Crystallography & Symmetry
patterns derived from their groups
The classification

The Alhambra question

Textbooks say the Alhambra contains all seventeen wallpaper groups. Careful analysts of the same building have counted eleven, thirteen, fourteen and seventeen — and the disagreement is not about the mathematics but about what counts as an instance.

Assumes The groups ornament actually uses and Two colours, and a symmetry that swaps them.

16 min read 7 figures Exactly this manyThe motif must be asymmetricSymmetry is decidable

The claim that the Alhambra contains all seventeen wallpaper groups is among the most repeated facts in popular mathematics. It appears in textbooks, in lecture courses, in museum captions, and in a great many articles about Escher, who visited in 1922 and again in 1936 and left with the sketchbooks that changed his work.

It is also not established. Analysts who have gone through the building carefully have published counts of eleven, thirteen, fourteen and seventeen, and they were not looking at different buildings. The mathematics is not in doubt: there are seventeen and no more, and the proof is finite. What is in doubt is what it takes for a wall to contain one.

How often a dot gives the wrong group. Every 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.
Fig. 1 The hazard behind part of the disagreement, measured for every group at once. For each of the seventeen, a dot is placed at every position on a grid inside the cell and the resulting pattern’s symmetry is detected. The bar is how often the pattern comes out with more symmetry than the group used to make it — so a design aiming at one group frequently lands on another.

This essay is about a question a classification cannot answer, and about being precise regarding which part of it a classification can.

Four disagreements, and none of them mathematical

The published counts differ for reasons that can be separated and named.

How much pattern counts as a pattern. A group is a property of an infinite repeating design. A tile panel is finite, frequently interrupted by an arch or a doorway, and often a fragment of a larger scheme. Deciding that a panel exhibits p31m rather than p3 means deciding that its mirrors would have continued past the edge, and no amount of looking at the panel settles that.

Whether colour counts. A design whose shapes have p6m symmetry and whose colouring breaks half of it is p6m if colour is ignored and something smaller if it is not. Both readings are defensible and they give different totals. This is the ambiguity with a definite resolution — the two-colour classification treats the colour swap as an operation, and in that classification the question has an answer — but it is a different classification with a different list, so an analyst has to say which one they are counting in.

Whether a near-symmetry counts. Hand-cut tile is not exact. Every analysis therefore applies a tolerance, and the number of groups found rises as the tolerance is loosened. That dependence is not a detail: the count of symmetry operations in measured coordinates is a staircase in the tolerance, and an analyst who does not state theirs has not stated their result.

Whether the design or the execution is being classified. A pattern laid out to a p4g scheme and executed with a symmetric floral motif in each tile produces a wall with more symmetry than the scheme has. Which of the two is “the pattern in the Alhambra” is a question about intention, and intention is not recoverable from tiles.

What can be measured here, and what it explains

This site cannot count patterns in a building. It can measure the reachability question that lies underneath one of those four disagreements, and the measurement is the figure at the top of this page.

For each of the seventeen, a single dot is placed at every position on a grid inside the cell, the orbit under the group’s operations is computed, and the resulting point set is handed to a detector that has never heard of the group. When the detected symmetry is larger than the group used to build it, the design has landed somewhere else.

The spread is wide and its ends are the informative part. Four groups fail at every position tried — p1, pm, pg and cm cannot be illustrated with a single symmetric element at all, because the midpoint between a dot and its own lattice translate is always a centre of inversion. One barely fails at all: p6m already has everything its lattice permits, so there is nothing left to gain.

That second end deserves the closer look, because it inverts the expectation a survey brings to a building. The naive reading is that the most symmetric group is the one a careless construction is most likely to arrive at, and the measurement says the reverse: p6m is the hardest of the seventeen to reach by accident, because reaching it by accident would require gaining an operation the hexagonal lattice does not have. The only route left is for the orbit to collapse onto a finer lattice, which needs the mark to land exactly on a rotation centre, and two of the hundred and twenty-one positions do.

