Learn · Essay · 13 min read · Updated September 2026

The Lo Shu square

Nine numbers in a grid, every line of which adds to fifteen. The figure behind the bagua map, the Flying Star charts and every practitioner who talks about “the one” or “the nine.” Where it came from and what the Han made of it, its companion the river chart, the path its numbers trace, the square in the mathematics of Song China and Renaissance Europe, and what it does when it is laid over a flat.

Oriented SW 216°

Earth

Fig. 01 A screen of repeated openings. The Lo Shu is the tradition’s grid of nine cells, one centre, and a number in each.

Take the digits one to nine and arrange them in a three-by-three grid so that every row, every column and both diagonals add to fifteen. There is essentially one way to do it. Chinese tradition has known that arrangement for at least two thousand years, gave it a legend and a name, and laid it over the compass. The result is the Lo Shu square, and it is the grid under every bagua map, the frame of every Flying Star chart, and the reason a practitioner can say “the two” and mean the south-west.

The figure

Fig. 01 The Lo Shu square. Left, the classical arrangement with south at the top: every row, column and diagonal adds to fifteen, which is what made the figure sacred. Right, the same square turned north-up, as the bagua grid appears on this site, with the direction each number carries in the Later Heaven arrangement. The numbers are the ones the Flying Star method plots onto a plan and the ones a practitioner means by “the one” or “the nine.” Five, at the centre, has no direction and belongs to Earth.

The property is arithmetical and exact: the square is the unique three-by-three magic square, up to rotation and reflection. Its centre is five, the middle of the sequence; the odd numbers sit on the cardinal points and the even numbers in the corners; every pair of numbers opposite each other across the centre adds to ten. Needham traces the figure through the early literature of correlative thought, where its symmetry was taken as evidence of an order in things: if nine could be arranged so that every line balanced, the world might be arranged the same way, and the square became a plan of it.

The Han and the nine palaces

By the Han the square had a second name, the Nine Palaces, and a use. The court diviners moved the spirit Taiyi through the nine cells in a fixed order, one palace at a time, and read the year by where it stood; the Yijing’s appendices had already spoken of the numbers of heaven and earth, odd and even, and the square gave them places. Needham’s account sets this in the wider habit of Han thought, which arranged everything in fives and nines and read correspondence as cause. The palaces are the cells of every later plan: when a practitioner divides a house into nine, the palaces are what the nine are called, and the order Taiyi walked is the order the numbers are still moved in.

From square to compass

The Later Heaven arrangement of the trigrams places Li in the south and Kan in the north, and the Lo Shu is laid over it with nine at the south and one at the north. Each number thereby acquires a direction, a trigram and an element: one is Kan and Water, nine is Li and Fire, three and four are the two Wood trigrams in the east and south-east, six and seven the two Metal trigrams in the west and north-west, two and eight the Earth trigrams in the south-west and north-east, and five is the Earth of the centre. The bagua page draws the arrangement; the Explorer turns it.

The path

Read from the centre outward in numerical order, the square’s numbers trace a line, and the line is the most useful thing on the page for a reader who wants to follow the compass schools.

Fig. 02 The path through the square. Start at the centre, five, and follow the numbers in order: six is at the north-west, seven at the west, eight at the north-east, nine at the south, one at the north, two at the south-west, three at the east, four at the south-east, and the path returns to the centre. The line is the order in which the Flying Star method moves any number through the nine palaces: put a period’s number in the centre and count on along the path, and the chart writes itself. Drawn north up, as the square appears everywhere on this site. The path is the Lo Shu’s own, the sequence its numbers make when they are read from the centre outward, and the Han diviners already traced it with the spirit Taiyi through the Nine Palaces. It is the one thing a reader has to know to follow a Flying Star chart, and it is easier to remember as a line than as a rule.

Five is at the centre; six is at the north-west; seven at the west; eight at the north-east; nine at the south; then one at the north, two at the south-west, three at the east, four at the south-east, and the line returns to the centre. That sequence, centre, north-west, west, north-east, south, north, south-west, east, south-east, is the order in which the Flying Star method “flies” any number through the nine palaces: put a period’s number in the centre, count on along the path, and the chart is written. Practitioners learn it as a rhyme; it is easier as a line, and once it is seen the charts on the Flying Stars page stop looking like arithmetic and start looking like a walk.

What the numbers do

In the ordinary bagua map the numbers are labels. In the Flying Star method they are variables: the square is flown along the path, forward or in reverse, according to a building’s facing and its period of construction, and the resulting pairs of numbers in each cell, the mountain star and the water star, are read as the qualities of that part of the house for a span of years. The method is intricate, Skinner’s and Wong’s manuals give it in full, and the schools essay says where it sits; this site describes it and does not compute it.

