Folk art in crystallography education

We have found it most fascinating to find analogies of space-group symmetries from outside science that can be used for pedagogical purposes. In the 1970s and 1980s, we were so immersed in our studies of symmetry that we saw its appearance everywhere. This is how we noticed the wealth of symmetries in folk art, including Hungarian folk art. We became acquainted with a most knowledgeable scholar and prolific author of Hungarian folk art, Ms Györgyi Lengyel, and this led to a systematic search for the appearance of the one-dimensional space-group symmetries and then the two-dimensional space-group symmetries in Hungarian folk art. Two brief communications resulted from these interactions in the Journal of Chemical Education in 1984 and 1985 [1, 2]. The editors of the journal valued our efforts, and as a token of appreciation, they prepared one of our patterns by hand and put it on the cover of the December 1984 issue of the journal. In 1986, we embedded a polyethylene chain with glide-reflection symmetry of a one-dimensional space group in the two-dimensional space group pattern with Pmg2 symmetry on the cover of the first edition of our monograph of Symmetry Through the Eyes of a Chemist [3].

We combined the folk-art patterns with patterns of an asymmetric black triangle, which is a widespread general basic motif for representing space group symmetries. We first met with this on the inside of the covers of Martin J. Buerger’s Elementary Crystallography [4]. We presented a more complete discussion of these illustrations in our monograph on Symmetry Through the Eyes of a Chemist [3]. When we decided to return to this topic, we wanted to involve Ms Lengyel and learned that she had passed away. Despite our efforts, we could not find anything more about her except that her popular folk art books are still in some libraries, but none of her books are in print anymore. Many of her numerous books are listed by second-hand bookstores and they accept pre-orders should the books become available.

Left: the cover of the December 1984 issue of the Journal of Chemical Education with the image of a hand-made reproduction of the pattern with a glide-reflection plane. Reproduced with permission (2025) from Thomas Holme, Editor, Journal of Chemical Education. Right: cover of the first edition of our Symmetry through the Eyes of a Chemist, 1986. It shows the polyethylene chain with glide-reflection symmetry embedded in the two-dimensional space group pattern with Pmg2 symmetry.

The seven one-dimensional space group symmetries with illustrations

The seven symmetry classes of one-sided bands.

Illustrations of the seven symmetry classes of one-sided bands by Hungarian needlework.

1. (a). A translation axis. The period of translation is the distance between two identical points of consecutive black triangles. The needlework is an edge decoration of a table cover from Kalocsa, southern Hungary. 

2. (a)·ã. A glide-reflection plane. The black triangle comes into coincidence with itself after translation through half of the translation period and reflection in a plane perpendicular to the plane of the drawing. The needlework is a pillow decoration from Tolna County, southwest Hungary. The editors of the Journal of Chemical Education liked this pattern so much that they produced the actual needlework, and its image appeared on the cover of the issue above.

3. (a):2. Translation on rotation through 180° around an axis perpendicular to the plane of the one-sided band. The needlework is a decoration patched onto a long embroidered felt coat of shepherds from Bihar County, eastern Hungary.

4. (a):m. Translation by transverse symmetry planes. The needlework is embroidered stripes edge decoration of a bed sheet from the 18th century. There are some deviations from the described symmetry in the lower stripes of the needlework.

5. (a)·m. A translation axis combined with a longitudinal symmetry plane. The needlework is a decoration on a shirt from Karad, southwest Hungary.

6. (a)·ã:m. Combination of a glide-reflection plane with transverse symmetry planes. A translation axis and twofold rotation axes are generated. The needlework is a pillow decoration pattern from Torockó [Rimetea], Transylvania, Romania. 

7. (a)·m:m. Combination of a translation axis with transverse and longitudinal symmetry planes. Twofold rotation axes are generated. The needlework is a grape leaf pattern from the region east of the river Tisza.

Here, so-called non-coordinate notation was given for all seven one-dimensional symmetry classes following Shubnikov and Koptsik’s symmetry monograph [5]. Here we list their so-called coordinate (international) equivalents:

(a)           P1   
(a)·ã        P1a1
(a):2        P112
(a):m       Pm11
(a)·m       P1m1
(a)·ã:m    Pma2
(a)·m:m   Pmm2

The seventeen two-dimensional space group symmetries with illustrations

P1 and P4. Indigo-dyed decorations of textiles for clothing. Selye, Baranya County, 1899.

P2. Indigo-dyed decoration with palmette motif for curtains; a popular pattern.

Pm. Decoration with tulip motif for tablecloths. Cross-stitched needlework from the turn of the last century.

Pmm2. Bed-sheet border decoration with pomegranate motif from northwest Hungary, 19th century.

P4mm. Pillowslip decoration with stars. Cross-stitched needlework from Transylvania, 19th century.

Cm. Pillowslip decoration with peacock tail motif. Cross-stitched needlework, much used throughout Hungary around the turn of the last century.

Cmm2. Bed-sheet border decoration with cockscomb motif. Cross-stitched needlework from Somogy County, 19th century.

Pg. Indigo-dyed decoration from Pápa, Veszprém County, 1856.

Pgg2. Children’s bag decoration from Transylvania, turn of the last century.

Pmg2. Pillowslip decoration with scrolling stem motif, used throughout Hungary around the turn of the last century.

P4gm. Blouse-arm embroidery from Bács-Kiskun County, 19th century.

P3, P6, P3m1, P3₁m and P6mm. Decorations with characteristic bird motifs from peasant vests; the basic motif — the bird — is genuine but the patterns were created by Györgyi Lengyel.

References

[1] Hargittai, I. & Lengyel, G. (1984). The seven one-dimensional space-group symmetries illustrated by Hungarian folk needlework. J. Chem. Educ. 61, 1033.

[2] Hargittai, I. & Lengyel, G. (1985). The seventeen two-dimensional space-group symmetries in Hungarian needlework. J. Chem. Educ. 62, 35.

[3] Hargittai, M. & Hargittai, I. (2009). Symmetry through the Eyes of a Chemist. 3rd ed. Springer Science+Business. [Hargittai, I. & Hargittai, M. (1986). Symmetry through the Eyes of a Chemist. 1st ed. Weinheim: VCH.]

[4] Buerger, M. J. (1967). Elementary Crystallography: an Introduction to the Fundamental Geometrical Features of Crystals. New York: Wiley.

[5] Shubnikov, A. V. & Koptsik, V. A. (1974). Symmetry in Science and Art. New York, London: Plenum Press.