When a feng shui book overlays a floor plan with a 3×3 grid, or a flying-star chart drops the numbers 1 through 9 into its nine cells, that frame is not the author’s invention. It is the Lo Shu square (洛书, Luoshu, “the writing of the Luo River”) — and beside it stands an older companion, the He Tu (河图, Hetu, “the diagram of the River”), which pairs the same numbers with the five phases. Cited together as one phrase — he tu luo shu (河图洛书) — the two are the numerical layer beneath the more famous charts: the post-heaven trigrams carry Lo Shu digits as their addresses, and every flying-star or nine-palace diagram is drawn on the Lo Shu grid. This page lays out both figures: the legend and the earliest text, the birth and completion numbers of the He Tu, the magic square of the Lo Shu, how the trigrams plug into it, where the two charts surface in practice, and how they differ.
What the two charts are#
The legend gives each diagram a river and a courier. A dragon-horse rose from the Yellow River wearing the He Tu on its back; a divine turtle rose from the Luo River with the Lo Shu patterned on its shell. Later tradition assigns them patrons: Fuxi, who is said to have drawn the eight trigrams from the river diagram, and Yu the Great, who read the “nine divisions” of statecraft from the turtle’s nine cells. Myth aside, the earliest surviving mention is sober and brief. The “Appended Statements” (《周易·系辞上》, Xici) — the ancient commentary attached to the Book of Changes — records the tradition in one line:
河出图,洛出书,圣人则之。 “The River brought forth the Diagram; the Luo brought forth the Writing; the sage took them as his model.”
The sentence cites both diagrams without describing either — by the time it was written, each was already authoritative enough to need no introduction. What the tradition did with them, though, divides the labor cleanly: the He Tu explains how the five phases acquire their numbers; the Lo Shu is the grid on which those numbers are put to work. One diagram is a theory of the numbers, the other is the numbers arranged in space — and nearly every numerical method in feng shui runs on the second.
He Tu: the birth and completion numbers#
The He Tu distributes the numbers 1 through 10 as five pairs, one pair per phase — each pair made of an odd birth number (生数, shengshu) and an even completion number (成数, chengshu), the completion always five more than the birth:
| Phase | Birth number | Completion number | Quarter |
|---|---|---|---|
| Water 水 | 1 | 6 | North |
| Fire 火 | 2 | 7 | South |
| Wood 木 | 3 | 8 | East |
| Metal 金 | 4 | 9 | West |
| Earth 土 | 5 | 10 | Center |
The odd digits are “heaven’s numbers,” the even ones “earth’s numbers,” and the traditional mnemonic recites the pairs in order — heaven one births water, earth six completes it; earth two births fire, heaven seven completes it; and so on through wood, metal, and earth, with earth’s own pair settled at the center. Two readings of the table are worth keeping. First, each phase is counted twice — water is both 1 and 6 — which is why classical texts can give one phase two numbers without contradiction: they are naming its birth and its completion. Second, Earth holds the center with 5 and 10, and 5 is exactly the amount that turns every birth number into its completion number — the center phase is, arithmetically, the mediator, the same pivot role the five phases assign to Earth in their cycles.
The same Xici chapter goes on to total the numbers:
天数二十有五,地数三十,凡天地之数五十有五。 “Heaven’s numbers are five-and-twenty; earth’s numbers are thirty; all the numbers of heaven and earth together are five-and-fifty.”
The odd digits 1+3+5+7+9 make 25, the even digits 2+4+6+8+10 make 30, and 55 is the sum of everything the He Tu arranges. The diagram is, among other things, a picture of that arithmetic.
Lo Shu: the 3×3 magic square#
The Lo Shu uses the numbers 1 through 9, one per cell, in a 3×3 square. Drawn the traditional way — south at the top, east at the left — it reads:
| 4 Southeast | 9 South | 2 Southwest |
|---|---|---|
| 3 East | 5 Center | 7 West |
| 8 Northeast | 1 North | 6 Northwest |
A Chinese mnemonic describes the figure from the inside: 戴九履一,左三右七,二四为肩,六八为足 — “nine wears the crown, one treads the floor; three on the left, seven on the right; two and four as the shoulders, six and eight as the feet,” with five holding the center.
