Crystallography & Symmetry
patterns derived from their groups
The classification

The groups ornament actually uses

Seventeen exist and decoration does not use them evenly. Which are common is an empirical question that published surveys answer differently, and part of the reason is a hazard this site can measure exactly.
15 min read 6 figures The motif must be asymmetricExactly this manyGenerated, not drawn

Seventeen groups exist. Decorative traditions do not use them in equal proportions, and they never have — some appear on pottery from every continent and others are close to absent from the entire historical record. That much is agreed. Which groups fall where is not, and the disagreements between published surveys are more interesting than any of their numbers.

How often a dot gives the wrong groupEvery position on a grid inside the cell, tried as a single-dot motif for each of the seventeen groups. The bar is how often the resulting pattern turned out to have more symmetry than the group it was made with — so the caption would have been wrong and nothing about the picture would have shown it.grouppositions at which a dot gives a different group (of 121)p1121 / 121pm121 / 121pg121 / 121cm121 / 121p385 / 121p3m167 / 121pgg57 / 121p457 / 121pmg41 / 121p4g41 / 121pmm40 / 121p637 / 121p31m29 / 121cmm20 / 121p4m20 / 121p28 / 121p6m2 / 1214 groups fail at every position · p6m fails at 2each orbit's symmetry detected from the point set11×11 grid
Fig. 1 Every position on a grid inside the cell, tried as a single-dot motif for each of the seventeen. The bar is how often the resulting pattern turns out to have more symmetry than the group it was made with — so a craftsman aiming at the group on the left, working with a symmetric element, lands somewhere else.

This site cannot settle an empirical question about ornament, and it can measure one thing that bears on it directly: how hard each group is to hit by accident. The measurement turns out to explain part of the pattern in the surveys, and part of the disagreement between them.

What the surveys say, and why they differ

The best-known claim in this area is that the Alhambra contains all seventeen wallpaper groups. It is repeated in textbooks constantly, and it is not established. Different analysts of the same building have produced counts of eleven, thirteen, fourteen and seventeen, and they are not looking at different buildings.

They differ for reasons that have nothing to do with mathematics.

How much pattern counts as a pattern. A group is a property of an infinite repeating design. A tile panel is finite, often interrupted by an arch or a doorway, and frequently a fragment of a larger scheme. Deciding that a panel exhibits p31m rather than p3 means deciding that the mirrors would have continued.

Whether colour counts. A pattern whose shapes have the symmetry of p6m and whose colouring breaks half of it is p6m if colour is ignored and p3 or p31m if it is not. Both readings are defensible, and the colour groups — the classification that treats a colour permutation as part of the symmetry — are a larger list of forty-six two-colour groups in which the question has a proper answer.

Whether a near-symmetry counts. Hand-cut tile is not exact. Every analysis applies a tolerance, and nobody states theirs.

So the honest summary is that the frequency of a group in a corpus is a measurement with a large and mostly undeclared methodological component. Washburn and Crowe’s 1988 book, which is the standard reference for applying the classification to archaeological material, spends most of its length on exactly this: a procedure for making the assignment reproducible between analysts.

What can be measured here

The measurable question is different and sharper: given a group, how likely is a careless construction to produce a different one?

The sweep behind the figure at the top of this page takes each of the seventeen, tries a single dot at every position on a grid inside the cell, computes the orbit under the group’s operations, and hands the resulting point set to a detector that has never heard of the group. When the detected symmetry is larger than the group used to build it, the pattern is of a different group from the one intended, and the position is counted as a failure.

The spread is large and the ends of it are the informative part.

Four groups fail at every position tried. p1, pm, pg and cm cannot be illustrated with a single dot at all. The reason is a construction rather than an accident: the midpoint between any dot and its own lattice translate is a centre of inversion, so the orbit of a lone dot always acquires a half turn, and a group without one is never what results.

