Equidissections of centrally symmetric octagons
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In 1970 Monsky proved that a square cannot be cut into an odd number of triangles of equal areas. In 1988 Kasimatis proved that if a regularn-gon,n ⩾ 5, is cut intom triangles of equal areas, thenm is a multiple ofn. These two results imply that a centrally symmetric regular polygon cannot be cut into an odd number of triangles of equal areas. We conjecture that the conclusion holds even if the restriction “regular” is deleted from the hypothesis and prove that it does forn = 6 andn = 8.
AMS (1980) subject classificationPrimary 52A45, 52A25
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- [B]Backman, G.,Introduction to p-adic numbers and valuation theory. Academic Press, New York, 1964.Google Scholar
- [I]Ireland, K. andRosen, M.,A classical introduction to modern number theory. Springer-Verlag, New York, 1982.Google Scholar
- [K]Kasimatis, E. A.,Dissections of regular polygons into triangles of equal areas. Discrete Comput. Geom. (to appear).Google Scholar
- [KS]Kasimatis, E. A. andStein, S.,Equidissections of polytopes. Discrete Math. (to appear).Google Scholar
- [Me]Mead, D. G.,Dissection of the hypercube into simplexes. Proc. Amer. Math. Soc.76 (1979), 302–304.Google Scholar
- [Mo]Monsky, P.,On dividing a square into triangles. Amer. Math. Monthly77 (1970), 161–164.Google Scholar
- [Sp]Spanier, E. H.,Algebraic topology. McGraw-Hill, New York, 1966.Google Scholar
- [S]Stein, S.,Mathematics, the man-made universe. 2nd Ed., Freeman, New York, 1969.Google Scholar