06/17) Yellow River Diagram, Lo Shu

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Hetu luoshu 河圖洛書 "The River Chart and the Inscription of the Luo"

The River Chart Hetu 河圖 The Inscription of the Luo Luoshu 洛書 The Yin/Yang diagram version of the Hetu and Luoshu The Hetu 河圖 "Yellow River Chart" and the Luoshu 洛書 (also written 雒書) "Inscription of the River Luo" are two cosmological diagrams used in ancient China. They are employed by both Daoists andConfucians and serve to explain the correlation of the hexagrams of theYijing 易經 "Book of Changes" with the universe and human life. They are also used in geomancy (fengshui 風水). The two diagrams are first mentioned in the chapter Guming顧命 of the Confucian Classic Shangshu 尚書 "Book of Documents", where it is said that the three precious jades and the hetu are stored in the Eastern Chamber. It can thus be assumed that the Hetu was a kind of jade stone the texture of which were interpreted as the eight trigrams (bagua 八卦) of theYijing. The Han period 漢 (206 BCE-220 CE) commentator Kong Anguo 孔安國 to theShangshu is the first who mentions the legend of a dragon horse (longma 龍馬) that emerged out of from the Yellow River and whose back was patterned with the shape of the eight trigrams. The diagram to be seen of the horse of the back was the so-called "River Chart" that was written down by Emperor Fu Xi 伏羲 and preserved as the eight trigrams. As an auspicious omen, the horse regularly appeared during the reigns of the virtuous rulers Yao 堯, Shun舜 and Yu 大禹. Confucius 孔子 lamented that during his lifetime, the horse (often translated as "unicorn") did not appear again, which was a bad portent of unlucky times. The Inscription of the River Luo is first mentioned in the bookGuanzi 管子, where it is said that a dragon turtle (longgui 龍龜) left the waters of the River Luo, so that an inscription was seen on its back, actually also a pattern of the shell, that could be interpreted as the eight trigrams in a constellation different to that on the Hetu. Similarly to the dragon horse, the turtle appeared during lucky times when virtuous rulers reigned the empire and ceased to be seen when bad and selfish men governed the world. Both inscriptions are mentioned in theXici 繫辭 commentary of the Yijing. The sage rulers read and interpreted the River Chart and the Luo Inscription and modelled their reign according to the evidence provided in the two diagrams. Yet the same text also says that Fu Xi invented the arrangement of the trigrams after observing the starry sky and all things on earth, without referring to the River Chart. The story of the two diagrams as omina of a golden age is repeated in the book Baihu tongyi 白虎通義 of the Han period, and the scholar Liu Xin 劉歆 said that the "Inscription of the Luo" was found by Yu the Great when he tamed the floods. He interpreted this inscription and came to the conclusions made in the correlative cosmology described in the chapter Hongfan洪範 of the Shangshu. While the Hetu is connected with the eight trigrams, the Luoshu is related to the Five Processes 五行 in this story. Archeological findings from the Yangshao 仰韶 and Dawenkou 大汶口 cultures have shown that the patterns to be found in the two charts and in the hexagrams date from the Neolithic age. The distribution of the numbers is also identical to that found in the numbers in the chart on the prognostication dish of the Great Unity of the Nine Palaces (taiyi jiugong zhanpan太乙九宮占盤) from the Warring States period 戰國 (5th cent.-221 BCE) found in Fuyang 阜陽, Anhui. Because of their enigmatic character, both charts were used by the apocryphal interpretation of the Confucian Classics that flourished during the Han period. The imperial bibliographyJingjizhi 經籍志 of the official dynastic history Suishu 隋書 says that there were nine chapters of text about the Hetu and 6 chapters about the Luoshu. It lists a book Hetu 河圖 with a length of 20 (including the Luoshu 24) juan, written during theLiang period 梁 (502-557) and already lost during the earlyTang period 唐 (618-907), as well as the books Hetu wei 河圖緯 and Luoshu wei 洛書緯 with a total length of together 45 chapters. Other books parts of which have survived until today are the Hetu longwen 河圖龍文, Hetu kuodi