Five lattices, and no others
There are five kinds of repeating grid in the plane. Not five that anybody has drawn — five that can exist, with an argument that fits on a page and leaves no sixth case anywhere.
The claim needs a definition to be precise, and the definition is the interesting part: two lattices are the same kind when they have the same holohedry — the same group of rotations and reflections mapping the lattice onto itself. Classify by symmetry rather than by appearance and the count comes out at five.
The classifying question
Every lattice, however skewed, has at least two symmetries: the identity, and a half turn about any lattice point. The half turn is unavoidable, because if is a lattice vector then so is , and negating every vector is exactly what a half turn does.
So the question is never “does this lattice have any symmetry” — it always has at least that — but “how much more than the minimum”. The answer is a finite group of order two, four, four, eight or twelve, and each value corresponds to exactly one lattice type.
That framing makes the enumeration tractable. Rather than trying to survey the infinitely many shapes a parallelogram can have, the argument asks which finite groups of rotations and reflections can map a discrete grid onto itself. The crystallographic restriction already limits the rotations to orders one, two, three, four and six, and the rest is bookkeeping.
Oblique: the general case
Take two vectors with no relationship between them — different lengths, an angle that is nothing in particular. The lattice they generate has the identity and the half turn and nothing else. Its holohedry has order two.
This is the generic case in the strict sense: choose two vectors at random and the result is oblique with probability one. Every other lattice type is a coincidence — a measure-zero subset of the possibilities — and the whole subject consists of asking what those coincidences make possible.
That the exceptional cases are the interesting ones is not unusual in mathematics. What is unusual is how few of them there are.
Rectangular: one right angle
Impose one condition — the two basis vectors are perpendicular — and the holohedry jumps from two to four. As well as the identity and the half turn, the lattice now admits reflection in each of the two axes.
The lengths are still free. A rectangular lattice may be nearly square or extremely elongated; nothing in the classification distinguishes them, because the holohedry is the same group of order four in both cases.
This is the first place where the classification’s convention shows itself. Two lattices with very different appearances are called the same type, and two with nearly identical appearances — a rectangular lattice and an oblique one with an angle of 89.9° — are called different. Symmetry is a discontinuous property, and a classification by symmetry inherits the discontinuity. That is a feature: symmetry is either present or absent, and a property that were to fade in gradually would not support theorems.
Centred rectangular: the one that looks like a mistake
Now the case that trips everybody up. Take a lattice whose two basis vectors are the same length but at an arbitrary angle — a rhombic lattice. Its holohedry is order four again: the identity, the half turn, and reflections in the two diagonals of the rhombus.
So there are two distinct lattice types with holohedry of order four, and they are genuinely distinct: in one, the mirror lines are along the basis vectors; in the other, they bisect the angle between them. No choice of basis converts one into the other.
The naming is where the confusion begins. This lattice is usually called centred rectangular rather than rhombic, because the convention is to describe it with a rectangular cell that has an extra lattice point at its centre — a cell containing two lattice points instead of one. That looks redundant. It is not: the rectangular description makes the mirror directions coincide with the axes, and every subsequent formula is simpler for it. The essay on unit cells takes up the trade in detail.
Two names for one object cause a specific and recurring error. A reader who counts “oblique, rectangular, centred rectangular, rhombic, square, hexagonal” arrives at six, and six is wrong, because the third and fourth are the same lattice described two ways.
Square: both conditions at once
Require the basis vectors to be perpendicular and equal in length, and the holohedry jumps to order eight: quarter turns in both senses, half turn, identity, and four mirrors — two along the axes and two along the diagonals.
That last observation is the mechanism by which lattices constrain groups. The three wallpaper groups containing a fourfold rotation — p4, p4m and p4g — must all sit on a square lattice, because there is nowhere else for a quarter turn to live. The lattice is not a stylistic choice that happens to accompany the symmetry; it is forced by it.
The converse fails, and it is worth being explicit. A pattern on a square lattice need not have fourfold symmetry: the motif may decline the opportunity, in which case the pattern’s group is one of the lower ones and the lattice is more symmetric than the pattern. The lattice permits; the motif disposes.
Hexagonal: the most symmetric
Take two vectors of equal length at . The lattice generated is the triangular grid, and its holohedry has order twelve — rotations by every multiple of , and six mirrors.
This lattice carries five of the seventeen groups — p3, p3m1, p31m, p6 and p6m — which is more than any other, and it is the one that shows up whenever equal circles are packed as tightly as possible. Bees, soap films, graphene and stacked oranges all end up here, for the good reason that the hexagonal arrangement is the densest packing of equal discs in the plane, a fact conjectured by Kepler and finally proved for the plane case by Axel Thue in 1910.
The relationship between the hexagonal and triangular descriptions confuses people, so: the lattice points form a triangular grid, and the region closest to each point is a hexagon. Both adjectives are used, sometimes in the same paragraph, and they refer to the same object seen two ways.
Which groups each lattice carries
The classification’s real use is as a filter, and the tally is worth having in one place.
The oblique lattice carries p1 and p2 — the two groups with no mirrors, no glides and no rotation beyond a half turn. The rectangular lattice carries pm, pg, pmm, pmg and pgg, which is to say every group whose mirrors or glides sit at right angles to one another. The centred rectangular lattice carries cm and cmm, the two whose mirrors alternate with glides in a way that only a centred cell describes cleanly. The square lattice carries p4, p4m and p4g. The hexagonal lattice carries p3, p3m1, p31m, p6 and p6m.
