engleski

Geometry of GS-quasigroups

A golden section quasigroup (shortly GS-quasigroup) is defined as an idempotent quasigroup which satisfies the mutually equivalent identities a(abc)c=b, a(abc)c=b. In this presentation identities and relations which are valid in a general GS-quasigroup will be researched. The geometrical meaning of the obtained identites will be given in the GS-quasigroup $C(1/2(1+\sqrt 5))$. Some interesting geometric concepts can be defined in a general GS-quasigroup. Namely, in a general GS-quasigroup the geometrical concept of the parallelogram, GS-trapezoid and some other geometric concepts can be introduced. The geometric concept of an affine-regular pentagon can be defined by means of GS-trapezoids. The concept of an affine-regular dodecahedron and affine-regular icosahedron can be obtained using the affine regular pentagons. Algebraic proofs of the statements about properties of the geometric concepts and the relationships between them in a general GS-quasigroup will be presented by means of the identities which are valid in a general GS-quasigroup. The geometrical representation of the introduced concepts and the obtained relations between them will be given in the GS-quasigroup $C(1/2(1+\sqrt 5))$.

GS-quasigroup; GS-trapezoids; affine-regular pentagon

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