COPYRIGHT 2002 -- George Baloglou

(AMS) Abstract






Crystallography Now


A web page dedicated to the memory of Kyparissos Stephanos (1857-1917)

4/20/02


WHAT IS THIS PAGE ABOUT, AND WHAT IS A WALLPAPER PATTERN?



In what follows I provide a brief description and rigorous classification of the seventeen planar crystallographic groups (also known as wallpaper patterns). This topic forms the core of my ISOMETRICA book, which is in turn an outgrowth of the SYMMETRIES course (MAT 103) that Margaret Groman introduced at SUNY Oswego in 1992 (as SYMMETRY AND CULTURE) and I taught for ten consecutive semesters prior to my 2000-2001 sabbatical leave. And this page will eventually be part of a much larger and varied web page closely associated with MAT 103. The mathematical background assumed is minimal -- after all, the description and classification offered here are purely geometrical and totally elementary. Missing details may be found in my book or worked out with the help of a more geometrically inclined friend or teacher. (At the other end, ambitious readers may like to extend my approach to the 'real', three-dimensional crystallographic groups!)

A wallpaper pattern may be defined as a subset of the plane invariant under translation in two, therefore -- by the familiar from Physics Parallelogram Law -- infinitely many directions; a simple example is provided by the p1 wallpaper pattern shown below:





It is also assumed that the translations that leave a wallpaper pattern invariant may not be of arbitrarily small length: this is known, in the words of eminent crystallographer Arthur Loeb, as Postulate of Closest Approach. Resources on wallpaper patterns are abundant, both on line and in print; in the latter direction I would particularly recommend Peter Stevens' highly readable, brilliantly written Handbook of Regular Patterns and Doris Schattschneider's comprehensive study on Escher's work in Visions of Symmetry.

In addition to translation, there can only exist three other types of planar isometries (distance-preserving transformations of the plane) that may, in particular, leave a wallpaper pattern invariant: rotation, reflection and, combining translation and reflection parallel to each other, glide reflection. The standard geometrical proof of this fundamental fact is available, for example, in Appendix I of Dorothy Washburn and Donald Crowe's Symmetries of Culture . I offer an alternative geometrical approach in Isometries come in circles, a manuscript devoted primarily to the six isometries of the Euclidean space (where the number of crystallographic groups rises from 17 to 230).



WHICH ROTATIONS, AND WHAT HAPPENS TO THEIR CENTERS?



The possibilities for wallpaper patterns are naturally limited by the ways the four types of isometries may interact with each other and, mostly, by the fundamental Crystallographic Restriction: the only rotations that may leave a wallpaper pattern invariant are rotations by 180, 120, 90, and 60 degrees; as we see further below, rotation by 60 degrees guarantees rotations by 120 and 180 degrees, as in the case of the beehive p6m pattern),





while rotation by 90 degrees guarantees rotation by 180 degrees, as in the case of the checkerboard p4m pattern).





Proofs may be found in H. S. M. Coxeter's Introduction to Geometry (Barlow's 1901 proof, p. 60) or Loeb's Color and Symmetry (chapters 2 and 3).

When it comes to useful interactions among isometries, it is obvious that the composition F*G of every two isometries F, G is again an isometry, and not quite as obvious that the 'image' F[G] of G under F is also an isometry. The latter fact is what I call Conjugacy Principle, a term justified by the identity F[G] = F*G*F', where F' denotes the inverse of F: this identity provides a proof at once, and is in turn established via simple case-by-case geometrical arguments. (An indirect reference to this principle appears in Coxeter's book (second edition), p. 61). Here is an illustration in the context of the following 60-degree 'half-beehive' p6m pattern:





Back to compositions, it is obvious that the composition of two rotations sharing the same center is a rotation by an angle equal to the sum of the two oriented angles. It is far less obvious that this still holds for two rotations of distinct centers (with the composition reduced to a translation in the case of two half turns or two opposite rotations): this is seen by representing each rotation as the composition of two reflections intersecting at the rotation center at half the rotation angle, and by choosing the two 'middle' reflections (out of four involved) to be the same (and therefore canceling each other):




It is easy now to see how two 60-degree rotations may produce a 120-degree rotation, how two 90-degree rotations may produce a 180-degree rotation, how one 120-degree rotation and one 180-degree rotation may produce a 60-degree rotation, etc; the corresponding rotation triangles may in turn explain the structure of the lattice of rotation centers of the beehive and the checkerboard as seen above.