The square laid on the year

Fig. 03 The square laid on the year. Each year has a ruling star, found by adding the digits of the year to one figure and taking it from eleven, and the star is set at the centre of the square and flown through the palaces by the Lo Shu order the essay gives, centre, north-west, west, north-east, south, north, south-west, east, south-east. The three years are drawn: 2024 with the 3 at the centre and the 5 in the west, 2025 with the 2 at the centre and the 5 in the north-east, 2026 with the 1 at the centre and the 5 in the south, which is where the almanacs and the practitioners put the year’s five yellow, the palace the school does not dig, drill or build in that year. The grids are computed by the rule and not copied, and the Flying Stars tool computes any year on the same rule. What the reader should take from them is the mechanism: one number, one order, nine palaces, and a square that has been flown this way over houses and years since the Song. The annual star is the school’s and the site records it as such; the year begins at the spring festival, early February, so that January belongs to the year before. The five yellow’s palace is the one thing about the annual chart most households ask a practitioner for, and the tool gives it.

The square is flown over houses, which the Flying Stars essay gives, and it is flown over years, which is the use most households meet first, because it is the one the almanac prints and the practitioner is asked about at New Year. Each year has a ruling star, found by adding the digits of the year to one figure and taking it from eleven: 2024 is 2 + 0 + 2 + 4, eight, from eleven three; 2025 nine, from eleven two; 2026 ten and then one, from eleven ten and then one. The star goes to the centre of the square and flies through the palaces in the order above, centre, north-west, west, north-east, south, north, south-west, east, south-east, and the drawing shows the three years side by side, computed by the rule rather than copied from a table. What the school reads in them is mostly one number. The 5, the five yellow, is the star it treats as the year’s affliction, and the palace it falls in is the palace the school will not dig, drill, renovate or hammer in that year: the west in 2024, the north-east in 2025, the south in 2026, which is where the almanacs of those years put it and where a practitioner would tell a household to leave the wall alone; the 2, the sickness star, gets the same caution at a lower pitch, and the 8, 9 and 1, the retiring, ruling and coming stars of Period 9, are read as the year’s good palaces. The house through the year essay gives the calendar the annual chart sits in, with the year beginning at the spring festival in early February rather than in January, and the Flying Stars tool computes the chart for any year by the same rule, laid over a house’s own. The site records the annual square as the school’s: it is arithmetic, the arithmetic is old and exact, and what it predicts has not been tested and cannot easily be. What the drawing teaches that is not the school’s is the mechanism itself, one number, one order and nine palaces, which is the whole of the Lo Shu’s working life from the Song to the tool on this site.

The square elsewhere

The three-by-three magic square is the only one of its size, up to rotation and reflection, and it turns up wherever people have played with numbers. In China itself the mathematician Yang Hui, in a treatise of 1275, set out magic squares of orders three to ten and the methods for building them, treating the Lo Shu as arithmetic rather than revelation, which by then it was to mathematicians and was not to geomancers. In the Islamic world the square was a talisman, engraved on amulets; in Europe it was known as the square of Saturn in the Renaissance tables of planetary squares, and the four-by-four square that Dürer engraved into his Melencolia I of 1514 belongs to the same family. What is Chinese is not the arithmetic but the reading: the sequence in which the numbers are placed became the path along which the Flying Stars move, the odd and even numbers became yang and yin, and the whole figure was laid over the compass so that each number acquired a direction, a trigram and an element.

In a room

Stripped of the calendar, the square gives the tradition a way of dividing any plan into nine and naming the parts, which is what the bagua map does. The centre, five, is asked to be open and stable; the eight around it are read by direction. The Bagua explorer shows the square at the centre of the ring, north up, with its numbers, so that turning the ring and reading the plan can be done in one look.

Plan 04 The Lo Shu square laid over a flat, north up: nine palaces stretched to the plan’s outline, and one of them missing The square is stretched to the plan’s bounding rectangle and its nine cells become the nine palaces, each named by its number, its direction and its trigram. The flat is L-shaped, so the south-west palace, two, Kun, falls outside the walls and is drawn ghosted: a missing palace, which the schools read and remedy in different ways. The centre, five, is the palace the tradition wants open. The classical square is printed south up; here, as everywhere on this site, north is up, so the numbers are turned round.

Laid over a flat, the square looks like the drawing: nine palaces stretched to the plan’s outline, each with its number and its direction, north up as everywhere on this site. The stretching is itself a decision the schools make differently: most lay the square over the bounding rectangle of the plan, as here; some divide the plan by its actual area; the Black Sect lays it from the door instead of the compass. The exercise shows two things at once. One is the centre, five, which the tradition wants open and which in most flats is a corridor or a wall. The other is what the outline does to the square: an L-shaped plan leaves a palace outside the walls, here the south-west, two, Kun, and the schools differ on what a missing palace means and what to do about it. The choosing a home guide draws the missing corner at the scale of a house; the Flying Star method plots its two stars into each of these nine.