Cyclic Averages of Regular Polygons and Platonic Solids

Mamuka Meskhishvili


The concept of the cyclic averages are introduced for a regular polygon \(P_n\) and a Platonic solid \(T_n\). It is shown that cyclic averages of equal powers are the same for various \(P_n(T_n)\), but their number is characteristic of \(P_n(T_n)\). Given the definition of a circle (sphere) by the vertices of \(P_n(T_n)\) and on the base of the cyclic averages are established the common metrical relations of \(P_n(T_n)\).


Regular polygon; Platonic solid; Circle; Sphere; Locus; Sum of like powers; Rational distances problem

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