Eight ways to leave a square alone

A square is a shape with a peculiar property: it can be picked up, moved and put down again so that nothing about the result betrays what happened. A quarter turn does it. A flip across a diagonal does it. Doing nothing does it, which sounds like a joke and turns out to be load-bearing. The interesting question is not whether such motions exist but how many there are, and the usual answer — eight — is normally stated rather than found.

The picture above finds it. Its subject is not the eight symmetries; it is the twenty-four candidates, of which the eight are a minority that had to earn its place.

What a symmetry is, stated so a machine can test it #

The definition worth using is not “a motion that leaves the shape looking the same” — looking is exactly the faculty being taken out of service. The workable version is about distances. Number the corners. A relabelling sends corner one somewhere, corner two somewhere else, and so on. It is a symmetry when every distance survives: the gap between the corners now called one and two is the gap that was there before, and the same for all six pairs.

That is a condition a build can check, and the figure checks it. There are twenty-four relabellings of four things. For each, the six pairwise distances are compared against the six original ones, and the relabelling is kept only if every comparison agrees. Eight survive.

The sixteen that fail are worth a moment, because they are what the count is a count against. Swapping two adjacent corners and leaving the others alone turns a square into a bow tie — the sides cross. Swapping two opposite corners folds it. None of these is a motion of the plane at all; each is a relabelling that no rigid movement could produce. The eight survivors are precisely the relabellings that a physical square, moved physically, could arrive at.

The eight, drawn as the things they do #

Named, they are four rotations — by nothing, a quarter, a half and three quarters — and four reflections. The reflections split into two kinds, which is the first fact about the square that a triangle does not share: two of the mirror lines run corner to corner, and two run through the midpoints of opposite edges. A square has two sorts of axis because it has two sorts of thing at the ends of an axis.

Rotation by nothing is on the list because the list has to be closed under a certain operation, and closure is the whole subject of the next section. Leaving it off would be like leaving zero out of the whole numbers: nothing is lost that anyone would miss until the arithmetic starts.

Doing one, and then another #

The eight motions are not merely a set. They compose. Turn the square a quarter, then flip it across a diagonal, and the combined effect is some single motion — necessarily, because the composite leaves every distance where it was, and the search already established that only eight relabellings do that.

The table is filled by composition, not from memory. Each cell takes two of the eight permutations, applies one after the other, and looks the result up in the list. If a lookup ever failed, the figure would stop the build — and what it would be reporting is that the eight are not closed, which would mean the search had missed something.

That is the shape of every claim on this site: the table is not an illustration of closure. It is sixty-four separate tests of it.

Where the order matters, and by how much #

Quarter turn, then a flip across the main diagonal. Or the flip first and then the turn. These are different motions, and the table says which: r then m₁ is m₄; m₁ then r is m₂ — two different reflections, in two different axes.

This is not a subtlety and it is not a defect. It is the property that makes a symmetry group of a shape interesting rather than a relabelled version of ordinary addition. Turning a dial by three and then by five gives the same result as five and then three, which is why clock arithmetic is so docile. Flipping and turning do not commute, and the failure is visible at the smallest possible scale: two motions, applied in two orders, landing in two places.

The four rotations on their own do commute — they are a copy of the clock with four hours on it. The whole of the non-commuting behaviour is in the interaction between a turn and a flip, and it can be summarised in one line: doing a flip and then a turn is the same as doing the turn backwards and then the flip. That single relation, plus the fact that four quarter turns get back to the start and a flip done twice does nothing, determines the entire table.

It is worth noticing how little is being claimed. The table has sixty-four entries; the three facts just listed have between them about twenty symbols. Everything else in the table is a consequence, and a consequence that can be derived without ever picking up a square. That compression — a whole multiplication table squeezed into three relations — is the reason groups are written down in this way in the first place, and it is the same economy that lets a finite field with four elements be specified by one irreducible polynomial rather than by two sixteen-cell tables.

What every row being complete guarantees #

Read across any row of the table. All eight motions appear, each exactly once. Read down any column: the same. The table is a Latin square, and the figure asserts that it is — sixteen separate checks, one per row and one per column.

That property is not decoration. A row containing every motion exactly once means that for each target motion there is exactly one thing to compose with the row’s motion to reach it. In particular the row contains do nothing, and whatever produces it is that motion’s inverse. Every motion is undoable, exactly one way.

The same argument runs on any finite set closed under an associative operation with no ambiguity about cancellation, which is the abstract statement that this square happens to be an instance of. The picture is not evidence for that general fact. It is one case of it, drawn at a size where every cell can be read.

The triangle, where the search finds nothing to reject #

Run the identical machinery on a triangle and something instructive happens: all six relabellings survive. Every permutation of three corners is a symmetry. The search rejects nothing.

This is the control the figure needs. A test that always finds a proper subset would be suspicious — perhaps the distance condition is doing no work and merely re-encoding the answer. On the triangle it does no work because there is no work to do: with three corners there is only one distance, so any relabelling preserves it trivially.

