Group In The Application Of Graph And The Procedure Realizes

This article first did to the group action and the permutation group has conducted the further research, discussed the displacement group and the rigid body group of motions above a these two knowledge foundation and obtained a very important conclusion:every rotation axis is through the space in certain spot straight line limited step rotation group only possibly is five kinds respectively is the step revolving circulation group, the dihedral group, the tetrahedral group, the cube group either the octahedral group and the ten dihedral groups or the icosahedral group。This in spatial five kind of regular polyhedrons which the Euler polyhedron formula obtains with the graph theory in: The regular tetrahedron, the cube, the octahedron, the dodecahedron and the two decahedron's conclusion is consistent.This article emphatically in the cube symmetry transformation, the rigid body transformation and are not equal to the cube apex dough making above the type which colors to carry on the discussion and the research。The cube symmetry transformation altogether had 24, the predecessors had thorough and the accurate research。. When conducts the research to the cube rigid body transformation this article has used two methods:①The cube rigid body transformation's research is carries on above the cube symmetry transformation foundation. The cube symmetry transformation is by identically equal revolving, circles three pair of opposites central revolving, to circle the opposite side center point segment revolving and circles to vertex revolving is composed. What cube rigid body transformation was increases the mirror surface above the cube symmetry transformation foundation to cast light upon, to reflect the antipodal point, to circle to vertex revolving and the mirror surface casts light upon the product and circles to vertex revolving and reflects the antipodal point the product. Thus obtained the cube rigid body transformation 48 replacement way. This method is concrete, and the image has unfolded the cube rigid body transformation.②Using the group function's this knowledge spot, the orbital integer which, stable subgroup's integer and the group step three stoichiometric relation the parallel connection got on well with others obtains the cube rigid body transformation to have 48 kinds finally. This way concrete has not obtained the cube rigid body transformation, was only obtained the final result through the abstract stoichiometric relation.Obtained the consistent conclusion with vivid and the abstract two ways is the cube rigid body transformation way altogether has 48 kinds.Then this article unified the group theory, the theory of numbers and the graph theory partial knowledge spot obtained the theorem and the counting formula. Then utilizes these two formulas and the related knowledge selects us to obtain carries on with red and the blue color two kind of colors to the cube apex is not equal colors altogether has 23 ways; Carries on with red and the blue color two kind of colors to the cube surface is not equal colors altogether has 10 ways.Finally this article the language unifies with matble the cube symmetry transformation and the computer program, in computer general cube symmetry transformation namely identically equal revolving, circles three pair of opposites central revolving, to circle the opposite side center point segment revolving and circles to vertex revolving can realize.