One group barely fails at all. p6m already has every operation its lattice permits, so there is nothing left to gain, and the only way for a dot to change the answer is for its orbit to collapse onto itself at a special position.

Between them the groups sort roughly by how much room they leave above themselves in the containment ordering — which is what the measurement is really detecting.

Which of the seventeen contain whichThe containment relations among the wallpaper groups, computed by comparing operation sets rather than read from a table. A group sits above every group it contains, and the height of a node is the number of operations in its cell.12346812p1p2pmpgcmpmmpmgpggcmmp4p4mp4gp3p3m1p31mp6p6m45 containments, 27 of them maximaloperations per cell on the leftdecided by set inclusion in a shared basishexagonal and square are not comparable
Fig. 2 The containments among the seventeen. A group low in this diagram has many groups above it and is correspondingly easy to overshoot; a group at the top has nowhere to go. The collapse measurement above is this diagram read as a hazard.

The numbers the sweep gives

The counts are worth setting out, because their ordering is not the one anybody would guess.

Out of 121 positions tried per group, four groups fail at all 121: p1, pm, pg and cm. Then the failures fall away — p3 at 85, p3m1 at 67, p4 and pgg at 57, pmg and p4g at 41, pmm at 40, p6 at 37, p31m at 29, cmm and p4m at 20 — and the two lowest are p2 at 8 and p6m at 2.

Two features of that list deserve comment.

Order does not predict the hazard. p2 has two operations per cell and fails 8 times; cm has two and fails 121. p4m has eight and fails 20; p4g has eight and fails 41. What matters is not how large the group is but how much room there is directly above it — how many groups contain it on the same lattice, and how easily a symmetric motif supplies the missing operations.

The failures are not small. When a dot betrays p3, the detected group frequently has 36 operations per cell against the 3 it was built with: the orbit has collapsed onto a much more symmetric arrangement, and what is on the page is not a slightly-wrong p3 but a completely different pattern. The site’s round trip refuses these outright rather than reporting a discrepancy, because there is nothing intermediate to report.

The p2 entry is the one that repays a second look. A half turn is the operation a lone dot’s orbit acquires for free — the midpoint construction gives it on any lattice whatever — so p2 is nearly the only group a dot can illustrate honestly, and its 8 failures are the special positions where the dot lands on a rotation centre and its orbit shrinks.

Why this bears on the surveys

The connection is a claim about craft rather than about mathematics, and it should be stated as such.

A decorative tradition working with symmetric elements — a rosette, a star, a six-petalled flower, a circular boss — is working with motifs whose own symmetry adds to the pattern’s. The result is not the group the layout intended but the group the layout and the motif produce together, which is always at least as large. So a tradition whose vocabulary is symmetric shapes will produce high-symmetry groups whether or not anybody was aiming at them.

That is a real effect and it is visible in the record. The groups that dominate most surveys — pmm, p4m, p6m, cmm — are the ones with the most operations per cell, which is exactly what a symmetric motif on a symmetric layout produces. The groups that are consistently rare — p1, pg, p2, pgg — are the ones that require the absence of symmetry in the motif, which is a deliberate act rather than a default.

There is a second and simpler explanation for the same observation, and both are probably true: high-symmetry patterns are easier to lay out and easier to get right by hand. A grid of mirror lines is self-correcting, and a p1 pattern has no internal check at all — which is the practical version of the point the notation makes when it gives p1 the shortest symbol and the least to say.

Why p3 cannot be drawn with dotsThe same group applied to a single dot and to a motif with no symmetry of its own. The dot's orbit turns out to have more symmetries than the group it was made with, so a figure drawn that way illustrates a different group from the one in its caption.a dot at (1/12, 1/12)6 symmetries per cella motif with no symmetry3 symmetries per cell85 of 121 dot positions give more symmetry than p3both generated with the 3 operations of p3the dot gains 3
Fig. 3 The mechanism, for a single group. A dot placed at most positions in a p3 cell produces a pattern with mirrors and glides the group never had — the picture is of p31m — and nothing about the drawing announces it.