xiang 河圖括地象,Hetu xiyao gou 河圖稽耀鉤, Hetu kaoling yao 河圖考靈曜 orLuoshu lingzhun ting 洛書靈準聼. Today, fragments of 120Hetu books are preserved, and those of about 20 writings to the Luoshu. Although the compilers of the Suishu purported that these books were compiled during the ages of the mythical rulers of the past, it is for sure that they date from the Han period or somewhat later. Tang period scholars were not very interested in the two charts, and they only gained prominence again during theSong period 宋 (960-1279). The Daoist scholar Chen Tuan 陳摶 is said to have received a dragon chart (longtu 龍圖) from the Daoist Master Mayi 麻衣道者 and transmitted it to his own disciples Zhong Fang 種放, Li Gai 李溉, Xu Jian 許堅, Fan Echang 范諤昌 and Liu Mu 劉牧. Another branch of disciples were Mu Xiu 穆修, Li Zhicai 李之才 and the mathematician Shao Yong 邵雍 who was one of the early Song Neo-Confucians. The great Southern Song period Neo-Confucian Zhu Xi 朱熹 took over Shao Yong's transmitted shape of the two charts and used them for his interpretation of the Yijing, the Zhouyi benyi 周易本義. Interestingly enough, Liu Mu's two charts just had the opposite configuration of the trigrams as that of Shao Yong, so that Shao Yong's Hetu corresponded to Liu Mu's Luoshuand vice versa. Apart from Shao Yong's and Zhu Xi's version, there is the Yinyang yutu 陰陽魚圖 "Fish chart of Yin and Yang", a very popular version of the constellation of the eight trigrams, with the trigrams forming the outer frame and a black (Yin) and white (Yang) field the centre. The two fields are shaped in curves and creeping into each other to express the permanent fluctuation between Yin and Yang during the seasons. It is the much more famous of the trigram charts and is widely used in Daoist circles, where it became THE symbol of Daoism. Zhu Xi's version of the two charts can be seen in the illustrations above. Numerologic speculation was very common among the Neo-Confucians. Zhu Zhen's 朱震 book Zhouyi guatu 周易挂圖 says that the white circles in the Hetu chart sum up to an odd number (25), the black circles to an even number (20), with a total sum of 45. The white circles in the Luoshu chart are 25, black circles 30, with a total sum of 55. While the Hetusymbolized the theory and substance (ti 體) behind all things, the congenital and innate (xiantian 先天) nature of things, theLuoshu symbolized that practical aspect (yong 用) and the state of things how they are and live (houtian 後天). The main number of the Hetu is 10; 1 and 6 express the ancestral (zong宗, Celestial) nature, 2 and 7 the Way (dao 道), 3 and 8 friendship (peng 朋), 4 and 9 mutual support (you 友), and 5 and 10 protection and safety (shou 守). The main number of the Luoshu in 9; head is 9, feet is 1, left is 3, right is 7, 2 and 4 are the shoulders, 6 and 8 the legs, and 5 is the number of the physical centre. The Hetu also expresses geographical directions, each of the nine regions of the empire (jiuchou 九疇, jiuzhou 九州) represented by one symbol of the chart. The cipher 9 also stands for the Nine Palaces (jiugong 九宮) of the earth, while the cipher 5 represented in the centre of theLuoshu symbolized the Five Processes. Qing period 清 (1644-1911) Confucians like Huang Zongxi 黄宗羲 or Hu Wei 胡謂 contradicted the cosmological interpretation of the Neo-Confucians. In his inscription Wan Gongze muzhi ming 萬公擇墓志銘 Huang Zongxi assumed that the Hetu andLuoshu were very crude geographical maps of ancient times. Yet it is more probable that the charts served as a theoretical illustration of the universe for the purpose of prognostication, or to symbolize the elements of which the cosm or the body were believed to consist. Sources: Chen Yongzheng 陳永正 (ed. 1991). Zhongguo fangshu da cidian 中國方術大辭典, p. 37. Guangzhou: Zhongshan daxue chubanshe. Gao Liushui 高流水 (1996). "Hetu luoshu 河圖洛書", in: Feng Kezheng 馮克正, Fu Qingsheng 傅慶升 (ed.), Zhuzi baijia da cidian諸子百家大辭典, p. 894. Shenyang: Liaoning renmin chubanshe. Zhao Fujie 趙馥潔 (1988). "Hetu luoshu 河圖洛書", in: Zhao Jihui 趙吉惠, Guo Houde 郭厚安 (ed.), Zhongguo ruxue cidian 中國儒學辭典, p. 755. Shenyang: Liaoning renmin chubanshe. Zhou Guidian 周桂鈿 (1993). "Hetu luoshu 河圖洛書", in: Pang Pu 龐樸 (ed.), Zhongguo ruxue 中國儒學, vol. 4, p. 50. Shanghai: Dongfang chuban zhongxin.

