engleski

An Affine Regular Icosahedron Inscribed in an Affine Regular Octahedron in a GS-Quasigroup

A golden section quasigroup or shortly a GS- quasigroup is an idempotent quasigroup which satisfies the identities $a(ab \cdot c)\cdot c=b$, $a\cdot(a\cdot bc)c=b$. The concept of a GS- quasigroup was introduced by Volenec. A number of geometric concepts can be introduced in a general GS-quasigroup by means of the binary quasigroup operation. In this paper, it is proved that for any affine regular octahedron there is an affine regular icosahedron which is inscribed in the given affine regular octahedron. This is proved by means of the identities and relations which are valid in a general GS-quasigrup. The geometrical presentation in the GS-quasigroup $\mathbb{;C}; (\frac{;1};{;2};(1+\sqrt 5))$ suggests how a geometrical consequence may be derived from the statements proven in a purely algebraic manner.

GS–quasigroup, GS–trapezoid, affine regular icosahedron, affine regular octahedron

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