Where the exactness stops

The sweep is exact and its scope is narrow, and the gap between the two needs stating plainly because this essay is one step closer to an empirical claim than the rest of this site.

The sweep measures dots, not ornament. A real decorative motif is not a point; it is a shape with its own symmetry, and its own symmetry group is what interacts with the layout. A dot is the extreme case — the most symmetric possible motif — so the numbers here are an upper bound on the hazard rather than an estimate of it.

The grid is finite. Positions are tried on an eleven-by-eleven grid of rationals, so the counts are counts over that grid and would change slightly on another. What does not change is which groups fail everywhere and which barely fail, since those are consequences of the group structure rather than of the sampling.

And no count of historical patterns appears on this page. Quoting one would mean adopting some survey’s methodology without being able to check it, and the disagreements described above are exactly about methodology. What is quoted instead is which groups are common and which are rare, which every survey agrees on even where the counts differ.

What a tradition has to do to reach p1

Turning the measurement round gives a practical reading: what does a decorative tradition have to do in order to produce each group?

To reach the high-symmetry groups, nothing in particular. Lay a symmetric motif on a symmetric grid and p4m or p6m arrives whether or not it was wanted. These groups are the default outcome of ordinary competence with a compass.

To reach the glide groups, a deliberate offset. pg, pgg and pmg all need alternate rows displaced by half a repeat and reversed. A footprint trail does this naturally, which is why the frieze version is common in representational borders and rare in geometric ones, and the two-dimensional versions are correspondingly uncommon.

To reach p1 or p2, an asymmetric motif and a deliberate refusal. There is no way to arrive at a pattern with no mirrors by accident: any symmetric element puts one back. A p1 design is a design where somebody decided not to have symmetry, executed it consistently across a whole surface, and had no self-correcting grid to work against.

That ordering matches what the surveys report, and it is worth being clear about the direction of the inference. The measurement does not show that any tradition reasoned this way. It shows that the outcomes are unevenly reachable, which is enough to predict an uneven distribution without anybody having intended one — the same shape of argument as a physical process producing a skewed distribution from unbiased inputs.

The generalisation

The same reasoning applies wherever a structure is built from symmetric components, and it is not confined to decoration.

In crystal structures the effect has a name: pseudosymmetry, where a structure is very nearly but not exactly of a higher-symmetry group. A molecule with its own symmetry sitting at a general position frequently produces a structure whose diffraction pattern is almost that of a more symmetric group, and refining it in the wrong one gives a fit that looks acceptable and coordinates that are slightly wrong. Detecting missed symmetry is enough of a standing problem to have dedicated software, and papers reporting corrections to published structures are a regular feature of the literature.

The mechanism is identical to the dot problem. A symmetric component contributes its own symmetry to the whole, the whole is then more symmetric than the arrangement was meant to be, and nothing about the result looks wrong.

Why p4 cannot be drawn with dotsThe same group applied to a single dot and to a motif with no symmetry of its own. The dot's orbit turns out to have more symmetries than the group it was made with, so a figure drawn that way illustrates a different group from the one in its caption.a dot at (1/12, 1/12)8 symmetries per cella motif with no symmetry4 symmetries per cell57 of 121 dot positions give more symmetry than p4both generated with the 4 operations of p4the dot gains 4
Fig. 4 The same mechanism on the square lattice. Of 121 dot positions, 57 give a pattern more symmetric than p4 — and the failures are large ones, with detected orders reaching 32 against the 4 intended.
The wallpaper group p1A pattern with the symmetry of p1, generated by applying the group's 1 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.p1oblique lattice · 1 operations per cell
Fig. 5 p1, the group with nothing but translations, drawn with the three-point motif it requires. No single dot produces this pattern at any position on any lattice: the midpoint between a dot and its translate is always a centre of inversion, so the orbit is p2 whatever the intention.
The seventeen wallpaper groupsEvery way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others.p1pgp2pggcmmp4mp6m7 groups, each generated and verified
Fig. 6 The two ends of the frequency distribution together: the groups a tradition has to work to achieve, on the left, and the ones that arrive by default, on the right. The difference between them is whether the motif has to be asymmetric.