http://www.chinaknowledge.de/Literature/Daoists/hetuluoshu.html

The Yellow River Map, Scheme, or Diagram (河圖, with variants for the second character) is an ancient Chinese concept. It is related to the Lo Shu Square. The origins of the two from the rivers Luo and Heare part of Chinese mythology. The development of the two are part of Chinese philosophy. (Wu:52)

Contents

  [hide] 

• 1 Geographical background
• 2 Mythological background
  • 2.1 Great Flood
  • 2.2 Fu Xi
  • 2.3 Yu
  • 2.4 He Bo
  • 2.5 Houtu
  • 2.6 Bagua
• 3 Historical evidence
• 4 Literature
• 5 Interpretation
  • 5.1 Tables
    • 5.1.1 Wuxing
    • 5.1.2 Cardinal directions
      • 5.1.2.1 Odd number order (1, 3, 5, 7, 9)
      • 5.1.2.2 Even number order (2, 4, 6, 8, 10)
• 6 Places
• 7 See also
• 8 References

Geographical background[edit]

Further information: Yellow River

Map of current configuration of Yellow River system, and the Luo (Lo) River.

The Yellow River or Huang He is the second-longest river in Asia, following the Yangtze River, and the sixth-longest in the world. The Yellow River has an estimated length of 5,464 km (3,395 mi). It has been and remains an important factor for human habitability of northern China. Anciently, it was often referred to just as He or "the River", and thus the Yellow River Map, just as "River Map" or "River Plan". The Yellow River has changed its course, settling in new beds, with different outlets to the ocean, many times in the past, often accompanied by death and devastation to the human population. Flowing through the yellow loess soil deposited as a deep, packed dust across much of northern China, it gets its name from the yellow color of resulting suspended solids. The Lo, or Luo, River is a major tributary.

Mythological background[edit]

Further information: Fu Xi, Great Flood (China), Houtu and Yu the Great

Myths of the Yellow River Map go back to earliest stages of the recorded history of Chinese culture.

Great Flood[edit]

Main article: Great Flood (China)

Further information: Great Flood

The Great Flood of China (Chinese: 大洪水; pinyin: Da Hongshuǐ, or just 洪水), also known as the "Gun-Yu myth" (Yang:74) was a major flood event that continued for at least two generations, which resulted in great population displacements among other disasters, such as storms and famine: according to mythological and historical sources, it is traditionally dated to the third millennium BCE, during the reign of the Emperor Yao. Treated either historically or mythologically, the story of the Great Flood and the heroic attempts of the various human and other characters to control it and to abate the disaster is a narrative fundamental to Chinese culture. Among other things, the Great Flood of China is important to understanding the history of the founding of both the Xia Dynastyand the Zhou Dynasty, it is also one of the main flood motifs in Chinese mythology, and it is a major source ofallusion in Classical Chinese poetry. Various divine or heroic persons or beings contributed to control or in some cases worsen the flooding, including the mysterious bird-turtles of the Heavenly Questions of the Chuci. One of the main (if not the main) rivers involved according to tradition was the Yellow River, and one of the keys to the eventual successful efforts to control the flood waters is traditionally the Yellow River Map.

Fu Xi[edit]

Main article: Fu Xi

Fu Xi (Paoxi) was said to have been inspired by spider webs and other natural phenomena. (Yang:120-121)

Fu Xi, also known as Paoxi or Baoxi, is still actively worshiped in modern China. Modern accounts of Fu Xi tend to focus on the population of the world by human beings, in the aftermath of a great flood. Fu Xi is a culture hero, credited with various inventions or discoveries. Among them are the use of knots (known sometimes as quipu, especially in the context of South American studies, where this system of recording information by knotting strings also has been subject to binary mathematics analysis) to record information historically and the mathematically binary bagua system of the 8 (2³ trigrams), which were later developed into 64 (2⁶), in which the equivalent of 1 and 0 bits were represented by solid (yang) lines and broken (yin) lines. The classic literary source for this system appears with various philosophical and mystical material suitable for oracular work, in the world-famous Confucian classic Yi Jing (or, in many older sources, I Ching). Various random or semi-random procedures can be used to generate the trigrams or hexagrams of this system.

Yu[edit]

Main article: Yu the Great

Further information: Yubu

Yu is often known as Yu the Great, especially in English language sources. He succeeded Gun in flood control. Yu the Great (Chinese: 大禹; pinyin: Da Yǔ, c. 2200–2100 BCE), was a legendary ruler in ancient China famed for his introduction of flood control, inaugurating dynastic rule in China by founding the Xia Dynasty, and for his acclaimed upright moral character.

He Bo[edit]

Main article: He Bo

The deity ("Count" or "Earl") of the Yellow River is He Bo (河伯). According to some accounts he was involved with the production of the Yellow River Map.

