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Liber XXXI · Taoism & Chinese Religion

Liber XXXI · Taoism & Chinese Religion · Chapter IX: Yin-Yang & the Five Phases

Hetu and Luoshu: The River Chart, Luo Writing and Lo Shu Square

The Hetu (River Chart) and Luoshu (Luo Writing) explained: dragon-horse and turtle legends, the Lo Shu magic square, the Nine Palaces and Song debates.

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The Hetu ("River Chart") and the Luoshu ("Luo Writing", often spelled Lo Shu) are two sacred diagrams of Chinese cosmology, said to have emerged from the Yellow River and the Luo River on the backs of a dragon-horse and a divine turtle. Since the Song dynasty they have been drawn as patterns of black and white dots: the Hetu arranges the numbers one to ten around a centre, and the Luoshu arranges one to nine in the famous three-by-three magic square in which every line adds up to fifteen.

The Legends: Dragon-Horse and Divine Turtle§

According to Chinese tradition, in the remote age of the sage rulers a creature called the dragon-horse (longma) rose from the Yellow River bearing markings on its back. The culture hero Fuxi studied them and, together with his observation of heaven, earth and the patterns on birds and beasts, created the eight trigrams that underlie the I Ching. Later, when Yu the Great was taming the great flood, a divine turtle emerged from the Luo River, a tributary of the Yellow River in modern Henan, with a pattern on its shell. From it Yu is said to have derived the "Great Plan" (Hongfan), the nine categories of government preserved in the Book of Documents.

The two diagrams were thus imagined as revelations from nature itself, given to sages at the dawn of civilisation, which established the numerical order underlying the cosmos and good rule. Their appearance was a sign of Heaven's favour toward a virtuous ruler; their absence a sign of decline.

Early References: The Analects, the Book of Changes and the Han Apocrypha§

The names Hetu and Luoshu are old, but early sources do not say what the diagrams looked like. The Book of Documents lists a "River Chart" among treasures displayed at the funeral of King Cheng of Zhou, perhaps a precious object rather than a diagram. In the Analects (9.9), Confucius laments the absence of auspicious signs; in James Legge's 1861 translation, "The FANG bird does not come; the river sends forth no map:—it is all over with me!"

The Great Commentary (Xici), one of the "Ten Wings" appended to the Book of Changes, states that the Yellow River brought forth a chart and the Luo River a writing, and that the sages took them as models. This sentence linked the two diagrams permanently to the Changes.

During the Han dynasty (206 BCE–220 CE) the names took on another meaning. A large body of "weft" or apocryphal texts (chenwei) circulated under the titles Hetu and Luoshu, containing prophecies, omens and esoteric interpretations of the classics. Usurpers and founders, including Wang Mang and Emperor Guangwu of the Eastern Han, used such prophecies to claim Heaven's mandate. Because they were politically dangerous, later dynasties repeatedly banned them, and the Sui emperor Yang ordered them burned; only fragments survive.

The Luoshu Magic Square and the Nine Palaces§

The diagram now called the Luoshu arranges the numbers one to nine in a square, odd numbers drawn as white dots and even numbers as black:
4 9 2
3 5 7
8 1 6

Every row, column and diagonal sums to fifteen. Five occupies the centre; the odd (yang) numbers stand at the four cardinal points and the even (yin) numbers at the corners. In the traditional orientation south is at the top, so nine is in the south (fire) and one in the north (water).

This arrangement was known long before it was called the Luoshu. The Records of Ritual of the Elder Dai (Da Dai liji), a Han compilation, describes the nine halls of the Bright Hall (Mingtang), the cosmic temple of the emperor, numbered in the same pattern. Han diviners spoke of the "Nine Palaces" (jiugong), through which the deity Taiyi was believed to circulate in a fixed sequence, and a lacquered divination board from a tomb at Shuanggudui, sealed around 165 BCE, appears to reflect this scheme. The square thus belongs to the correlative cosmology described in yin and yang and the five phases, in which numbers, directions, seasons and elements correspond.

The Luoshu is the oldest known magic square in the world. Chinese mathematicians studied it explicitly; Yang Hui, in 1275, gave methods for constructing it and larger squares. The same 3×3 square appears in Islamic and later European magic, where Agrippa assigned it to Saturn (see magic squares and planetary kameas).

