# Cyclic Averages of Regular Polygons and Platonic Solids

## DOI:

https://doi.org/10.26713/cma.v11i3.1420## Keywords:

Regular polygon, Platonic solid, Circle, Sphere, Locus, Sum of like powers, Rational distances problem## Abstract

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 \(P_n(T_n)\).

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## References

T. M. Apostol and M. A. Mnatsakanian, Sums of squares of distances in m-space, American Mathematical Monthly 110(6) (2003), 516 – 526, DOI: 10.2307/3647907.

T. M. Apostol and M. A. Mnatsakanian, Sums of squares of distances, Math Horizons 9(2) (2001), 21 – 22, DOI: 10.1080/10724117.2001.12021858.

R. Barbara and A. Karam, The rational distance problem for equilateral triangles, Communications in Mathematics and Applications 9(2) (2018), 139 – 145, DOI: 10.26713/cma.v9i2.659.

R. Barbara, Points at rational distance from the vertices of a unit polygon, Bulletin of the Iranian Mathematical Society 35(2) (2009), 209 – 215, URL: http://bims.iranjournals.ir/article_40.html.

J. Bentin, 79.15 Regular simplicial distances, The Mathematical Gazette 79(484) (1995), 106, DOI: 10.2307/3620008.

J. Bentin, 81.32 Regular polygonal distances, The Mathematical Gazette 81(491) (1997), 277 – 279, DOI: 10.2307/3619212.

T. G. Berry, Points at rational distance from the corners of a unit square, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4 17(4) (1990), 505 – 529, URL: http://www.numdam.org/item/?id=ASNSP_1990_4_17_4_505_0.

M. Gardner, Mathematical circus. More puzzles, games, paradoxes, and other mathematical entertainments from Scientific American, revised reprint of the 1981 edition with a preface by Donald Knuth, MAA Spectrum, Mathematical Association of America, Washington, DC (1992), URL: http://www.logic-books.info/sites/default/files/k08-mathematical_circus.pdf.

R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, New York ” Berlin (1981), DOI: 10.1007/978-1-4757-1738-9.

R. K. Guy, Tiling the square with rational triangles, Number Theory and Applications, R. A. Mollin (ed.), NATO Advanced Study Institute Series C 265 (1989), 45 – 101.

B. J. McCartin, Mysteries of the Equilateral Triangle, Hikari Ltd., Ruse (2010), URL: http://www.m-hikari.com/mccartin-2.pdf.

P.-S. Park, Regular polytopic distances, Forum Geometricorum 16 (2016), 227 – 232, URL: http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf.

P. Tangsupphathawat, Algebraic trigonometric values at rational multipliers of (pi), Acta et Commentationes Universitatis Tartuensis de Mathematica 18(1) (2014), 9 – 18, DOI: 10.12697/ACUTM.2014.18.02.

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*Communications in Mathematics and Applications*,

*11*(3), 335–355. https://doi.org/10.26713/cma.v11i3.1420

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