The same method shows, by representing a translation as the composition of two parallel reflections and having the rotation triangle(s) degenerate, how to compose a rotation and a translation (and their inverses). This is demonstrated below for each of the four rotation angles allowed by the Crystallographic Restriction and the smallest translation allowed by the Postulate of Closest Approach; what you see is what I call the 'Big Bang' phase of the lattice of rotation centers (bound to expand indefinitely through rotation*rotation and rotation*translation compositions and/or the Conjugacy Principle):




WHAT HAPPENS IN THE ABSENCE OF ROTATION?



Extending the previously mentioned result on reflections, it is possible to show that the composition of two non-parallel glide reflections is a rotation by twice their angle of intersection ... and even find out where its center is! An immediate consequence of this fact is that rotation-free wallpaper patterns may have (glide) reflection in at most one direction (and along, thanks to the pattern's translation and some 'Physics', infinitely many axes). In the absence of (glide) reflection we obtain with no effort the p1 type already seen above.

The main question now is: what type of parallel (glide) reflections may coexist in a rotation-free wallpaper pattern? Before we answer this question, it is important to learn to view a reflection as a glide reflection: not merely as a glide reflection of zero gliding vector, but also as a genuine glide reflection consisting of the given reflection and a parallel to it translation implied -- see (a) further below -- by the pattern's 'diagonal' translation. As an example, consider the following pm pattern:





Notice that, although there exist 'two kinds' of reflection axes in the pm pattern above, the associated glide reflections share the same minimal (non-zero) vector. The same holds true in the following pg pattern, obtained by 'cutting in half' the pm pattern above and featuring genuine glide reflection only:



So the natural question arises: can a rotation-free pattern have two (by necessity parallel) glide reflections of distinct minimal gliding vectors? (If that is the case then the pattern's two distinct glide reflections will have to alternate -- a mere consequence of the Conjugacy Principle!) A simple argument shows that this is possible only in case precisely one of the glide reflections is in fact a reflection -- a possibility that is not merely theoretical, as the following cm wallpaper pattern (obtained from the pm pattern above through a 'perfect shifting' of every other 'row') demonstrates:



As it turns out, the four types of wallpaper patterns presented so far (p1, pm, pg, cm) are the only possible rotation-free types. Moreover, and leaving the 'trivial' ((glide)-reflection-free) p1 case aside, the cm type is characterized by the existence of a valid translation T the components of which in the parallel and perpendicular to (glide) reflection directions (T2 and T1, respectively) are not valid translations. This characterization follows from our descriptions of the three types above, the fact that every valid translation's 'vertical' (i.e. parallel-to-reflection) component is an integer multiple of T2, and from the following simple observations about the minimal (non-zero) vector T0 associated with the pattern's (glide) reflection MG:

a) If MG is a reflection, then either T0 = T2 or T0 = 2xT2.

b) If MG is a glide reflection, then either T0 = (1/2)xT2 or T0 = T2.

c) If there is reflection only, T0 = 2xT2 is impossible.

d) If there is glide reflection only, T0 = T2 is impossible.

e) In the presence of reflection M and glide reflection G, the combinations T0,G = (1/2)xT2 & T0,M = T2, T0,G = (1/2)xT2 & T0,M = 2xT2, and T0,G = T0,M = T2 are all impossible.


This completes the discussion of the four rotation-free patterns. As we shall see, the characterization of the cm type is destined to play a crucial role in the classification of the 'higher' patterns; but we must first deal with the 'singular' case of the half turn patterns.



