What Are Hà Đồ and Lạc Thư: Number Arrangement, Links to the Tiên Thiên and Hậu Thiên Bagua, and the Nine Stars

What are Hà Đồ and Lạc Thư?

Hà Đồ and Lạc Thư are two ancient diagrams of the I Ching tradition, regarded as early maps describing how heaven and earth operate. Hà Đồ has 55 black and white dots arranged in number pairs across five directions and five elements. Lạc Thư has nine numbers from 1 to 9 placed in a 3x3 square, forming the only standard third order magic square, where every row, every column and both diagonals sum to 15.

The two diagrams usually go together in I Ching studies, feng shui, numerology and even early natural science research in ancient China. They provide the set of numbers, images and interaction relations between basic factors, forming the foundation for later calculation and interpretation.

How does Hà Đồ arrange birth and completion numbers by direction and element?

In the common explanation, each element has a birth number (yang, belonging to Heaven) and a completion number (yin, belonging to Earth). The birth number is the starting number for the formation of that element, the completion number marks that it has taken shape. The I Ching records: number 1 belongs to Heaven and starts Water, number 6 belongs to Earth and completes Water; number 2 belongs to Earth and completes Fire, number 7 belongs to Heaven and completes Fire; number 3 belongs to Heaven and starts Wood, number 8 belongs to Earth and completes Wood; number 4 belongs to Earth and completes Metal, number 9 belongs to Heaven and completes Metal; number 5 belongs to Heaven and starts Earth, number 10 belongs to Earth and completes Earth. This wording shows Heaven with Earth, Yin with Yang interacting, and through this the five elements are created.

The table below summarizes how Hà Đồ arranges birth and completion number pairs by direction and element:

A quick way to remember: birth and completion numbers always differ by exactly 5, meaning the birth number plus 5 gives the completion number. Odd numbers 1, 3, 5, 7, 9 are yang, even numbers 2, 4, 6, 8, 10 are yin. This arrangement also shows Hà Đồ places its focus at the central palace with the pair 5 and 10, while the other four elements spread out to the four directions.

How exactly is Lạc Thư a 3x3 square where every row sums to 15?

Lạc Thư places the nine numbers 1 to 9 into a 3x3 grid so that every horizontal row, every vertical column and both diagonals sum to 15. This is the only standard third order magic square, meaning there is only one arrangement if rotations and reflections are not counted. In the presentation commonly used by Vietnamese and Chinese people, number 1 sits at the bottom and number 9 at the top, because ancient maps place the South at the top and the North at the bottom.

In this square, number 5 sits at the center. The four even numbers 2, 4, 6, 8 sit at the four corners, the four odd numbers 1, 3, 7, 9 sit at the four middle edges, forming a cross around number 5. One notable property: because 5 is at the center, the sum of two cells symmetric through the center always equals 10, for example 4 with 6, 9 with 1, 2 with 8, 3 with 7. This number 10 equals the sum of the birth and completion pair of Earth in Hà Đồ, that is 5 plus 5.

How does Lạc Thư link to the Hậu Thiên bagua and the nine stars?

In the common mapping, the eight outer cells of Lạc Thư correspond to the eight trigrams of the Hậu Thiên bagua. This mapping matches the directions of the Hậu Thiên bagua: Kham in the North, Ly in the South, Chan in the East, Doai in the West, Can in the Northwest, Khon in the Southeast, Can in the Northeast, Ton in the Southeast. Because ancient maps place the South at the top, number 9 at the top corresponds to Ly and the Fire element, number 1 at the bottom corresponds to Kham and the Water element.

Because the eight outer cells correspond to the eight trigrams and number 5 is at the center, Lạc Thư is also called the Nine Palace Diagram or the nine palace diagram. In Huyền Không Phi Tinh feng shui, these nine numbers are used as names for the nine stars, and the flight path of the stars is based on their original positions in Lạc Thư. Feng shui texts note: Hà Đồ with its 55 dots shows the five elements and yin yang operating by the principle of mutual generation, while Lạc Thư with its 9 numbers shows that the world, after heaven and earth are formed, is balanced and orderly.

How does Hà Đồ link to the Tiên Thiên bagua?

According to legend and the familiar explanation, King Phục Hi looked at Hà Đồ and established the Tiên Thiên bagua and the 64 hexagrams. The mapping of Hà Đồ to the Tiên Thiên bagua is based on birth and completion number pairs by direction: the pair 1 and 6 in the North corresponds to Khon, the pair 2 and 7 in the South to Can, the pair 3 and 8 in the East to Ly, the pair 4 and 9 in the West to Kham. This is the common mapping in many I Ching books, though some books present slightly different details in matching each trigram.

One Vietnamese folk I Ching source pushes the comparison further: it holds that Hà Đồ gives rise to the Tiên Thiên bagua and follows the forward direction, Lạc Thư gives rise to the Hậu Thiên bagua and follows the reverse direction. The forward direction is explained as clockwise, where yin and yang do not oppose each other, taking compassion and righteousness as the standard. The reverse direction is counterclockwise, taking conflict and opposition as the basis. This interpretation is heavily colored by cultivation philosophy, is not the only way, and other books may stop at matching numbers with trigrams.

Why is Hà Đồ said to be Tiên Thiên and Lạc Thư Hậu Thiên?