That also explains why the square is the first interesting case. Four corners have two distinct distances between them — the side and the diagonal — and a relabelling has to respect the difference. Five corners have two distances again, six have three, and from four onward the symmetry count is always 2n against n! candidates, a proportion that collapses rapidly. For an octagon it is sixteen out of forty thousand three hundred and twenty.

There is a second reason the triangle deserves the visit. Its six symmetries are all the permutations of its corners, which means the symmetry group of an equilateral triangle and the group of rearrangements of three objects are the same object seen twice. That coincidence stops at three. For the square the eight symmetries sit inside the twenty-four rearrangements as a strict part, and the sixteen left over are not defects or degenerate cases — they are perfectly good rearrangements of four labels that simply do not correspond to any way of moving a rigid square. The distinction between permuting labels and moving a shape becomes visible for the first time at four corners, and stays visible from there on.

The parts that are wholes #

Some subsets of the eight are worlds of their own: compose two members and the answer is still inside. The four rotations are such a subset. So is the pair consisting of doing nothing and one particular flip. The figure finds them all by brute force — two hundred and fifty-six subsets, each tested for closure — and reports ten.

The sizes are the finding. One, two, four, eight. No subset of three motions is closed; none of five, six or seven is either. Every closed subset has a size dividing the size of the whole, which is Lagrange’s theorem, and here it arrives as an observation about a list rather than as a theorem with a proof attached.

The reason is worth stating even though the picture cannot draw it. Fix a closed subset. Every motion of the whole group carries that subset to a translated copy of itself, the copies are all the same size, and every motion lies in exactly one copy. So the group is partitioned into equal blocks, and the block size divides the total. The eight-element square is small enough that the partition can be checked by hand, and large enough that the fact is not obvious in advance.

The subgroup picture is also where a warning belongs. Ten closed subsets is a small number, and it is small because eight is small. The count does not grow gently: the twelve motions of the hexagon have sixteen closed subsets, and by the time a group has a few dozen elements the lattice is no longer something a page can hold. Exhaustive walks over subsets cost two to the power of the group’s size, so the method that settles the square in two hundred and fifty-six tests would need a million million for a group of forty. Everything here is drawn at the size where brute force still fits, which is the same discipline the thirty-six officers essay works under and the same wall it eventually meets.

The hexagon, and what carries over #

Nothing in the machinery is about squares. Run it on a hexagon and twelve motions come back; on a pentagon, ten. The pattern — n turns and n flips — is not a coincidence but a short argument: a symmetry is determined by where it sends one chosen corner and whether it flips, and there are n choices for the first and two for the second.

Which is a proof, and it is a proof of the kind this site is suspicious of, because it is a proof that could easily be wrong and look right. The search is the check on it. The two agree at every size the search can reach, which is what makes the argument trustworthy at sizes it cannot.

The hexagon also carries something the square does not: it contains the triangle. Three of its turns are exactly the turns of an equilateral triangle inscribed in it, and that copy is one of the sixteen closed subsets the exhaustive walk finds. Shapes with more symmetry contain the shapes with less, and the subgroup lattice is where that containment is visible as arithmetic.

What the picture cannot show #

Three things, and they matter more as the ladder gets taller.

The figures draw a finite group. The symmetries of a circle are a continuum — every rotation by every angle — and no drawing of a table can hold them. Almost everything interesting about groups in the rest of mathematics happens in the infinite case, and none of it is here.

The figures also draw a group as motions of a particular shape. The eight are an object in their own right, with a multiplication table that other things also satisfy: the symmetries of a rectangle-with-a-twist, certain sets of matrices, certain permutations of eight coins. Two systems with the same table are the same group wearing different clothes — the site’s same thing twice motif, in its cleanest form — and a picture of the square cannot indicate which of its features are about the square and which about the table.

And the picture cannot show why anyone should care about the composition at all, rather than about the eight motions as a list. The answer to that is in the next rung, where the group stops being the object of study and becomes the instrument.

Where this goes next #

The eight motions are useful for the same reason a ruler is: they measure sameness. Two things are the same if a motion carries one to the other, and once that is agreed, counting changes character — because the question is no longer how many objects there are but how many there are up to the group.

That question has an answer with a formula behind it, and the formula is strange: to count the classes, count instead how many objects each motion leaves untouched, and average. The next rung draws both counts side by side, on colourings of these same four corners, and finds that two entirely different computations return the same integer.

Some of the ground here is already elsewhere on the site under other names. The rotations alone are the clock face of numbers that wrap; the counting of things up to rotation is what makes the necklace argument in Fermat’s little theorem work at all; the classification of the five regular solids is a symmetry argument that never uses the word. The reflections turn up again wherever an invariant direction is at issue, since a reflection is the map whose invariant directions are its axis and its perpendicular.

What the square adds is smallness. Eight is a number that fits on a page, twenty-four is a number that can be searched, and two hundred and fifty-six is a number that can be walked. Everything claimed here has been checked at that size, and that is exactly what makes the claims that hold at every size worth stating separately.