Houtu[edit]

Main article: Houtu

Houtu is a he or her divinity (or neither), depending on source, although the image of Sacred Mother earth deity is common . Houtu is currently worshiped in Chinese popular religion, with her birthday on 18 March, lunar calendar. Sacrifice and prayer to Houtu is supposed to be efficacious for problems of weather, reproduction and family, wealth, and boating safety on the Yellow River. (Yang, et al:136–137) According to one account, when Yu "the Great" was attempting to channel the Yellow River and so to avoid its flooding, Yu began by trying to open it to the west (towards the mountains and away from the sea, thus completely the wrong way). Observing this, Houtu is said (in this version) to have created and studied the Yellow River Map, after which she sent her divine messenger birds to Yu "the Great" to tell him to open up the river to the east, instead. Yu's new dredging was a success, the flood waters drained into the eastern sea, and Yu's former dredging project toward an impossible western drainage was named "River Wrongly Opened". (Yang, et al:137) In this version of the story, Houtu and the Yellow River Map were key to the successful engineering solution to the flood problem.

Bagua[edit]

Main article: Bagua

Derivation of the bagua, as conceived philosophically according to a system in which original unity differentiates into 8 trigrams, each uniquely named by an associated character word.

Bagua is a main concept in Chinese combinatoric philosophical thought: 8 figures of mythical origin and emblematic significance that are specifically said to be related to the Yellow River Map and the Luoshu Square. The origin of these concepts is prehistorical, been written about them since, evolving a complex body of literature, some of it more esoteric, and some less so. Derivation of the bagua has been conceived philosophically according to the taijior other system in which original unity, symbolized by the bottom circle first differentiates into yin and yang symbolized by solid versus dashed lines. 8 possible unique groupings of these lines into three-line sets are possible. These sets of 3 are known as "trigrams". Each trigram has its own proper name, in Chinese, and is also considered to possess or to symbolize various qualities of the natural, human, or heavenly worlds. Certain traditions suppose that the Yellow River Map and the Luo River Writing reveal all of these things to one who knows how to read them.

Historical evidence[edit]

The Yellow River Map is attested to in the Gu Ming section of the Book of Documents (one of the "new text" sections). Supposedly, the Yellow River Map was put on display during the Zhou dynasty; however, this has also been interpreted to mean a depiction of the 8 trigrams (bagua). (Wu:52–53, and note 14, 102) This incident is recorded to have been during the reign of King Kang of Zhou, about 1020–996 BCE or 1005–978 BCE.

Literature[edit]

Further information: Yi Jing

The Yi Jing (also known as the Book of Changes, or I Ching) cites the Yellow River Map and the Lo Shu Square(or "script"). (Wu:52)

Interpretation[edit]

Wolfram Eberhard (sub "Square", 276) says that the River Plan is proven "beyond a reasonable doubt" to be amagic square. He connects it to the mingtang halls of worship, saying that they share a division into 9 fields: these in turn are correlated with the 9 "planets" (Sun, Moon, Mercury, Mars, Venus, Jupiter, Saturn, Rahu, andKetu), introduced from and according to Indian astronomy. Other sources emphasize these points for the Luo River Writing. Another interpretation of the River Diagram has to do with the 5 "elements" (wuxing) and the 5cardinal directions.

Tables[edit]

Wuxing[edit]

	fire (火) 7 （extinction） 2（generation）
wood (木) 8 （extinction） 3 （generation）	earth (土) 5 （generation） 10（extinction）	metal (金) 4 （generation） 9 （extinction）
	1 （generation） 6 （extinction） water (水)

Notes:

Extinction is: 成數, which could also be translated as "completion".

Generation is: 生數, which could also be translated as "birth".

10 is represented in the Chinese (as are the other numerals) with a (different) single character: 十.

Cardinal directions[edit]

Odd number order (1, 3, 5, 7, 9)[edit]

x	7（west, 西）	x
3（south, 南）	5（center, 中）	9（north, 北）
x	1（east, 東）	x

Even number order (2, 4, 6, 8, 10)[edit]

x	2（west, 西）	x
8（north, 北）	10（center, 中）	4（south, 南）
x	6（east, 東）	x

Places[edit]

Certain places in modern China use Hetu (河图) as part of their proper place names. These include 河图镇 (岳西县), 河图镇 (保山市), and 河图乡.

See also[edit]

• Loess Plateau, the highly erodable land through which the Yellow River finds its course.
• Longma, dragon-horse creature, mythological delivery beast of the Yellow River Map.
• Lo Shu Square, companion piece to the River Map (actually named the "Luo" (or "Lo") "Script" or "Writing").
• Luo River (Henan)
• Luo River (Shaanxi)
• Mount Buzhou, an important geographic feature, in relevant mythology.