Then the arithmetic. The nine digits total 45, so any three-cell line must average 15 — and in this square all eight lines, the three rows, three columns, and both diagonals, hit 15 exactly. Check any line of the table above — 4+9+2, 9+5+1, 4+5+6 — the sum never moves. That property makes the Lo Shu a magic square, and on the standard account the earliest one recorded anywhere, known in China centuries before the idea surfaces in India, the Islamic world, or Dürer’s 1514 engraving. One further fact does most to explain why the tradition treated the figure as given rather than designed: there is essentially only one 3×3 magic square. Arrange the digits 1–9 so that all eight lines share a sum, and every solution is a rotation or reflection of the Lo Shu — which is also why 5 has no choice but to occupy the center. The square is not one clever arrangement among many; it is the arrangement, up to symmetry.
Lo Shu and the trigrams: where the bagua numbers come from#
The post-heaven (King Wen) arrangement of the eight trigrams assigns each trigram a compass point; the Lo Shu then numbers each point, and in doing so gives each trigram a number: Kan 坎 1 north, Kun 坤 2 southwest, Zhen 震 3 east, Xun 巽 4 southeast, Qian 乾 6 northwest, Dui 兑 7 west, Gen 艮 8 northeast, Li 离 9 south — with 5 left at the center as the seat of the taiji. These are the “trigram numbers” of feng shui literature, and they are not a separate numbering scheme: the post-heaven bagua is the Lo Shu with trigram names substituted for digits. The full direction–trigram–element–life-area stack is laid out in Feng Shui Bagua Map Explained; read it with this page in hand and its cells acquire street addresses, so that “the Gen palace” and “palace 8” name the same corner.
Where you meet them#
You do not have to go looking for these charts — they are already in the tools people use:
- Flying star charts. The time-based school of feng shui (玄空飞星, xuan kong fei xing) draws its chart as a Lo Shu grid: nine “stars,” numbered 1 through 9, occupy the nine palaces, and the star standing at the center changes with the twenty-year period. A flying star feng shui chart, in other words, is a Lo Shu square with time-varying contents — when you ask where its 3×3 frame and its nine digits come from, the answer is this page’s square. (How the school casts and reads a chart is a discipline of its own; the frame underneath is what is supplied here.)
- The Western feng shui grid. The “bagua map” of Western feng shui books — eight life areas around a center, overlaid on a floor plan — is the Lo Shu frame with the numbers dropped, which is why the same object is often sold as a “lo shu grid.” Nine zones, one center, eight rim cells: whatever the packaging, the geometry is this square.
- Qi Men Dun Jia. The nine-palace board of qimen dun jia — stars, doors, deities and all — is laid out on the Lo Shu palaces; cast a chart for any hour and watch the nine cells fill in exactly this geometry.
He Tu vs Lo Shu: what actually differs#
Put side by side, the two diagrams sort into a clean division of labor:
| He Tu 河图 | Lo Shu 洛书 | |
|---|---|---|
| Numbers used | 1–10 | 1–9 |
| Shape | five pairs at four quarters and the center | one number per cell of a 3×3 grid |
| What it explains | why each phase carries two numbers (birth and completion) | how numbers sit in space — the working grid |
| What runs on it | the numerology of the five phases | flying stars, the nine palaces, the Western bagua grid |
The classical one-line summary calls the He Tu the substance (体, ti) and the Lo Shu the function (用, yong): the He Tu is where the number–phase pairs come from, the Lo Shu is what feng shui and qimen actually compute on. A practical test for any text in front of you: if it gives each phase two numbers, it is speaking He Tu; if it fills a nine-cell square, it is speaking Lo Shu.
A note for overseas readers#
Three things are worth knowing before comparing notes with Chinese diagrams. First, the Lo Shu usually reaches the West as a magic square — a recreational-math curiosity, a cousin of the 4×4 square Dürer etched into Melencolia I. In the Chinese arts the same nine cells are not a puzzle but a coordinate system: each cell is a direction, and the square is the board the arts are played on. Second, Chinese diagrams put south at the top and east at the left, so a traditionally drawn Lo Shu is mirrored relative to a north-up Western drawing — check the orientation before deciding that two diagrams disagree. Third, the “Lo Shu grid” of Western birth-date numerology, which counts the digits of a birth date into the nine cells, is a modern adaptation with no classical counterpart; the classical square orients space, not personality.
Further reading: What Is Feng Shui · Feng Shui Bagua Map Explained · back to the Foundations section.