SYMMETRY, 2-FOLD AXIS

Last Update 18/ 05/ 2000

in English/ in Esperanto/ in French/ in Portuguese

The special presentation of ordered repetition forms along one or more directions in space either following straight lines, planes or several curves attracted the attention of people. The organization of diverse materials in vegetable, animal and mineral reign is explained by well defined mathematical rules, its study enables comprehension of a nature's face. In chemistry symmetry related properties are studied with analytical objectives, such that the sequential exhibition of the parts is expected to lead to information on the focalized object and simplifies its identification. One of the investigation possibilities uses to rotate the sample. If the same image is observed repeatedly before the 360 o or 2p radians turn is completed, a symmetry axis is present. Several other symmetry elements exists beyond the rotation axis, as for example the symmetry plane, inversion centre, inversion and rotation axis and reflection and rotation axis, screw axis and glide planes, including translation in space. After any object is rotated 2p radians around an axis the same image is observed as before the execution of this operation. This is a trivial identity operation. If the object develops the same image after turning p radians around an axis, it is named two fold rotation axis, represented by C2 or 2, according to Schoenflies or to Hermann-Maugin notation. The figure shows the ethene molecule, represented by two black spheres in the place of the two carbon atoms double bonded, two blue spheres bonded to each carbon atom, representing the four hydrogen atoms. Any line represents a bond from one atom to another.  Different groups, indicated by different colours on the spheres can replace the original blue spheres by clicking on the buttons labeled A=X; A=B=X; A=C=X; A=D=X; A=X, B=Y; A=X, C=Y; A=X, D=Y. Different sphere colours indicate different chemical groups. If you push and drag the mouse properly when the cursor is over the green square, you can animate the molecule to a sequence of rotation positions. Half turn on the molecule can be performed after a click on the button 0 o and 180 o. It will be interesting to verify the existence or absence of C2 in the given orientation in the eight kinds of selected molecules, with four equal groups or with one or two different groups at your choice by clicking on the buttons at the right.

Bibliography

SHUBNIKOV, A.V. and KOPTSIK, V.A., Symmetry in Science and Art, Plenum Press, N.Y., 1974.

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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
Mirror planes and Miller indices game - tetrahedron
Orientations of the cube
p2mm
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
Oxidation and reduction
Resources of chemical-ICT: water, health and symmetry
Solid and liquid gold