Song Dynasty Diagrams: Liu Mu, Zhu Xi and the Two Charts§

The dot diagrams familiar today were first published in the Song dynasty (960–1279), when scholars fascinated by the Book of Changes produced charts to explain its numbers. Song accounts trace their transmission to the Taoist recluse Chen Tuan (d. 989), the same figure linked to the early history of the taijitu, though the chain of transmission is uncertain.

Liu Mu (1011–1064) presented the nine-number square as the River Chart and the ten-number diagram as the Luo Writing. Others disagreed, and the matter was settled for orthodox scholarship by Zhu Xi (1130–1200), who, with his student Cai Yuanding, reversed Liu Mu's labels in Introduction to the Study of the Changes (Yixue qimeng, 1186). Zhu Xi printed both charts at the beginning of his commentary on the Changes, and through his authority in Neo-Confucianism they became standard for the next seven centuries.

In the early Qing dynasty, critical scholars attacked them. Huang Zongxi and his brother Huang Zongyan, and above all Hu Wei in his Clarification of the Diagrams of the Changes (Yitu mingbian, 1706), argued that the diagrams had no basis in the ancient classics and had entered Confucian learning from Taoist circles in the tenth century. Most modern historians accept that the diagrams in their dot form are Song creations, though the Luoshu arrangement of numbers is much older.

The Hetu, the Five Phases and Cosmological Numbers§

The Hetu, as fixed by Zhu Xi, places the numbers one to ten in pairs around the centre: one and six in the north, two and seven in the south, three and eight in the east, four and nine in the west, five and ten in the centre. Each pair joins a "generating" number (one to five) with a "completing" number (six to ten), and each is assigned to one of the five phases: water in the north, fire in the south, wood in the east, metal in the west and earth in the centre.

The scheme draws on Han theories that associated the phases with numbers, recorded in the "Monthly Ordinances" and later commentaries. Odd numbers are heavenly and yang, even numbers earthly and yin; heaven's numbers total twenty-five and earth's thirty, making the "number of heaven and earth", fifty-five, mentioned in the Great Commentary. Song thinkers saw in the Hetu the static, generative order of the cosmos, and in the Luoshu its dynamic transformations.

Zhu Xi and his followers also matched the two diagrams to the two classic arrangements of the eight trigrams, the "Prior Heaven" sequence attributed to Fuxi and the "Later Heaven" sequence attributed to King Wen. The Later Heaven trigrams, laid over the Luoshu, became a basic framework of Chinese divination.

Uses and Afterlife: Feng Shui, Ritual and Divination§

The Luoshu and its Nine Palaces underlie many later systems:

  • In feng shui, the "Flying Stars" method moves the nine numbers through the sectors of a building according to time, and the rings of the luopan compass include the Nine Stars.
  • In calendrical divination such as Qimen dunjia, the board of nine palaces is the field on which the cycles of heaven and earth are plotted. Japan's popular "Nine Star Ki" (kyūsei) system also derives from it.
  • In Taoist ritual, the "Pace of Yu" (Yubu), a limping step said to imitate Yu the Great, is performed along patterns of stars and the nine palaces.

Modern popular numerology sometimes uses a "Lo Shu grid" to plot the digits of a birth date. This is a recent adaptation, not a traditional Chinese practice, though it borrows the ancient arrangement. As historical objects, the Hetu and Luoshu show how Chinese thinkers sought a single numerical order linking the heavens, the land, the state and the human body. For the wider history of number symbolism, see the guide to numerology.

Frequently asked questions§

What is the Lo Shu square?§

It is a three-by-three arrangement of the numbers one to nine in which every row, column and diagonal adds up to fifteen. In legend it appeared on the shell of a turtle from the Luo River; historically it is the oldest known magic square and is linked to the Nine Palaces of Chinese cosmology.

What is the difference between the Hetu and the Luoshu?§

The Hetu, or River Chart, arranges the numbers one to ten in pairs around a centre and is associated with the five phases and with Fuxi. The Luoshu, or Luo Writing, arranges the numbers one to nine in a magic square and is associated with Yu the Great and the Nine Palaces.

Are the Hetu and Luoshu really ancient?§

Their names are ancient, appearing in the Analects and the Great Commentary on the Book of Changes. The dot diagrams, however, were first published in the Song dynasty, and many scholars since the Qing have argued that they were created then, although the Luoshu arrangement of numbers is attested in the Han.

How is the Luoshu used in feng shui?§

It provides the framework of nine sectors used in Compass School methods, especially Flying Stars, where the numbers shift through a building's sectors over time. It is also used in Qimen dunjia and in Japan's Nine Star Ki.

Sources and further reading§