References[edit]

• Christie, Anthony (1968). Chinese Mythology. Feltham: Hamlyn Publishing. ISBN 0600006379.
• Eberhard, Wolfram (2003 [1986 (German version 1983)]), A Dictionary of Chinese Symbols: Hidden Symbols in Chinese Life and Thought. London, New York: Routledge. ISBN 0-415-00228-1
• Wu, K. C. (1982). The Chinese Heritage. New York: Crown Publishers. ISBN 0-517-54475X.
• Yang, Lihui, et al. (2005). Handbook of Chinese Mythology. New York: Oxford University Press. ISBN 978-0-19-533263-6

Lo Shu: Definition, Nature and Histor

Lo Shu Square (simplified Chinese: 洛书; traditional Chinese: 洛書 ; literally: Luo (River) Book/Scroll) or the Nine Halls Diagram (simplified Chinese: 九宫图; traditional Chinese: 九宮圖), often in connection with the Ho Tu (河圖) figure and 8 trigrams, is the unique normal magic square of order three. Lo Shu is part of the legacy of the most ancient Chinese mathematical and divinatory (Yi Jing 易經) traditions, and is an important emblem in Feng Shui (風水, translate as "wind-water"), the art of geomancy concerned with the placement of objects in relation to the flow of qi (氣), 'natural energy'.

Actually, the first Chinese magic square is believed to have been created by Fuh-Hi, the mythical founder of Chinese civilization, who lived from 2858 to 2738 B.C. The scroll is a 3x3 magic square, where odd numbers are expressed as white dots, or yang symbols, and even numbers are expressed as black dots, or Yin symbols. The odd numbers are supposed to be symbols of heaven, while even numbers are symbols of the earth.

四	九	二
三	五	七
八	一	六

Chinese Lo Shu
with true Chinese numbers

Chinese Lo Shu square
Image by Jenny Olive

Modern Lo Shu square

In the Chinese Lo Shu square above, we have added colour here to make the distinction between the odd and even numbers stand out more clearly. In fact the yellow blobs should be white, being Yang symbols or emblems of heaven, and the red blobs should be black, being Yin symbols or emblems of earth.

Feng Shui is a form of qi divination. The retention or dissipation of qi is believed to affect the health, wealth, energy level, luck and many other aspects of the occupants of the space. Color, shape and the physical location of each item in a space affects the flow of qi by slowing it down, redirecting it or accelerating it, which directly affects the energy level of the occupants.

The qi (Chinese 氣, Japanese 気, Korean 氣, prāna in indian (sanskrit प्राण)) is an active principle forming part of any living thing; literal translation is "air", "breath" or "spirit". Qi is a didactic concept in many Chinese, Korean and Japanese martial arts.

Location of 7 chakras

Lo Shu is also connected to the Chakras and stimulate them. Chakra (derived from the Sanskrit cakraṃ चक्रं, wheel or disc. There are seven major chakras or energy centers, located within the subtle body, from the base of the spine to the top of the head, at major branchings of the human nervous system, beginning at the base of the spinal column and moving upward to the top of the skull and expressed life force energy (biophysical energy or prana of the human body). The seven major chakras are: Sahasrara, सहस्रार, Ajna, आज्ञा, Vishudda, विशुद्ध, Anahata, अनाहत, Manipura, मणिपूर, Svadhishthana, स्वाधिष्ठान and Muladhara, मूलाधार. Moreover placebo effect, there is a relationship between the positions and functions of the chakras and of the various organs of the endocrine system.

The odd and even numbers alternate in the periphery of the Lo Shu pattern, the 4 even numbers are at the four corners, and the 5 odd numbers form a cross in the center of the square. The sums in each of the 3 rows, in each of the 3 columns, and in both diagonals, are all 15 [fifteen is the number of days in each of the 24 cycles of the Chinese solar year; Xia Li (夏曆) or "Yin Calendar" - Gregorian calendar is the "Yang Calendar"]. This pattern, in a certain way, was used by the people in controlling the river. Since 5 is in the center cell, the sum of any two other cells that are directly through the 5 from each other is 10 (the number of the Ho Tu). The even numbers are Yin, the feminine principle. The odd numbers are Yang, the male principle. The symbolism of the square Lo Shu interprets 5 elements: earth, fire, metal, water, and wood. Like the Ho Tu, the Lo Shu square is used as a mandalic representation important in Feng Shui. For Chinese, the magic square symbolizes the harmony of the universe.

Bagua with name and nature (King Wen "Later Heaven" order)

Mathematics of Lo Shu Magic Square

From any magic square of order 3, by pivoting the square, you can get three other magic squares by rotation and four other magic squares by symmetry. These eight magic squares are considered as equivalent, but the center cell in a 3×3 normal magic square must be 5.