Two, five, two, three, five: seventeen.
The distribution is uneven in an informative way. The hexagonal lattice, with the largest holohedry, carries the most groups; the oblique lattice, with the smallest, carries the fewest. That is not a coincidence but close to a tautology: the holohedry is the ceiling on what a pattern can have, and a higher ceiling leaves room for more distinct arrangements underneath it.
The uneven distribution also explains a practical fact about ornament. Patterns on hexagonal lattices are over-represented in historical decoration relative to their share of the seventeen, partly because the underlying triangular grid is easy to lay out with a compass, and partly because the groups that are hardest to reach by construction are the ones with essential glides — and those sit on rectangular lattices.
Why the list closes
The argument that there is no sixth type runs as follows, and it is short enough to give in full.
The holohedry of a plane lattice is a finite group of isometries fixing a point, so it is either cyclic or dihedral. Its rotations are restricted to orders one, two, three, four and six. Every lattice has a half turn, so the rotation part contains at least order two, leaving orders two, four and six as candidates for the maximum.
If the maximal rotation is a half turn, the holohedry is order two if there are no mirrors and order four if there are — and in the mirror case the mirrors either lie along a basis or bisect it, giving rectangular and centred rectangular respectively.
If the maximal rotation is a quarter turn, mirrors are forced by composition and the holohedry is order eight: the square lattice.
If the maximal rotation is a sixth turn, mirrors are again forced and the holohedry is order twelve: the hexagonal lattice. A threefold maximum is not a separate case, because the half turn every lattice has composes with the threefold rotation to give a sixfold one.
Two, four (twice), eight, twelve. Five lattices, and the argument has nowhere left to branch.
The count that is not five
A natural question at this point is why textbooks so often say fourteen. The answer is that they are talking about three dimensions.
Auguste Bravais ran the same argument in space in 1848 and got fourteen lattice types, grouped into seven crystal systems. The extra structure comes from centring having more options — a three-dimensional cell can be centred on its body, on all its faces, or on one pair of faces — and from the greater variety of point symmetries available.
Fourteen is a slightly awkward number and the awkwardness is instructive. Some plausible-looking cases turn out to be duplicates: a body-centred tetragonal lattice and a face-centred tetragonal lattice are the same thing described with different axes, so only one appears in the list. Bravais’s enumeration is careful about exactly this, and Moritz Frankenheim’s earlier attempt of 1842 got fifteen because he missed one such coincidence. The plane’s five have the same hazard in miniature, which is why “rhombic” and “centred rectangular” must not be counted twice.
What the enumeration does not settle
Three limits are worth naming, because the number five gets quoted well beyond its scope.
It classifies lattices, not patterns. Five lattices carry seventeen wallpaper groups, and the gap is everything the motif contributes. Knowing a pattern’s lattice type narrows its group to a handful of candidates and never to one — and a motif with symmetry of its own can make the pattern come out more symmetric than intended, which is a hazard with its own essay.
It classifies lattices, not structures. A crystal structure is a lattice plus the contents of a cell. Two substances with nothing in common may share a lattice type, and the statement “there are fourteen kinds of crystal” is false for exactly this reason.
It says nothing about aperiodic order. A Penrose tiling has no lattice at all, so no entry in the list describes it, and the fact that it has none is not an oversight in the enumeration — it is the definition of what makes it aperiodic.
How the classification is checked here
The five types are not typed into a template on this site. They are enumerated, and the enumeration is a gate.
Each lattice type’s holohedry is computed from first principles — all integer matrices of determinant that preserve the lattice’s metric — and the orders come out as two, four, four, eight and twelve. The figure that draws all five asserts that there are five and that each has the holohedry order claimed for it; if either changed, the figure would throw and the build would stop.
That is deliberately more work than writing the number down. The point of the exercise is that no count on this site is a memory. It is the output of running the argument, and if the argument breaks the site does not build.
Who did it first
Bravais’s 1848 memoir is the reference, and it is worth noting what he was doing it for. He was not classifying abstract objects; he was trying to explain why crystals cleave along particular planes and grow particular faces, questions raised by Haüy’s stacking hypothesis sixty years earlier and unanswered since.
The plane case is a simplification that came later and mostly for teaching. Its five lattices are usually introduced as a warm-up for the fourteen, which slightly undersells them: the plane case is where the classification is short enough to check entirely by hand, and where a reader can see the whole argument at once rather than trusting a summary of it.
Where the ladder goes next
The natural sequel is the convention question: the unit cell is a choice, several are in use, and the centred cell that looked wasteful above is doing genuine work.
The natural consequence is the restriction that made the enumeration finite. Lattices tolerate rotations of order one, two, three, four and six, and nothing else at all, by an argument one line long — with a second and quite different proof available for the five-fold case, which is worth having because a fact this load-bearing deserves two independent routes.
And the payoff is the seventeen, which is what happens when these five lattices are paired with every point group they can carry.
What the pictures here cannot show. A lattice type is defined by its holohedry, which is a group, and a group is not visible. The figures show representative lattices at particular proportions; the claim that two lattices are of the same type is a claim about their symmetry groups being equal, and it is checked by comparing groups rather than by comparing drawings.