(Plane or 2D) Space group of heart (or, dart?) tiling


Thanks Karthik for appreciating my Escher talk given so long ago. As you must have realised in creating this red heart pattern (or, in the spirit of Escher, is it a white dart pattern?), that creating such patterns is much more challenging than just analysing their space group. Still I enjoy the latter; so, I spent a happy half-hour determining the space group, the result of which is shown above. I already see a solution, P3m1 at your post. However, I suspect that the answer is probably P31m p31m; and, the teacher in me encourages me to explain how it is arrived at.

The central parallelogram is the unit cell, showing the symmetry elements. The continuous lines are mirror lines which are easy to identify. The mirror line of each heart in the pattern is also the mirror of the whole pattern. A set of three-fold axes are at the intersections of these mirror planes (points where the sharp corners of three hearts meet — shown in the figure with triangles, as is customary). Another set of three-fold axes without mirror planes can also be easily identified (shown in the figure). The tricky symmetry elements are the glide planes (shown in broken lines) which, honestly, I also didn’t see till I consulted the space group tables (I used the one in Kelly, Groves  and Kidd[1]).

There are three space plane groups with three fold axes, namely, P3, P3m1, and P31m (No. 14, 15, and 16) p3, p3m1 and p31m (No. 13, 14, 15) . P3 p3 has no mirrors. In P3m1 p3m1, mirror planes pass through all the three-fold axes. However, in P31m p31m, there are two kinds of three-fold axes — with and without mirrors. Since the heart pattern has both of these kinds, I believe the space plane group is P31m p31m. In fact, that this is P31m p31m is what made me realise that there should also be glide planes in the pattern, which, as I noted above, I figured out after consulting the tables.

Update on 25.02.2009: Corrections shown by strikethroughs are inspired by the comments of Cid (see below) on the original version.


[1] A Kelly, G W Groves and P Kidd, Crystallography and crystal defects, John WIley, 2000; pp. 85-87. A very good reference for basic crystallography.

PS: By the way, I would like to thank Guru who was persuading me for a long time now to join or to contribute to Materialia Indica (lively discussions with whom is always a joy!).

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3 Responses to (Plane or 2D) Space group of heart (or, dart?) tiling

  1. Abi says:

    @Rajesh: Welcome to Materialia Indica!

    Thanks for the explanation about the difference between P3m1 and P31m (and alerting us about the glide plane as well). In my attempt, I figured it was just (P) 3m. Now, I know better! And it feels better!

  2. Cid says:

    …except that it should be p31m, not P31m, as we are dealing with 2D, not 3D, images. Thus, we are not dealing with space groups 14, 15 and 16, but rather with plane groups 13, 14 and 15!

  3. rajesh prasad says:

    Dear Cid

    Thanks for pointing out three important errors in my description:

    1. I was blissfully ignorant about the distinction between p (for plane group) and P (for space group). So thanks for the education.

    2. The numbers 13, 14, 15 were a mix up. Thanks for the correction.

    3. I take your point for using plane group instead of space group for 2D patterns (although one may call plane groups as 2D space groups!).

    Thanks for your comments.


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