4	9	2
3	5	7
8	1	6

8	3	4
1	5	9
6	7	2

6	1	8
7	5	3
2	9	4

2	7	6
9	5	1
4	3	8

Original

Rotate 90º

Rotate 180º

Rotate 270º

8	1	6
3	5	7
4	9	2

6	7	2
1	5	9
8	3	4

2	9	4
7	5	3
6	1	8

4	3	8
9	5	1
2	7	6

Vertical reflections of squares immediately above

In these transformations of normal magic square of order 3, and in all cases:

1. The number 5 is always at the center of the square, at the intersection of the 2 diagonals.
2. The 5 odd numbers are always placed cross in the central 2 lines (vertical and horizontal) of the square.
3. The 4 even numbers are always in the 4 outside corners of the square.
4. The sums in each of the 3 rows, in each of the 3 columns, and in both diagonals, are all 15 (15 = magic constant).

 
Applet‎ by Yasusi Kanada

The applet‎ searches for a magic square. If you used this applet‎ in its initial state, you can track the process by your eye in some extent. "Swapping rule" exchanges two integers in columns and "rotation rule" rotates three integers in three columns. If you change the option value, which is "medium speed (20 rps)" in its initial state, to "full speed", the computation will be done as quick as possible. The BEST is to use "slow speed" (3 rps) with "Swapping rule". [20 rps means that the rule is applied 20 times per second (rps = reductions per second). However, the real rps is less than 20]. You can start the computation again using the "restart" button. You probably find a different solution each time because random numbers are used, and the computation time is also different each time.

Moreover, the magic constant of a normal magic square depends only on n and has the value M = (n³ + n)/2. Here is the proof. Given an  normal magic square, suppose M is the number that each row, column and diagonal must add up to. Then since there are n rows the sum of all the numbers in the magic square must be . But the numbers being added are 1, 2, 3, ... n², and so 1 + 2 + 3 + ... + n² = . In summation notation, . Using the formula for this sum, we have , and then solving for M gives . Thus, a Lo Shu's  normal magic square must have its rows, columns and diagonals adding to , a Albrecht Durer's  to M = 34, a Benjamin Franklin's  to M = 260, and so on.

Legend of turtle and worldwide diffusion

According to the Chinese book Yih King: Emperor Yu walked along the river Lo. Then he saw the magic square on the carapace of a turtle sacred. This square is called the diagram of the river Lo or Lo Shu.

Here is another version of the story of 'Lo Shu'. In the ancient time of China, there was a huge flood. The people tried to offer some sacrifice to the 'river god' of one of the flooding rivers, the 'Lo' river, to calm his anger. However, every time a turtle came from the river and walked around the sacrifice. The river god didn't accept the sacrifice until one time, a child noticed the curious figure on the turtle shell. Hence they realized the correct amount of sacrifice to make (15). Lo is name of a river and the word 'Shu' means "book".
According to Philip I.S. Lei - Hong Kong.

After several centuries, the 3x3 magic square made its way out of China and entered the Indian subcontinent. From India, it traveled on to Arabia and into medieval Europe. Since, magic squares have fascinated humanity throughout the ages, and have been around for over 4,120 years. They are found in a number of cultures, including Egypt and India, engraved on stone or metal and worn as talismans, the belief being that magic squares had astrological and divinatory qualities, their usage ensuring longevity and prevention of diseases. In the ninth century, Arabian astrologers used magic squares to interpret horoscopes.

The 3x3 magic square was used as part of rituals in India from vedic times, and continues to be used till date. The Kubera-Kolam is a floor painting used in India which is in the form of a magic square of order three. It is essentially the same as the Lo Shu Square, but with 19 added to each number, giving a magic constant of 72. 

23	28	21
22	24	26
27	20	25

In Egypt, magic squares were used to represent the difference between order and chaos. Squares made up of two or four cells were said to represent chaos because they were incapable of forming magic squares. Magic squares 3x3 or larger were dedicated to the sun, moon, and planets in the form of talismans. The talismans were made by taking a magic square and placing it in a polygon with the number of sides of the polygon equal to the root of the square (i.e. a 3x3 magic square was placed in a triangle, a 5x5 was placed in a pentagon, etc...). These polygons were then placed in a circle, and in between the sides of the polygon and the circle were inscribed signs of the zodiac. Then, the "good" or "evil" name of the corresponding planet was written on the talisman. It is rumored that Pythagoreas, who traveled through Egypt at that time (500 B.C.), was greatly influenced by the Egyptian philosophy on magic squares and numbers.