Why p6m cannot be drawn with dots. The 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.
Fig. 2 The safe end of the range. Only two of the 121 dot positions spoil p6m, and both are special positions where the orbit collapses rather than places where an operation is gained — the group already carries everything the hexagonal lattice permits. Set beside p3 below, which is spoiled at eighty-five positions on the same lattice, this is the measurement’s central point: what makes a group easy to hit by accident is how much room sits above it, not how many operations it has.

The other end of the same lattice is where the hazard is worst, and the contrast is the whole of the argument. p3 sits on the identical hexagonal lattice as p6m and carries three operations against its twelve, so almost everything the lattice permits is still available to be acquired — and a dot acquires most of it, at eighty-five positions in a hundred and twenty-one. Two groups, one lattice, opposite behaviour, and the difference is entirely how much room each leaves above itself.

Why p3 cannot be drawn with dots. The 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.
Fig. 3 The mechanism for one group. A dot placed at most positions in a p3 cell produces a pattern with mirrors and glides that p3 never had — so the picture is of p31m — and nothing about the drawing announces it.

The consequence for a corpus is a bias rather than an error. A tradition whose vocabulary is symmetric shapes — rosettes, stars, six-petalled flowers — produces high-symmetry groups whether or not anyone was aiming at them, because the motif’s own symmetry adds to the layout’s. The groups that dominate most surveys are exactly the ones with the most operations per cell; the groups that are consistently rare are the ones that require the absence of symmetry in the motif, which is a deliberate act rather than a default.

What the numbers in the sweep actually are

The counts behind the bar chart are worth setting out, because their ordering is not the one anybody would guess and it is the ordering that does the explanatory work.

Out of a hundred and twenty-one positions tried per group, four fail at all hundred and twenty-one — p1, pm, pg and cm. Then the failures fall away through p3, p3m1, p4 and pgg in the middle of the range, and the two lowest are p2 at eight failures and p6m at two.

Order does not predict the hazard. p2 has two operations per cell and fails eight times; cm has two and fails at every position. p4m has eight and fails twenty times; p4g has eight and fails forty-one. What matters is not how large the group is but how much room sits directly above it in the containment ordering — and that explanation is itself only half right, which is worth knowing before it is used to explain anything about a building. It gets the extremes exactly right and the middle of the range wrong, because a dot has a second route to extra symmetry that the ordering cannot see: its orbit can collapse onto a finer lattice, and a pattern on a finer lattice is not above the intended one in the ordering at all. It is not even close to it — a spoiled p4g reaches sixty-four operations where the square lattice carries nothing above eight.

The containment ordering explains half the hazard and not the other half. Each group with the groups that contain it on its own lattice, beside how many of the 121 dot positions spoil it and the largest number of operations a spoiled pattern turned out to have. Two directional claims hold and are checked here: every group a dot spoils at every position has something above it on its own lattice, and every one of the 8 groups with nothing above it is spoiled at fewer than half the positions. What does not hold is the monotone reading. pgg has nothing above it and is spoiled 57 times; p31m has p6m above it and is spoiled 29. The last two columns say why: the order a spoiled pattern reaches runs far past anything the group's own lattice carries — sixty-four for p4g, whose lattice offers nothing larger than eight — because the orbit has collapsed onto a finer lattice and left the ordering behind. It is always an exact multiple of the group's own order, since the orbit is invariant under the group by construction, and that multiple is the honest measure of how far the accident went.
Fig. 4 The containments set beside the failures, which is the tempting reading put to the test. Two directional halves of it hold: every group a dot spoils at every position has a larger group above it on its own lattice, and every group with nothing above it is spoiled at fewer than half the positions. The monotone reading does not — pgg has nothing above it and is spoiled fifty-seven times where p31m, which has p6m above it, is spoiled twenty-nine. The last two columns say why: the order a spoiled pattern reaches runs far past anything its own lattice carries, because the orbit has collapsed onto a finer lattice and left the ordering behind.

The failures are not small. When a dot betrays p3, the detected group frequently has twelve times the operations it was built with. What is on the wall is not a slightly-wrong p3 but a different pattern, and an analyst classifying it correctly as p31m is not making an error — the maker’s intention has simply left no trace in the tiles.

