engleski

Some geometric concepts in GS-quasigroups

GS-quasigroup is an idempotent quasigroup which satisfies the (mutually equivalent) identities a(ab.c).c = b, a.(a.bc)c=b. The identities of mediality, elasticity, left and right distributivity and some other identities and equivalencies are also valid in a GS-quasigroup. The concept of GS-quasigroup is introduced by V.Volenec. Let C be the set of points in the Euclidean plane. If groupoid (C, .) is defined so that for any two different points a, b in C we define ab=c if the point b divides the pair a, c in the ratio of golden section, then (C, .) is a GS-quasigroup. That quasigroup will be denoted C((1+sqrt(5))/2) because we have c=(1+sqrt(5))/2 if a=0 and b=1.The figures in this quasigroup C((1+sqrt(5))/2) can be used for illustration of "geometrical" relations in any GS-quasigroup. In the general GS-quasigroup the concept of the parallelogram and midpoint can be introduced. The concept of the affine regular pentagon will be defined by means of GS-trapezoid. The concept of the affine regular dodecahedron and affine regular icosahedron is introduced by means of the affine regular pentagon. The concept of the affine regular decagon is introduced by means of the GS-deltoid. The geometrical representation of all introduced concepts will be given in the GS-quasigroup C((1+sqrt(5))/2) and the connection between them will be investigated.

quasigroup

nije evidentirano