Lo Shu and Cardinal Points

Lo Shu is often drawn in the form of a 9-squares chart with each square representing a direction in the compass (see diagram below) namely North, South, East, West, Northeast, Northwest, Southeast, Southwest and the center (total 9 locations). The south is always represented by the number 9, and the north by the number 1.

Lo Shu is also connected to the Bagua and the eight trigrams. The Bagua (Chinese: 八卦; pinyin: bā gua; Wade-Giles: pa kua; literally "eight symbols") are eight diagrams used in Taoist cosmology to represent a range of interrelated concepts. Each consists of three lines, each either "broken" or "unbroken," representing a yinline or a yang line, respectively. Due to their tripartite structure, they are often referred to as "trigrams" in English. The eight trigrams are: Qian 天, "Heaven", Xun 風, "Wind", Kan 水, "Water", Gen 山, "Mountain", Kun 地, "Earth", Zhen 雷, "Thunder", Li火, "Fire" and Dui 泽, "Lake".

Wind
Xun 風
Southeast
☴

Fire
Li 火
South
☲

Earth
Kun 地
Southwest
☷

East ☳
Zhen 雷
Thunder

4	9	2
3	5	7
8	1	6

☱ West
Dui 泽
Lake

☶
Gen
山
Northeast
Mountain

☵
Kan
水
North
Water

☰
Qian
天
Northwest
Heaven

It is important here to remind where does the Chinese Yin Yang symbols come from. The Chinese calendar divides the year cycle into 24 solar segments, including the Vernal Equinox, Autumnal Equinox, Summer Solstice and Winter Solstice, using the sunrise and Dipper positions; that is according to the sun positions on the tropical zodiac (Similar to western astrology). So, the proper origin of Yin Yang is the pattern of 24 solar segments.

Yin Yang - 24 solar segments
by Allen TsaiTaijitu, rough English translation "diagram of ultimate power" (Taiji tu of Zhou Dunyi, 1017-1073), is best known in western cultures as Yin and Yang (literally meaning "dark and light")

Lo Shu Numbers and Their Meanings

The numbers in each square of the Lo Shu embrace a variety of meanings. Just to name a few:

Lo Shu Numbers	Representing Location	Representing Colour	Representing Element
1	North	White	Water
2	Southwest	Black	Earth
3	East	Pure Green	Wood
4	Southeast	Light Green	Wood
5	Central	Dull Yellow	Earth
6	Northwest	White	Metal
7	West	Red	Metal
8	Northeast	White	Earth
9	South	Purple	Fire

From D L Wang / hiakz.com

The sum of each line (whether vertical, horizontal or diagonal) always equals fifteen. In other words, if you ignore the 5 in the center, you will notice that the number opposite each other all sums up to 10. (The River Map uses the difference while the Lo Shu Square uses the sum).

Similar to the River Map, the odd numbers (white dots) represents Yang and the even numbers (black dots) represents Yin. Apart from the 5 in the center, the Lo Shu numbers are arranged such that after every Yang number, a Yin number follows, and, after every Yin number comes a Yang number. In the I Ching, yin (陰) and yang (陽) are represented by broken and solid lines: yang is solid (—) and yin is broken (– –). These are then combined into trigrams, which are more yang or more yin depending on the number of broken and solid lines (e.g. ☰ is heavily yang, while ☷ is heavily yin)

Some examples of Yin and Yang:

The Yin and Yang must be balanced. Yin and Yang are complementary and inseparable. It seems impossible to imagine one without the other. So if at any moment Yin is too strong Yang will be in the future and seek balance. For example, in summer, Yang peaked, immediately after, Yin began to grow while Yang declines gradually. So in winter, the Yin peaked and Yang its lowest point. The fact of reaching the highest point, whether the Yin or Yang, marks the beginning of their decline. The Yin and Yang can be found everywhere, even in the human body. For example the upper body is Yang while the bottom is Yin, a muscle contraction is Yang while a muscle at rest is Yin, even spirit can be Yin and Yang. Indeed, the mind in sleep state is Yang and spirit in a state of sleep is Yin.

	Yin	Yang
Number	Even 2, 4, 6, 8	Odd 1, 3, 5, 7, 9
Position	Bottom	Top
Time	Night	Day
Season	Autumn, Winter	Spring, Summer
Direction	North, West	South, East
Element	Water	Fire
Gender	Female	Male
Heat	Cold	Hot
Humidity	Wet	Dry
Brightness	Dark	Light
Color	Black	White
Planet	Moon	Sun
View	Invisible	Visible
Planet	Earth	Heaven

Yin and Yang

LoShu, Sudokus and Bimagic Squares

This section is based on findings of Christian Boyer on his website multimagie.com.