Why the colour question is the sharpest

Of the four disagreements, the colour one is the only one with a mathematical resolution available, and it is worth seeing why the others do not have one.

p4m, two-coloured (1 of 7). One of the 7 two-colourings of p4m. 16 of the 32 operations in the quotient preserve the colours and 16 exchange them, so the colour-preserving half is a subgroup of index two. The 96 points drawn split 48 to 48 — exactly even, because a colour-reversing operation matches each point of one colour with a point of the other. The colouring repeats over two cells rather than one wherever a translation is colour-reversing.
Fig. 5 A two-coloured p4m. Ignoring the colours, this pattern has eight operations per cell. Counting them, half of those operations exchange black and white, and the colour-preserving group is one of p4m’s index-two subgroups. Neither reading is wrong and they give different names.

Colour is in the object. Whether a tile is black or white is a fact about the wall, so a classification that includes colour is asking a question with a definite answer, and the seventy-four two-colourings are the vocabulary for stating it. An analyst who declares “colour counted” or “colour ignored” has made their result reproducible.

The other three cannot be fixed that way. Whether a panel’s mirrors continue past a doorway is a counterfactual. Whether a wobble in a hand-cut tile is a broken symmetry or an executed one is a question about a threshold. Whether the layout or the finished wall is the pattern is a question about what somebody meant. None of these is decided by looking harder, and all three are decided differently by careful people.

What the honest form of the claim is

There is a version of the Alhambra statement that survives all of this, and it is worth stating because it is more interesting than the version that does not.

The Alhambra’s decoration is extraordinarily varied in its symmetry, drawing on a wide range of the possible plane groups in a single building, several centuries before any of them were enumerated. That is not in dispute, it does not depend on a convention, and it is remarkable.

The count of exactly seventeen is a claim about a methodology as much as about a building. Anybody quoting it should be able to say whose count they mean and what that analyst counted. The best-known systematic treatments differ, and the differences are traceable to the four questions above.

The standard reference for applying the classification to archaeological material — Washburn and Crowe’s 1988 book — spends most of its length on exactly this problem: not the mathematics, which is settled, but a procedure for making the assignment reproducible between two analysts looking at the same object. That is the right response to this whole family of difficulties, and it is a procedural response rather than a mathematical one.

A dot cannot illustrate 4 of the seventeen; three points illustrate all of them. For every wallpaper group: how many of the 121 dot positions across the cell produce a pattern with exactly that group's symmetries, and then whether the standard three-point motif truncated to two points, and at full length to three, does so. The dot column is a question about the best case rather than about one dot — p1, pm, pg, cm have no working position anywhere in the cell, so those four groups cannot be drawn with a single mark however carefully it is placed. Two points verify everything but p1, and that exception is forced rather than unlucky: inversion through the midpoint of any pair of points carries the pair onto itself and commutes with the translations, so the orbit of two points always has a half turn and can never be a pattern with no symmetry at all. Three points in no particular arrangement have no such forced coincidence, and every row of the last column was checked by generating the orbit and handing the bare point set to a detector that was not told which group made it.
Fig. 6 How much motif each group needs, which is the question a craftsman faces when choosing an element rather than a question about mathematics. Four groups have no working dot position anywhere in the cell, so nothing round, nothing star-shaped and nothing with a mirror of its own will illustrate them however carefully it is placed; two points fix every group but p1, and three fix all seventeen. Any survey of a real corpus is measuring a distribution shaped by that asymmetry as well as by whatever the makers intended.

Escher, and what he took away

The building’s best-known consequence is worth a paragraph, because it is often told in a way that gets the mathematics backwards.

Escher visited in 1922 and again in 1936, copying panels into sketchbooks both times. What he took from them was not a classification — he had none, and the seventeen were not part of his vocabulary — but a method: a way of dividing the plane into interlocking figures with no gaps and no overlaps, which is what the Moorish designs do with abstract shapes and what he went on to do with recognisable ones.

