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Algebraic Surfaces - CUBISM 2            ---        by Gerhard Brunthaler

Polynoms with cubic symmetry

The figures here show polynoms with certain cubic symmetries. It has to be said that there exists more than one cubic symmetry. If a body has three to each other perpendicular 180-rotation axis and four 120-rotation axis through the body diagonal of a cuboid, then it it's sides have to be squares and a body with such a symmetry is said to have cubic symmetry. This means that the faces of cube can still have some differences between left and right or between one corner and another - nervertheless it has already cubic symmetry.
The next figure shows the five cubic symmetry types. As one can see, the Oh symmetry (it's a mathematical group) has the highest symmetry, represented by a single colored cube. All symmetry operations which bring the cube into an equivalent position, are the allowed ones - there exist 48 different. Instead of the white color one can give the cube also a patten which has the full cubic symmetry. This is done with the surfaces shown in Cubism.
The other cubic symmetries can be presented by cubes with a colored (black and white) surface. One sees that in these cases not so many symmetry operations are allowed. E.g., the O-group has no inversion symmetry. If one would spatially invert the cube around it's center, it would come into a position which is not equivalent to the original one (i.e. the black parts would come onto white ones and vice versa). Also, there are no mirror operations allowed for the O-symmetry as every mirror transformation is identical to an inversion followed by a 180-rotation.
Another prominent cubic symmetry is the Td group as it presents the symmetry of a tetrahedron. This group is of special importance in crystallography because diamond as well as the most important semiconductor material silicon has it's atoms arranged in that symmetry.
The figure showing the five different cubic symmetries by black-white coloring of a cube are taken from the book of N.W. Ashcroft and N.D. Mermin about "Solid State Physics".

Cubic crystallographic point groups_Ashcroft-Mermin.png
Cubic crystallographic point groups_Ashcroft-Mermin.png
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The first image below, shows a sphere with a polynom of O-symmetry added. So this surface has cubic O-symmetry, although it is difficult to see. The other images show cubic surfaces or frames which are distorted by polynoms of O-, Td- and Th-symmetry. Their symmetry might be compared with the above black and white schemes of the cubic symmetries.

Sym_O_02.png
Cubic O-Symmetry 1
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Sym_O_04b.png
Cubic O-Symmetry 1b
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Sym_O_06c.png
Cubic O-Symmetry 2
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Sym_O_07.png
Cubic O-Symmetry 3
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Sym_Td_03.png
Cubic Td-Symmetry 1
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Sym_Td_03b.png
Cubic Td-Symmetry 2
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Sym_Td_04c.png
Cubic Td-Symmetry 3
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Sym_Td_05.png
Cubic Td-Symmetry 4
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Sym_Td_05b.png
Cubic Td-Symmetry 5
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Sym_Th_02.png
Cubic Th-Symmetry 0
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Sym_Th_02c.png
Cubic Th-Symmetry 1
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Sym_Th_02e.png
Cubic Th-Symmetry 2
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e-mail: Gerhard.Brunthaler@jku.at