action of the group. The entry “4,4,2 turnover” means the surface of a triangular puff pastry with corner angles Æ´ ½¼ Æ µ, Æ and ¼ Æ . The entry “4,4,2 turnover slit along 2,4” means the same surface, slit along the edge joining a Æ vertex to the ¼ Æ vertex. Open edges are silvered (mirror lines); such edges occur exactly for those groups whose Conway notation includes a £ . The last column of the table gives the dimension of the space of inequivalent groups of the given type (equivalent groups are those that can be obtained from one another by proportional scaling or rigid motion). For instance, there is a group of type Æ for every shape parallelogram, and there are two degrees of freedom for the choice of such a shape (say the ratio and angle between sides). Thus, the Æ group of page 309 (top left) is based on a square fundamental domain, while for the Æ group of page 311 (top left) a fundamental parallelogram would have the shape of two juxtaposed equilateral triangles. These two groups are inequivalent, although they are of the same type. Look on page Conway Cryst Quotient space Dim 309 top left; 311 top left Æ p1 Torus 2 309 top right ¢¢ pg Klein bottle 1 309 middle left ££ pm Cylinder 1 309 middle right ¢£ cm Möbius strip 1 309 bottom left ¾¾ ¢ pgg Nonorientable football 1 309 bottom right ¾¾ £ pmg Open pillowcase 1 310 top left ¾¾¾¾ p2 Closed pillowcase 2 310 top right ¾ £ ¾¾ cmm 4,4,2 turnover, slit along 4,4 1 310 middle left £ ¾¾¾¾ pmm Square 1 310 middle right ¾ p4 4,4,2 turnover 0 310 bottom left £ ¾ p4g 4,4,2 turnover, slit along 4,2 0 310 bottom right £ ¾ p4m 4,4,2 triangle 0 311 top right ¿¿¿ p3 3,3,3 turnover 0 311 middle left £ ¿¿¿ p3m1 3,3,3 triangle 0 311 middle right ¿ £ ¿ p31m 6,3,2 turnover, slit along 3,2 0 311 bottom left ¿¾ p6 6,3,2 turnover 0 311 bottom right £ ¿¾ p6m 6,3,2 triangle 0 The gures on pages 309–311 show wallpaper patterns based on each of the 17 types of crystallographic groups (two patterns are shown for the Æ , or translationsonly, type). Thin lines bound unit cells, orfundamental domains. When solid, they represent lines of mirror symmetry, and are fully determined. When dashed, they represent arbitrary boundaries, which can be shifted so as to give different fundamental domains. One can even make these lines into curves, provided the symmetry is respected. Dots at the intersections of thin lines represent centers of rotational symmetry. Some of the relationships between the types are made obvious by the patterns. For instance, on the rst row of page 309, we see that the group on the right, of type ¢¢ , contains the one on the left, of type Æ , with index two. However, there are more relationships than can be indicated in a single set of pictures. For instance, there is a group of type ¢¢ hiding in any group of type ¿ £ ¿. © 2003 by CRC Press LLC
Æ p1 ¢¢ pg ££ pm ¢£ cm ¾¾ ¢ pgg ¾¾ £ pmg © 2003 by CRC Press LLC
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