round heaven and square earth.' The Guqin has thirteen hui (markers), representing the 12 months and the leap month of a year. Initially, it had five strings, symbolizing Metal, Wood, Water, Fire, and Earth. Later, two more strings were added. Legend has it that this was to commemorate the sons of King Wen and King Wu and to enhance the momentum of overthrowing Zhou, but in doing so, it actually conforms to the Seven Celestial Bodies.
For more detailed numerical principles regarding the Guqin, one can refer to 'Biluozi's Guqin Making Method' and related materials, which will provide a deeper understanding of the application of numbers.
It needs to be pointed out that the use of numbers is only a standard, and in specific applications, there must be variations. 'The subtlety of application lies in the mind.' This is because the Way of Heaven has waxing and waning, and nature is constantly changing. Different eras, different aesthetic views, and different climatic conditions all affect the selection of natural materials. For example, when imitating Warring States period qins, using later wood often does not achieve the effect of directly using Warring States wood. In details, making a Guqin also requires attention to the generation and restriction of numbers. Through the combination of numbers and the Five Elements, the qin sound formed by this generation and restriction must conform to the aesthetic view in reality, and this aesthetic view depends on the aspirations of the human heart.
Simply put, numbers that are too natural and objective, although representing the Way of Heaven, are not necessarily what people like. Because 'heaven generates and heaven kills' is the principle of the Tao. As an objective principle, everyone can understand it, but this does not mean that understanding leads to love. Therefore, the number system has its own independent system, but in practical application, to invoke human resonance, adjustments must be made according to the expectations of the human heart.
Other things follow similar principles. The previous content mentioned ducks. If you have questions about why the cycle for American ducks is different, or why the cycle for half-fan ducks (hybrids of American ducks and Chinese river ducks) differs from ordinary ducks, please understand it yourself from the differences in geographical location and climatic conditions.