From the 8 LoShu Squares, you can get two Sudokus (each 3x3 subsquare contains the nine integers from 1 to 9, and each row and each column contains the nine integers from 1 to 9):

2	5	8	1	4	7	3	6	9
1	4	7	3	6	9	2	5	8
3	6	9	2	5	8	1	4	7
8	2	5	7	1	4	9	3	6
7	1	4	9	3	6	8	2	5
9	3	6	8	2	5	7	1	4
5	8	2	4	7	1	6	9	3
4	7	1	6	9	3	5	8	2
6	9	3	5	8	2	4	7	1

Sudoku A

2	9	4	6	1	8	7	5	3
7	5	3	2	9	4	6	1	8
6	1	8	7	5	3	2	9	4
9	4	2	1	8	6	5	3	7
5	3	7	9	4	2	1	8	6
1	8	6	5	3	7	9	4	2
4	2	9	8	6	1	3	7	5
3	7	5	4	2	9	8	6	1
8	6	1	3	7	5	4	2	9

Sudoku B

These Sudokus have nice supplemental properties, i.e. if we move one or more columns from one side to the opposite side, they remain Sudokus: the new 3x3 sub-squares contains again all the integers from 1 to 9. Same result if we move rows from one side to the opposite side.

Now, construct a 9x9 square in which each cell use the two cells of Sudokus A and B with the formula:

9(A - 1) + B

Then you get a bimagic square constructed with the Tarry-Cazalas method. ALL the 9x9 bimagic squares constructed with the Tarry-Cazalas method are a combination of 2 Sudokus. By definition, a bimagic square is a magic square that remains magic if all the numbers it contains amounted to squares.

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

Bimagic square constructed
from Sudoku A and Sudoku B

This square is bimagic:

1. consecutive integers from 1 to 81;
2. same sum S₁ = 369 for each of the 9 rows, 9 columns and 2 diagonals;
3. after you have squared each number, the square remains magic, same sum S₂ = 20 049 for the 9 rows, 9 columns and 2 diagonals.

And it has supplemental bimagic properties:

1. again the same sum S₁ = 369 for each of the nine 3x3 sub-squares (sum of the nine numbers in each sub-square);
2. after you have squared each number, again the same sum S₂ = 20 049 for each of the nine 3x3 sub-squares (sum of the nine squared numbers in each sub-square).

You can get another bimagic square using the other formula: 9(B - 1) + A

Of course, all couples of Sudokus do not give a bimagic square, and all bimagic squares (those not constructed by the Tarry-Cazalas method) are not coming from a couple of Sudokus.

Luo River: tributary of the Yellow River

The Luo River (Chinese: 洛河;) is a tributary of the Yellow River in China. It rises in the southeast flank of Huashan in Shaanxi province and flows east into Henan province, where it eventually joins the Yellow River at the city of Gongyi. The river's total length is 420 km. Although not a major river by most standards, it flows through an area of great archaeological significance in the early history of China. The Luo's main tributary is the Yi River, which joins it at Yanshi. During the era of the Three Kingdoms, Cao Zhi wrote a famous poem to the goddess of the Luo River as an indirect expression‎ of love for a deceased lover.

The Yellow River or Huang He / Hwang Ho (Chinese: 黃 河) is the second-longest river in China (after the Yangtze River) and the sixth-longest in the world at 5,464 kilometers (3,398 mi). The Yellow River is called "the cradle of Chinese civilization", as its basin is the birthplace of the northern Chinese civilizations and was the most prosperous region in early Chinese history. Early Chinese literature refers to the Yellow River simply as He, or "River". The yellow color comes from loess suspended in the water.

REFERENCES

• Lo Shu Square, by Wikipedia.
• Feng Shui, Wikipedia.
• Qi, by Wikipedia.
• Chakra, Wikipedia.
• Yin and Yang, Wikipedia.
• Taijitu, Wikipedia.
• I Ching trigrams, Wikipedia.
• Luo River (Henan), by Wikipedia.
• Bagua concept, Wikipedia.
• Catherine Polet, The Feng Shui Concept
• Karen Rauch Carter, Feng Shui Palace

Bibliography:

• Andrews, W.S. "Magic Squares and Cubes", The Open Court Publishing Company, Chicago 1908.
• Falkner, Edward. "Games Ancient and Oriental and How To Play Them", Dover Publications, Inc., New York 1961.
• Kraitchik, Maurice. "Mathematical Recreations", Dover Publications, Inc., New York 1953.
• Stern, David. "Math Squared", Teachers College Press, New York 1980.

http://www.taliscope.com/LoShu_en.html