Where the orbit of p4g collapses: 48 special positions of 576. Every point of a 24 by 24 grid across one cell of p4g, with its orbit computed. A small pale mark is a general position, whose orbit has all 8 points the group can give it; a large mark is a position some operation leaves alone, whose orbit is shorter by exactly the size of its stabiliser. 48 of the 576 positions are special and the shortest orbit found has 2 points, so the largest stabiliser has 4. The marks fall on lines and at points rather than anywhere, because a stabiliser is the set of operations fixing a place and the places an operation fixes are its mirror line or its rotation centre. Orbit length times stabiliser size was required to equal the order at every position drawn, which is what makes this a measurement of the collapse rather than a picture of it.
Fig. 7 The construction underneath both practices, drawn as the redundancy it is. Every position in a p4g cell has its orbit computed: a small pale mark is a general position, whose eight images are seven copies of information already present, and a large mark is a position an operation leaves alone, whose orbit is shorter by the size of its stabiliser. Take one point from each orbit and what remains is an eighth of the cell — a fundamental domain, which is the piece Escher drew as a shape rather than as a region. His method and the classification are two descriptions of the same eighth, arrived at seventy years and one discipline apart.

The mathematics reached him later and from a different direction: through his brother, a geologist, who sent him crystallographic papers, and through a long correspondence with Coxeter. By then the work was already made. That order of events is the interesting part of the story and it is usually reversed into a claim that he was applying a classification he did not have.

The same problem, without the tiles

The pattern of difficulty here is not peculiar to decorative art, and the crystallographic version of it is instructive because the stakes are different.

A crystal structure very nearly of a higher-symmetry group presents the identical dilemma: the higher group fits acceptably, produces slightly wrong coordinates, and nothing in the residual announces the error. The discipline’s answer is not a better tolerance but a different kind of test — impose the candidate symmetry, refine again with fewer parameters, and ask whether the fit worsened by more than the lost freedom explains.

Ornament has no equivalent, because there is no model being refined and no residual to test. What it has instead is a convention, stated by the analyst, and a corpus large enough that a systematic bias in one direction shows up as a distribution rather than as a single wrong answer. That is why the reachability measurement is worth having: it predicts the direction of the bias without needing to settle any individual panel.

What a reproducible procedure looks like

If the disagreements cannot be settled by mathematics, the remaining question is what a defensible analysis looks like, and the archaeological literature has an answer worth describing.

The standard treatment for applying the classification to real material — Washburn and Crowe’s 1988 book — is essentially a flowchart. The analyst is asked a short series of yes-or-no questions in a fixed order: is there a rotation, and of what order; is there a mirror; is a mirror parallel to a translation direction; is there a glide that is not the product of a mirror and a translation. Each answer narrows the list, and the sequence terminates at one of the seventeen.

Three things make that procedure valuable, and none of them is mathematical.

It is a fixed order. Two analysts working the same panel ask the same questions in the same sequence, so their answers can be compared question by question rather than only at the end. A disagreement is then localised to one decision rather than being a disagreement about a name.

It forces the conventions to be stated. The flowchart cannot be entered without deciding whether colour counts and what the boundaries of the pattern are, because the first questions depend on both. An analysis that never wrote down its conventions has not started.

It produces a record of the doubtful steps. Where a panel makes an answer genuinely uncertain — a mirror that may or may not have continued past a break — the step is marked rather than resolved, and the result is reported as a pair of possibilities.

That is what a discipline does when its objects do not admit exact answers, and it is worth contrasting with the approach on the rest of this site. Here a claim is checked by a computation that can fail. There a claim is made reproducible by fixing the procedure that produced it. Both are ways of making an assertion accountable, and the second is the only one available when the object is a wall.

What is actually agreed

It is worth ending the survey of disagreements by writing down what nobody disputes, because the list is longer than the argument suggests.

The building contains many distinct plane groups. No analysis has found fewer than eleven, which is a large fraction of seventeen and far more than any accidental variety would produce.

High-symmetry groups dominate. Every count agrees that the groups with many operations per cell are the common ones, and that p1 and the glide groups are rare or absent. That agreement holds across corpora and across analysts, and it is the observation the reachability measurement explains.

The craftsmen had no classification. The seventeen were enumerated in 1891. Nothing in the fourteenth century could have been aiming at completeness, so any count near seventeen is a fact about what the design vocabulary reaches rather than about an intention to exhaust a list.

The disputed part is a single number, and the undisputed part is everything that makes the building interesting.

