PLANE SYMMETRY GROUPS

Last Update 05/ 5/ 2007

in English/ in Esperanto/ in Portuguese

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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Table of subjects.
Presentation
Chemistry Analytical Chromatography
Elemental organic analysis
Volumetric analysis, simulation
Crystallography 3 fold screw axis
4 fold inversion axis on tetrahedron
5 fold rotation axis absent in crystallography
Binary axis and reflection plane in stereographic projection
Bravais lattices
Conic sections under symmetry operators
Converting from spherical coordinates to stereographic projection
Crystal lattice and unit cell
Determination of unit cell
Elements of symmetry in action - animation
Elements of symmetry in action - cube game
Elements of symmetry in action - dodecahedron game
Elements of symmetry in action - icosahedron game
Elements of symmetry in action - octahedron game
Elements of symmetry in action - tetrahedron game
Ewald sphere and crystal measurements
Extinctions
Five classes in the cubic system
Five classes in the rhombohedral system
From tetrahedron to prism
Gnomonic projection
Improper symmetry axis
Miller indices
Miller indices - animation
Miller indices - cube game
Miller indices - octahedron game
Miller indices - rhombic dodecahedron game
Miller indices - tetrahedron game
Mirror plane
Orientations of the cube
Plane symmetry groups
Question on point group
Rotation axis in octahedron and Werner compounds
Rotation axis on tetrahedron and organic molecules
Rotation of objects about an arbitrary axis
Rotation of the parallel and stereographic projections of the cube
Rotation of the stereographic and parallel projection of the cube III
Seven faces in stereographic projection
Seven classes in the hexagonal system
Seven classes in the tetragonal system
Six elements of symmetry in seven orientations
Spherical projection of the octahedron
Stereographic projection
Stereographic projection of six polyhedra in different orientations
Straight line equations and symmetry elements
Symmetry, 2 fold axis
Symmetry, 2, 3 and 6 fold axis in benzene
Symmetry, 3 fold axis in the cube
Symmetry, 4 fold axis in the cube
Symmetry, 4 fold axis in the unit cell of gold
Symmetry elements and Miller indices game
Symmetry elements and Miller indices game - octahedron
Symmetry in art and in crystallography
Three classes in the monoclinic system
Three classes in the orthorhombic system
Twin crystals
Two classes in the triclinic system
Unit cell in hexagonal net
General Butane conformations
Density
Electrochemical cell
Ethane conformations
Resources of chemical-ICT: water, health and symmetry
Solid and liquid gold