The surprising part

The hardest group to illustrate is the one with no symmetry at all.

p1 has a single operation per cell — the identity — and it is the only one of the seventeen that a single dot can never produce, at any position, on any lattice. Every other group is reachable from a dot somewhere. The group that asks for nothing is the one that cannot be had by accident, because the accident always adds something.

The connection worth carrying is to the comma on the nineteenth-century ornament plate. Those plates draw their motif as a comma rather than a dot, and the convention long predates anybody being able to say why. The reason is exactly this measurement: a dot cannot represent the general position, because it has symmetry of its own, and a pattern illustrated with one is illustrating the wrong group about two thirds of the time. The draughtsmen who adopted the comma had presumably discovered by eye that their p3 plates kept coming out looking like their p31m plates, and fixed it without a proof.

Which is a good description of what a craft tradition is: the right answer, arrived at by iteration, with the reason unavailable.

Who counted, and how

Washburn and Crowe’s Symmetries of Culture (1988) is the standard reference for symmetry analysis of decorated artefacts, and its contribution is a flowchart rather than a count — a decision procedure for assigning a group to a real, damaged, finite pattern in a way another analyst can reproduce.

The archaeological application predates it. Cultural traditions turn out to be strikingly consistent in which frieze and wallpaper groups they use, consistent enough that the symmetry profile of a ceramic assemblage can distinguish periods and production sites. The classification’s advantage for this purpose is that it does not depend on what the motif depicts, so it survives translation between cultures in a way that iconographic description does not.

The Alhambra question, meanwhile, remains open in the specific sense that it depends on choices nobody has agreed. Grünbaum and Shephard’s Tilings and Patterns (1987) treats the claim with scepticism and gives the reasons this essay has given; more recent surveys with explicit criteria have reported counts short of seventeen. The interesting fact is not the number but that a question this concrete can stay unsettled for decades because its terms were never fixed.

Where the ladder goes next

The hazard measured here is the subject of the motif must be a comma, where the same sweep is run for a single group and the mechanism is worked through.

The list being used unevenly is the seventeen, and the ordering that explains which groups are easy to overshoot is the classification’s containments.

The one-dimensional version, where the same analysis is done on borders and where the archaeological application is strongest, is the seven friezes, and the machinery that decides any of it is the orbit and its detector.

What the pictures here cannot show. The bar chart on this page measures dots on a grid, and no figure here shows a piece of ornament. Whether the mechanism it measures accounts for the distribution in any real corpus is a claim about history, argued in the prose from what the surveys agree on, and it is not something these figures establish.

Where this goes next

a dot at (1/12, 1/12)8 symmetries per cella motif with no symmetry4 symmetries per cell57 of 121 dot positions give more symmetry than p4both generated with the 4 operations of p4the dot gains 4 The motif must be a comma A dot is too symmetric to illustrate most wallpaper groups. Its orbit acquires mirrors nobody asked for, and the resulting figure is quietly of a different group from the one in its caption. p1p2pmpgcmpmmpmgpggcmmp4p4mp4gp3p3m1p31mp6p6m17 groups, each generated and verified The seventeen Every way of repeating a pattern across a flat surface, exhibited rather than tabulated. The number is a theorem with a finite proof, and the proof is walkable in an afternoon. p1hopp11gstepp1m1sidlep2spinning hopp2mgspinning sidlep11mjumpp2mmspinning jumpsolid line: a mirror · dashed: a glide · lens: a half-turn centreseven, and no eighth Seven friezes The same classification argument on a strip instead of a plane, where it is short enough to check by hand. Seven ways to repeat a motif along a line, with names like hop, step and sidle.