HALF TURNING THINGS AROUND



As we showed above, a single rotation center combined with a translation creates a quincux of rotation centers, extendable into an infinite lattice by way of isometry compositions and the Conjugacy Principle. The one exception was the case of the half turn: a single center combined with a translation creates just two new centers at a distance equal to half the translation vector's length from the old center; the three centers are collinear, and all other, likewise formed centers are destined to lie on the same line. As it turns out, another translation (guaranteed to exist by the very definition of a wallpaper pattern) is needed to get a full lattice of half turn centers; this is illustrated in the following example of p2 pattern -- a wallpaper pattern that has nothing but translation (in two, therefore infinitely many directions) and half turn:





These observations on the interplay between half turn centers and translations are valid in all 180-degree wallpaper patterns. But a major difference between the new patterns we are going to 'discover' and the p2 type is the 'allignment' of their half turn centers along 'vertical' and 'horizontal' directions (and translations): this is a consequence of the way the (glide) reflections (and associated translations as studied above) of the 'vertical subpattern' are combined with the pattern's half turn(s).

More specifically, everything follows from the easily proven fact that the composition of a reflection M and a half turn R at a distance d from M is a glide reflection G passing through R, perpendicular to M and of gliding vector of length 2d. Of course the identity R*M = G yields the equally useful identity R*G = M: the composition of a glide reflection G of gliding vector of length D and a half turn center R lying on G's axis is a reflection M perpendicular to G and at a distance D/2 from R; and in case R lies at a distance d from G's axis, a little more effort shows the composition to be a glide reflection of vector of length 2d and of axis lying at a distance D/2 from R:




Let us now assume that a given wallpaper pattern has half turn and some (glide) reflection. Based on what we discussed right above, as well as our earlier observations about the composition of two glide reflections, we conclude that the given pattern must have (glide) reflection in precisely two directions, perpendicular to each other. Starting from the 'vertical' direction 'subpattern', we know that there are precisely three possibilities (pg, pm, cm). The next question is: how are the vertical (glide) reflection axes positioned with respect to the half turn centers? By appealing to the Conjugacy Principle, we see that a half turn center may either lie on an axis or half way between two parallel, adjacent axes. Moreover, since the combined effect of any two half turns is a translation, and half turn centers are translated by the pattern's translations (Conjugacy Principleagain), a closer look at the structure of the three types (and the composition of (glide) reflection and translation as seen in 'Physics' above) shows that there may be precisely two possibilities: either all half turn centers lie on vertical (glide) reflection axes, or all half turn centers lie half way between vertical (glide) reflection axes.

A plausible consequence of the preceeding observations is that there may be at most 3x2 = 6 types of 180-degree wallpaper patterns with (glide) reflection. As we shall see, the actual number of such patterns is 4. While the interactions between (glide) reflection axes and half turn centers that determine this number are best studied in the 'abstract' context of symmetry plans, it is more convenient here to start with a specific pg/pm/cm pattern in the vertical direction and see what the half turn creates out of it in the horizontal direction.

Let us begin with a vertical pg and off-axis half turn centers in order to get a pgg pattern (glide reflection in both directions):





Next comes a vertical pg combined with on-axis centers to produce, a pmg pattern (reflection in one direction, glide reflection in the other):





Starting with a vertical pm and on-axis centers leads, with some overlapping between 'old' and 'new' subpatterns, to a pmm pattern (reflection in both directions):





Fortunately or unfortunately, the combination of a vertical pm and off-axis centers produces nothing new, bringing us back to another representative of the pmg type:





Since there is no essential difference between the 'vertical' and 'horizontal' directions, it is rather safe to guess at this point that the cm type is a loner, capable of coexisting only with itself (if at all). The first step in confirming this is to notice that half turn centers may not lie half way between a glide reflection axis and an adjacent, parallel to it reflection axis, mapping glide reflections to reflections and vice versa (Conjugacy Principle). Starting now with a half turn center either on a reflection axis or on a glide reflection axis and 'multiplying' axes and centers by way of isometry compositions and the Conjugacy Principle, we are bound to arrive at the same 'symmetry plan'; less abstractly, the cm pattern exhibited above is 'doubled' (through heavy motif overlapping, which necessitates a closer look at 'interior lines' rather than shapes) into the following cmm pattern (with reflection and in-between glide reflection in both directions):





This completes our discussion of the 180-degree wallpaper patterns and shows that there exist five such types altogether: p2, pgg, pmg, pmm, and cmm.