Where the exactness stops

Nothing here is a count of anything in Granada. The figures on this page measure dots on a grid. They do not show a single piece of ornament, and the conclusions they support are about which groups are easy to reach by accident, not about which groups a building contains.

The sweep measures dots, not motifs. A real decorative element is a shape with its own symmetry group, and that group is what interacts with the layout. A dot is the extreme case — the most symmetric motif there is — so the measured hazard is an upper bound rather than an estimate.

The historical claims are attributed, not derived. Which groups are common and which are rare is what the surveys agree on even where their counts differ, and that agreement is the only historical statement this essay makes.

Where the ladder goes next

The measurement this essay leans on, worked through for its own sake, is the groups ornament actually uses.

The classification that gives the colour question a definite answer is two colours, and a symmetry that swaps them.

The reason the tolerance question has no clean answer, and what the count of symmetries actually does as a threshold moves, is near-symmetry and the tolerance that is not here.

What the pictures here cannot show. Every figure on this page is generated from a group, and the subject of the essay is a building none of them depicts. That is deliberate — a photograph of a tile panel would invite exactly the judgement the essay argues cannot be made from looking — but it means the connection between the measurement and the historical claim is made in the prose and rests on an argument about craft rather than on anything the figures establish.

What would settle it

The disagreements are about conventions, so the question is not which count is right but what a count would have to publish before it could be compared with another. That is a short list and none of the published Alhambra counts satisfies it.

The panels, individually. A count of seventeen is a claim about seventeen specific pieces of wall, and reconciling two counts means comparing them panel by panel. A total with no list attached cannot be checked, cannot be corrected, and cannot be disagreed with productively — the two analysts simply report different numbers and there is nowhere to look.

The assignment for each. Which group, and on what evidence: which operations were seen, at what extent, with colour counted or not.

And the images. A symmetry assignment is a reading of an object, so the object has to be available to a reader who wants to check the reading.

None of that is difficult and none of it is expensive, and it is the ordinary standard for a measurement in any other subject. A crystallographic structure determination deposits its coordinates; a survey of ornament that deposits nothing but a total is asking to be believed.

That is the practical form of this essay’s conclusion. The mathematics is not in dispute and never was; what is in dispute is a set of readings that were never published in a form allowing them to be compared. The count of seventeen may well be right. What is certain is that nobody can currently check it, and that the checking would be a smaller job than any of the counts already were.

Where this goes next

How often a dot gives the wrong group. Every 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. 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. Two-colourings of the seventeen. How many ways each of these 17 plane groups can be two-coloured so that every symmetry either preserves the colours or exchanges them. 74 in all, each one a subgroup of index two enumerated by trying every assignment of colours to a generating set and keeping the assignments that turn out to be consistent. p3 admits none: a homomorphism onto a group of order two has nothing to send a three-fold rotation to but the identity, and once the rotation and its conjugates are killed nothing is left to reverse the colours. pmm admits the most, with 15. Every count is one less than a power of two because the homomorphisms of a group onto the two-element group are the non-zero elements of a vector space over that field. Two colours, and a symmetry that swaps them A chessboard and a grid of identical squares have the same group, which is plainly not what anybody sees. Admitting the colour swap as an operation gives a finer classification — and one of the seventeen turns out to admit no two-colouring at all. p4m, displaced and then measured. Every atom of a p4m pattern moved by up to 1.2 per cent of a cell edge, and the resulting positions examined at 26 tolerances. At zero tolerance only the identity survives, so the structure has no exact symmetry whatever. Between 0.021 and 0.094 the count sits at 8, which is the group that was displaced. Above that it climbs to 15, accepting operations no version of this pattern has. The climb is not even steady: at 4 of the 25 steps the count falls as the tolerance is loosened, because operations accepted separately at one threshold merge into one at the next. The correct answer is a step on a staircase and nothing in the coordinates says which step. Near-symmetry, and the tolerance that is not here Every claim on this site is decided by integer arithmetic, so no threshold is ever chosen. Measured coordinates do not arrive that way, and the moment a tolerance is introduced the answer stops being a fact about the structure and becomes a fact about the threshold.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

Accidental symmetryThe AlhambraColour symmetryGroup frequencyOrnamentPattern analysisTolerance