Wallpaper Patterns 2

Math 311. Wallpaper Patterns 2 Name: ¶ 1. Crystallographers have developed a notation for wallpaper groups. It is now well established and bellow we give a flow chart that allows to determine the name of a given wall paper group. The flowchart itself does not reveal all the features of a wallpaper pattern, and it is thus convenient to identify them here. A wallpaper group always contains translations in two independent directions, and then perhaps other rigid motions, like rotations and glide reflections. This group of translations is called the lattice group of the wallpaper pattern: a lattice is obtained by taking a point of the pattern and all other points obtained from it by translations of the pattern. A parallelogram whose vertices are points of this lattice and contains no other points of the lattice inside it or on its sides is called a primitive cell. Notice that neither the lattice not the primitive cell are uniquely determined by the pattern. Sometimes the whole pattern can be easily obtained from this primitive cell. But sometimes a primitive cell is not easily seen, so the analysis of the pattern is better done by studying the rigid motions that preserve it rather than by studying a primitive cell. Usually, a primitive cell is chosen with centers of rotation of the highest orders as its vertices. There are two cases (when the primitive cell is a rhombus) in which a centered cell is chosen; this has size double that of the primitive cell. There are five kinds of primitive cells that arise: parallelogram, rectangular, square, rhombic, and hexagonal. The reason is that a pattern can only have rotational symmetry of orders 2, 3, 4 and 6. If there are no rotational symmetries but there are mirror lines and glide reflection lines, then the lattice will have parallel rows in perpendicular directions. Crystallographers choose primitive cells as pictured below; in the case of the rhombic cell, a centered cell is chosen which is made of two unit cells. Parallelogram Square Rectangular Rhombic Hexagonal A primitive cell may overlap itself only along parts of its border when translated by the translation of the symmetry group of the pattern. However, if other symmetries are present (like rotations, and glide reflections), then some of those will take the primitive cell into positions that have regions in common with the original position. A minimal region withe the property that symmetries of the pattern take it to others that overlap along portions of its boundary, or not at all, is called a fundamental domain. 1 Math 311. Wallpaper Patterns 2 Name: ¶ 2. Pictured below are primitive cells of the 17 wallpaper groups according to standard crystallographic notation. Each cell contains markers identifying the several symmetries of the pattern. A double line (==) indicates a mirror line. A dashed segment (- -) indicates an axis of a glide reflection. A point marked with a circle indicates a center of 180◦ rotation, a  indicates a center of 90◦ rotation, a N indicates a center of 120◦ degree rotation, a hexagon indicates a center of 60◦ degree rotation. 2 2 2 2 2 2 2 2 2 p111 2 2 p2mg 2 2 2 2 2 p4gm 2 2 p211 2 2 2 2 p2gg p311 p1m1 c2mm p3m1 2 2 p1g1 2 p31m 2 p411 2 c1m1 2 2 2 p611 2 2 2 2 2 2 p2mm 2 2 2 p4mm p6mm 2 Math 311. Wallpaper Patterns 2 Name: ¶ 3. Wallpaper patterns: Crystallographic Notation. Crystallographers have standard notation for the 17 different wallpaper patterns. The full notation consists of 4 symbols: c p 1 2 3 4 6 m g 1 m g 1 (a) The first symbol is either a c (for centered cell), or a p (for primitive cell) otherwise (b) The second symbol indicates rotational symmetry. It is either of order 1, 2, 3, 4, or 6, corresponding to 360, 180, 120, 90, or 60 degrees. The symbol is the largest applicable number: for example, if symmetries of 360, 120 and 60 degrees are present, then the second symbol is 6. (c) The third symbol is either m, g, or 1, corresponding to the presence of mirror, glide, or no reflection symmetry. (d) The fourth symbol is either m, g, or 1, and describes the symmetry relative to an axis at an angle to the symmetry axis of the third symbol. That angle would be determined by the second symbol: if that second symbol is 1 or 2, then look for reflection or glide line at angle 180◦ ; if the second symbol is 4, then look for lines at angle 45◦ , and if the second symbol is 3 or 6, then look for lines at angle 60◦ . This notation is summarized in the following flowchart: Yes 360 Reflection? No 180 Yes Reflection? No Smallest Rotation? 120 Yes Reflection? No 90 60 Yes Reflection? Reflection? Glide Reflection in axis which is not a reflection axis? Glide Reflection? Reflections in two different directions? Glide Reflection? All rotation centers on reflection axes? Yes c1m1 No p1m1 Yes p1g1 No p111 Yes All rotation centers on reflection axes? No p2mg Yes No p2gg Yes p2mm No c2mm p211 Yes p3m1 No p31m Yes p4mm No p4gg p311 Reflections in lines at 45 deg? No p411 Yes p6mm No p611 3 Math 311. Wallpaper Patterns 2 Name: ¶ 4. Determine the crystallographic notation for the wallpaper groups of the following patterns. Then find a primitive cell and fundamental region for each. ¶ 5. Determine the crystallographic notation for the wallpaper groups of the following patterns. Then find a fundamental region for each. 4 Math 311. Wallpaper Patterns 2 Name: ¶ 6. Determine the crystallographic notation for the wallpaper groups of the following patterns. Then find a fundamental region for each. ¶ 7. Determine the wallpaper group of the following patterns. 5 Math 311. Wallpaper Patterns 2 Name: ¶ 8. Use the flowchart on the previous page to identify the crystallographic notation of the following wallpaper patterns. ¶ 9. Out of the 17 possible wallpaper patterns, there are 2 missing. Can you identify them and draw a section of each of them? 6 Math 311. Wallpaper Patterns 2 Name: ¶ 10. The crystallographic description can be used to design wallpaper patterns with a given symmetry group. In the figure below, for each wallpaper group two diagrams are given. On the left figure, a primitive cell and a minimal set of generators is given, and a fundamental domain is shaded. On the right figure, a set of generators that includes the primitive cell unit translations is given. p111 2 4 4 p411 p211 4 p4mm p1m1 4 4 p4gg p1g1 3 c1m1 p311 3 3 p3m1 p2mm 2 2 p2mg 2 3 p31m p2gg p611 c2mm p6mm 3 6 7 Math 311. Wallpaper Patterns 2 Name: ¶ 11. Use the figure in the previous Problem to designs patterns with symmetry group p3m1 and p31m 8 Math 311. Wallpaper Patterns 2 Name: ¶ 12. M.C.Escher (1898–1972) was a Dutch artist whose work illustrates in a dramatic way all the wallpaper patters. Use the crystallographic pattern flowchart to identify the following patterns from Escher’s work. 9