FOURFOLD CENTERS RULE



Unlike in the case of a 180-degree center, a single 90-degree center together with the pattern's minimal translation (the existence of which is guaranteed as usual by the Postulate of Closest Approach) create another four fourfold centers (as seen above) that lead in turn to the full lattice of fourfold and twofold (half turn) centers; this is best illustrated in the context of the following (glide) reflection-free p4 wallpaper pattern:





What happens in the presence of (glide) reflection? Unlike in the case of the 180-degree patterns, the lattice of p4 rotation centers is not at all affected ('alligned') by the pattern's (glide) reflection; on the contrary, it is the lattice of fourfold and twofold centers that determines what kind of (glide) reflection axes are allowed to pass through it. Moreover, fundamental facts about composition of two glide reflections (already discussed) and composition of a 90-degree rotation and a (glide) reflection (equal to another (glide) reflection at a 45-degree angle to the original (glide) reflection) show that there can be at most four and at least four, respectively, directions of (glide) reflection; in other words, any 90-degree pattern with some (glide) reflection must have (glide) reflection in precisely four directions, determined in fact by the p4 lattice: 'vertical', 'horizontal' (these are the directions of the pattern's minimal translation) and 'diagonal' (NW-SE and NE-SW). Finally, it follows from the Conjugacy Principle that the structure of (glide) reflection in the vertical and horizontal directions must be identical, and the same holds true between the two diagonal directions of (glide) reflection.

At this point, we return for a moment to the 180-degree methodology: starting with an 'initial' ('vertical') direction, we ask (I) what type of 'subpattern' (among pg, pm, cm) could be possible in that direction, and (II) how could it be positioned in relation to the lattice of rotation centers. We need to ask these questions in two 'initial' directions, that of the pattern's minimal translation ('vertical-horizontal' direction, on the right below), and that of the pattern's minimal 'diagonal' translation ('diagonal' direction, on the left below). Question (I) is answered below:





That is, we can only have a cmm subpattern in the 'diagonal' directions and either a pgg or a pmm subpattern in the 'vertical'-'horizontal' directions. Answering (II), and arguing as in the case of the cmm pattern, we see that the diagonal (glide) reflection axes of the cmm subpattern may only pass through either the fourfold centers or the twofold centers of the lattice of rotation centers (already obtained in the case of the p4 pattern right above); and there exist precisely two possibilities:


(i) reflection axes through the fourfold centers and glide reflection axes through the twofold centers:




(ii) glide reflection axes through the fourfold centers and reflection axes through the twofold centers:




In case (i), the composition of fourfold rotations with the reflections that pass through their centers creates two extra (vertical and horizontal) reflections passing through each of those fourfold centers: we conclude that the vertical-horizontal subpattern must be a pmm. Such a wallpaper pattern, known as p4m, does indeed exist:





In case (ii), the composition of fourfold rotations with the glide reflections that pass through their centers creates two extra (vertical and horizontal) glide reflections not passing through those fourfold centers: we conclude that the vertical-horizontal subpattern must be a pgg. Such a wallpaper pattern, known as p4g, does indeed exist:





This completes the discussion of the 90-degree wallpaper patterns; there are precisely three types of them: p4, p4m, and p4g.



THREEFOLD + THREEFOLD = SIXFOLD



The closely related cases of 120-degree and 60-degree patterns are not that different from the case of 90-degree patterns. Again, the lattice of the threefold centers, shown below in the context of a (glide)-reflection-free p3 pattern (embedded in the familiar p6m pattern) determines the kind of (glide) reflections that could possibly pass 'through' it.





