Concepts · The nine palaces
The Lo Shu square
The square
Drawn the conventional way, with south at the top:
Every line adds to 15. The three rows, the three columns, and both diagonals. Odd numbers occupy the centre cross, even numbers the corners, and 5 sits in the middle with the other eight arranged symmetrically around it, each opposite pair summing to 10.
One more thing is true of it, and it is the reason the square has fascinated mathematicians as much as geomancers: up to rotation and reflection, this is the only way to do it. There is exactly one 3 by 3 magic square. Larger sizes have thousands; this one is unique.
The turtle, and an honest word about dating
The traditional account has the pattern appearing on the shell of a turtle that surfaced from the river Luo, read by Yu the Great during his work controlling the floods. It comes paired with the He Tu (河圖), the Yellow River Diagram, and the two are mentioned together in the classics.
The names are genuinely ancient. Which diagram is which, and which number arrangement belongs to each, is another matter: that was argued over for centuries and only settled into its familiar form relatively late, in the Song. So the Lo Shu is old, but a page telling you this exact grid has been fixed since mythic antiquity is compressing a long and contested history.
Number, trigram, direction
The step that turns a magic square into a feng shui instrument is this table. Each number owns a palace, each palace owns a trigram and a compass direction:
| Number | Trigram | Direction |
|---|---|---|
| 1 | Kan 坎 | N |
| 2 | Kun 坤 | SW |
| 3 | Zhen 震 | E |
| 4 | Xun 巽 | SE |
| 5 | None 中 | Center |
| 6 | Qian 乾 | NW |
| 7 | Dui 兌 | W |
| 8 | Gen 艮 | NE |
| 9 | Li 離 | S |
Note the fifth row. Five sits in the centre and the centre faces nowhere, which is why it has no trigram. That single fact explains something readers usually meet as an unexplained quirk: there is no kua number 5, and anyone who computes one borrows 2 or 8 instead. The gap is not a patch. It is the geometry.
Why the stars fly the way they do
Flying star practice moves numbers between palaces in a fixed order: centre, northwest, west, northeast, south, north, southwest, east, southeast. Presented cold, that looks arbitrary, and most English sources present it cold.
It is just counting. Start at the centre, which holds 5, and walk up through the numbers: 6, 7, 8, 9, then round to 1, 2, 3, 4. Read off the palace each number owns from the table above and you get exactly that sequence:
| Step | Lo Shu number | Palace |
|---|---|---|
| 1 | 5 | Center |
| 2 | 6 | NW |
| 3 | 7 | W |
| 4 | 8 | NE |
| 5 | 9 | S |
| 6 | 1 | N |
| 7 | 2 | SW |
| 8 | 3 | E |
| 9 | 4 | SE |
The flight path is ascending Lo Shu order and nothing more. Once you see that, the annual and monthly charts stop being a table to memorise and become something you can derive, which is the whole reason this page exists.
| Term | Pinyin | Chinese | Meaning |
|---|---|---|---|
| Lo Shu | Luòshū | 洛書 | The Luo River Writing. The 3 by 3 magic square underlying the nine palaces. |
| He Tu | Hétú | 河圖 | The Yellow River Diagram, the Lo Shu's traditional counterpart. |
| Nine palaces | jiǔ gōng | 九宮 | The nine cells of the grid, each with a number, a trigram, and a direction. |
| Center | zhōng gōng | 中宮 | The fifth palace. It holds 5, has no trigram, and has no direction. |
| Flying the stars | fēi xīng | 飛星 | Moving numbers through the palaces in Lo Shu order. |
The honest note
The Lo Shu is a real and elegant piece of mathematics, and its place in the history of combinatorics is not in dispute. What it is not is evidence for anything built on top of it. A magic square is a fact about numbers; assigning those numbers to compass sectors and reading fortune from the result is a cultural system, and an internally consistent one, but the consistency is the system agreeing with itself rather than with the world. Our evidence page holds the two apart.
Frequently asked questions
What is the Lo Shu square?
The Lo Shu (洛書) is a 3 by 3 arrangement of the numbers 1 to 9 in which every row, every column, and both diagonals add to 15. It is the oldest magic square known anywhere, and in feng shui it is the coordinate system underneath the bagua grid, the nine palaces, kua numbers, and the flying stars.
Where does the turtle legend come from?
Tradition says the pattern appeared as markings on the shell of a turtle emerging from the river Luo, and was read by Yu the Great while he was taming the floods. It is paired with the He Tu (河圖), the Yellow River Diagram. Both names are ancient and appear in the classics, but the specific number arrangements attached to those names took their standard form much later.
Why does the flying star flight path go center, northwest, west, northeast?
Because it is simply counting through the Lo Shu. Each number sits in a fixed palace: 1 north, 2 southwest, 3 east, 4 southeast, 5 center, 6 northwest, 7 west, 8 northeast, 9 south. Start at the center, which is 5, then go to 6, 7, 8, 9, 1, 2, 3, 4, and the palaces you visit are center, northwest, west, northeast, south, north, southwest, east, southeast. The flight path is not an arbitrary sequence; it is ascending Lo Shu order.
Why is there no kua number 5?
Because 5 sits in the center of the square and the center has no compass direction. A system built on assigning people a direction cannot use it, so by convention men who compute 5 take 2 and women take 8. The gap is not an exception bolted on; it falls straight out of the geometry.
Is the Lo Shu mathematically special?
Yes, in a small but genuine way. Up to rotation and reflection there is exactly one way to arrange 1 through 9 in a 3 by 3 grid so that every line sums to 15. The Lo Shu is not one magic square among many; for that size it is the only one there is.
Sources & further reading
- Yijing (I Ching), Xici commentary, and the Shujing (Book of Documents), where the names He Tu and Lo Shu appear. Full texts at the Chinese Text Project.
- Joseph Needham, Science and Civilisation in China, vol. 3 (Cambridge University Press, 1959). On the Lo Shu as the earliest recorded magic square and on the later systematisation of the diagrams.
- The number-to-palace mapping used here is the same table this site's flying star and natal chart code runs on, which is unit-tested against published charts.