SPHERICAL PROJECTION OF THE OCTAHEDRON

Last Update 02/ 11/ 2001

in English/ in Esperanto/ in French/ in Portuguese

Interaction in this environment will allow observation of the normal to the crystal faces producing the perspective presentation of the spherical projection and the stenographic projection in a convenient orientation.

Nile's Sternness (1638-1687) recognized in 1669 the constancy of the dihedral angles in crystals of a material grown at equal temperature and pressure. This property, known as Stoniness law, has been used for analytical purposes. It is a handy work to draw the contour of the crystal faces when there are many of them. On practice the dihedral angle is substituted by its complement to trace the normal to the faces from an origin point. In the initial figure on this application the normal to the octahedral faces drawn in perspective pass through the origin point arbitrarily chosen as the octahedral centre. The intersection of any of the mentioned normal with a spherical surface centralized on the origin point will be defined as face pole. The set of eight face poles obtained is named spherical projection of the octahedron. The orientation of the octahedron related with the north south axis of the sphere (signed by N an S) can be modified by a push and drag mouse action on the green rolling bar. When the octahedral is oriented with two vertices along the NS axis then the respective stereographic projection is shown on the application. After a click on button b, the colour of each point in the stereographic projection is equal to the colour of the respective normal used to generate the face pole. Thus the green normal produces the green face pole in stereographic projection. As shown after a click on c, the spherical projection depends only on the normal to each face. However the representation of the octahedral without the normals to the faces is unable to build the spherical projection and consequently will not generate the stereographic projection, as observed after a click on d. The initial figure can be restored after a click on a.

Bibliography

1. Woolfson, M.M., X-ray Crystallography, Cambridge University Press, London, 1970.

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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