Indeed it is easy to see that there exist precisely six possible directions of (glide) reflection, as easy to determine as the four directions through the p4 lattice, while of course a 120-degree pattern may have (glide) reflection in precisely three of them. Once again the Conjugacy Principle shows that the rotation-free subpatterns in every other direction must be the same, so all we need to do is determine what kinds of subpatterns are possible in each of the two kinds of directions. Choosing two perpendicular representatives of these two directions and appealing again to the characterization of the rotation-free patterns, we reach these two goals at once:





It is not that difficult now to 'guess' what each of the two cm's should look like: the Conjugacy Principle shows that reflection axes may only pass through 'rows' of rotation centers, while glide reflection axes may only pass half way between such rows; distances between adjacent parallel reflection axes must be half the length of the minimal translation vector in the perpendicular to them direction, while the length of gliding vectors must be half the length of the minimal translation vector in their direction. We conclude that there exist two possible types of 120-degree patterns with (glide) reflection, one for each (triplet of) directions:





Each of the two 'predicted' types does indeed exist, and may be derived, just as the p3 type above, 'within' the p6m 'half-beehive' pattern:






The only thing that prevents the p3m1 and p31m structures from co-existing inside a single wallpaper pattern is the smallest rotation angle's size: a 120-degree angle permits only three directions of (glide) reflection. But if the smallest rotation angle is 60-degree, which is the last case allowed by the Crystallographic Restriction, then the two types (and their 'symmetry plans' as well) merge rather naturally into the p6m type we have already seen (without ever daring to exhibit its symmetry plan in full). Moreover, a closer look at our work shows that the p6m is the only 60-degree pattern with (glide) reflection: the lattice of rotation centers, as shown in the context of the (glide)-reflection-free p6 pattern below, is essentially the same as the p3 lattice, despite the presence of twofold centers (that hardly play any role in the classification process) and the 'upgrading' of some threefold centers to sixfold centers:




The same 'vector analysis' applied in the 120-degree case remains valid, and so do its consequences; the only new element is that the reflection axes in the 'scarce' p31m direction(s) must pass through the sixfold centers (Conjugacy Principle):





So the only types in the families of 120-degree and 60-degree patterns are the p3 , p6, p3m1, p31m, and p6m: this completes the classification of the seventeen wallpaper patterns.



TIDBITS OF HISTORY



The traditional classification of the 17 types of wallpaper patterns employs the tools of Group Theory. I am not sure where the presentation of that algebraic approach is done in the most accessible manner, but I would suggest to try chapter 2 in Thomas Wieting's most remarkable The Mathematical Theory of Chromatic Plane Ornaments. For an alternative geometrical approach there is always Loeb's book already mentioned, while for a deeper look into and beyond the subject I would recommend B. Grunbaum & G. C. Shephard's classic Tilings and Patterns and Marjorie Senechal's more recent Quasicrystals and Geometry . Finally, you may check my old Wesleyan officemate David Feldman's Pathways for a friendly exposition of John Conway's topologicalapproach (The orbifold notation for surface groups, London Mathematical Society Lecture Series 165 (1992), pp. 438-447).

When did humanity know that there exist precisely 17 types of wallpaper patterns? It seems to be an open question whether or not the Alhambra Moors had produced all 17 types in their Palace. I am not aware of other major developments between then and the late 19th century, but two independent (re)discoveries (with proof) of the 17 types are recorded in Coxeter (p. 50): E. S. Fedorov's in Zapiski Imperatorskogo S. Peterburgskogo Mineralogicheskogo Obshchestva (2), 28 (1891), pp. 345-390 and G. Polya & P. Niggli's in Zeitschrift fur Kristallographie und Mineralogie 60 (1924), pp. 278-298.

Ten years before Fedorov's derivation, that most crucial (even if in its degenerate form) to my 'Big Bang' approach rotation triangle had surfaced as a theorem proven by Kyparissos Stephanos, a mathematician from the Cycladic island of Kea to whose memory this page is dedicated:


[This reference has been supplied by yet another Greek geometer,

my father Christos (who passed away half a year after this page's completion).]