This wording comes from Hà Đồ being considered earlier, linked to the time of Phục Hi and the establishment of the Tiên Thiên bagua, while Lạc Thư is linked to the time of Great Yu controlling floods and establishing the Hong Pham Cu Tru, after which King Van established the Hậu Thiên bagua. In philosophical interpretation, Hà Đồ is seen as the symbol of Tiên Thiên and Non action, while Lạc Thư is the symbol of Hậu Thiên and Action.

Note that this is a philosophical assignment, not a verifiable historical event. Different schools may emphasize different aspects of the two diagrams, some leaning toward numerology, some toward cultivation, some toward feng shui. Learners should know clearly which interpretation they follow instead of mixing the different wordings together.

What do the Long Mã and sacred turtle legends say about the two diagrams?

Legend tells that in the time of King Phục Hi, a Long Mã appeared at the Ha River, carrying a diagram on its back, hence the name Hà Đồ. King Phục Hi then established the Tiên Thiên bagua and the 64 hexagrams. As for Lạc Thư, according to the Book of Documents, it was granted by Heaven to King Great Yu after he successfully controlled the floods, and from it the king devised the Hong Pham Cu Tru, the method of governing society. King Van used Lạc Thư to establish the Hậu Thiên bagua.

In another variant, Lạc Thư appeared on the shell of a sacred turtle rising from the Lac River. This legend is also linked to King Yu controlling floods and the formation of the numbers. An English Wikipedia document on Lạc Thư also records the variant of a sacred turtle rising from the Lac River with the Lạc Thư figure on its shell, the round dots representing integers from 1 to 9 arranged in a 3x3 grid. The variants are not fully consistent with each other about the animal, the river and the king, so this is a place to present as legend rather than fact.

Why are the diagrams only clearly seen from the Song dynasty and why is there a story of reversed names?

A point rarely mentioned in popular articles: Hà Đồ and Lạc Thư as we see them today are not images from the time of Phục Hi or Great Yu. One source on Tử Vi records that these two diagrams were lost before Confucius edited the I Ching, and the present Hà Đồ and Lạc Thư were redrawn by a Han dynasty man named Khổng An, twelfth generation descendant of Confucius, without clear basis. This wording reflects the view that the current diagrams are products of later generations.

Another source on experiential Tử Vi also mentions that passages in the I Ching contradict each other when discussing the origin of the bagua, forcing later people to propose many hypotheses. One theory holds that Phục Hi observed nature and drew them, another that he redrew them from a diagram on the bank of the Ha River, another that it came from the pattern on a turtle shell rising from the Lac River.

On naming, modern researchers often note that the Hà Đồ and Lạc Thư diagrams only appear clearly from the Song dynasty, linked to names such as Trần Đoàn and Lưu Mục. One notable point is that Lưu Mục and Chu Hy sometimes called the two diagrams by reversed names, meaning the diagram one book calls Hà Đồ another may call Lạc Thư. This is where documents disagree, and readers should check each book's convention before comparing.

Some theories hold that Hà Đồ and Lạc Thư have ancient Viet roots, how should this be understood?

Some authors, notably the research direction in the book Thuat so Lạc Thư, hold that the Lạc Thư table together with the Nine Palaces and the Hậu Thiên bagua belong to the Lạc Việt people, appearing before 2357 BCE, that is before King Nghiêu in the North. Ancient Chinese history records that in the Mậu Thân year of King Nghiêu's reign, envoys of the Việt Thường people from the South came to the Bình Dương capital, presenting the king a sacred turtle with the Khao Đẩu script on its shell recording events since the opening of heaven and earth, and the king had it copied and named the Turtle Calendar. From this these authors infer that the turtle calendar of the Việt Thường people must use the eight Hậu Thiên trigrams of the Lạc Thư table, and that the original meaning of Lạc Thư is the book of the Lac people.

This is a hypothesis of some authors, not a conclusion generally accepted by researchers. The reasoning relies heavily on inference from one passage of ancient history and on assigning meaning to the name, so when presenting it one should state clearly that this is the view of that group of authors. Readers may note it as one line of explanation, and should not treat it as settled historical fact.

Distinguishing Hà Đồ and Lạc Thư from Hà Lạc ly so

Hà Lạc ly so is a separate discipline, using Hà Đồ and Lạc Thư as the number foundation to convert the four pillars, that is the year month day and hour of birth, into I Ching hexagrams, then interpret them. It differs from simply reading the two diagrams as maps of numbers and elements. An I Ching document describes Hà Lạc ly so as the combination of Hà Đồ, Lạc Thư and I Ching principles, in which each number and each position in the two diagrams carries meaning about the five elements, yin yang and palace position. If one is only interested in Hà Đồ and Lạc Thư as two ancient diagrams, there is no need to go deep into this method of converting the four pillars into hexagrams.

What should be noted when using Hà Đồ and Lạc Thư?

First, books disagree on some details, especially the mapping of Hà Đồ to the Tiên Thiên bagua and the reversed naming of the two diagrams between Lưu Mục and Chu Hy. Learners should follow one clear convention and state which convention they follow.

Second, one should not assign these numbers or diagrams an absolute auspicious or inauspicious value. The same number can be interpreted in different ways depending on school and context. Applications in feng shui, Tử Vi and Hà Lạc are all interpretive systems, not formulas giving fixed results for everyone.

Third, one must clearly distinguish legend, historical record and research hypothesis. The Long Mã and sacred turtle legends belong to the legend layer; the fact that the diagrams only appear from the Song dynasty belongs to the historical record layer; and the ancient Viet roots theory belongs to the hypothesis layer of some authors. Mixing these three layers leads to poorly grounded conclusions.