Last Update 05/ 5/ 2007
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At least one example of two-dimensional pattern for each of the 17 plane symmetry groups can be observed after a mouse click on a yellow button with the respective Hermann-Mauguin symbol on the program above. Options begin with symbol p1 (plane group number1) to c2mm (plane group number 9) on the first line and Hermann-Mauguin symbol p4 (plane group 10) to p6mm (plane group 17) on the second line. A click on the X button replaces full disks with empty rings or vice versa.The lattice symbol p defines a primitive type of unit cell and c the centered type. The suggested unit cell is presented in a red parallelogram. There are seven crystal systems in three dimensional crystals but only four in two dimensional crystals: an oblique parallelogram for groups p1 and p2, a rectangle for groups from pm to c2mm as given in the upper line of the button sequence, a square for groups from p4 to p4gm and a parallelogram with 120o angle for groups p3 to p6mm as in the next line of buttons. Due to the lack of not trivial symmetry element in group p1 its asymmetric unit is the complete unit cell. In group p2 the asymmetric unit is only 1/2 of the unit cell, the complete unit cell can be obtained by operating the symmetry elements of this group, the binary axis, on the asymmetric unit. Find complete information on the International Tables.
The graphical error on the pattern map may be as large as 2 coordinate units or about 2 degrees in the expected angles.
Application in paintings: if the amount of ink used for each unit cell is known, the total required amount of ink for the complete wall paper with N unit cells is calculated by a simple multiplication.
Exercise
Define the plane symmetry group and suggest the coordinates for the corners of the most suitable unit cell for the pattern obtained after a mouse click under the green rectangle. Dismiss any less interesting pattern map generated and get another one after a new click under the green rectangle.
Reference
International Tables for Crystallography (2005). Vol. A, edited by Th. Hahn, Dordrecht: Springer.
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