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Dissections of regular polygons into triangles of equal areas. (English) Zbl 0675.52005
The following theorem is proved: Let n5 be an integer. Then a regular n-gon is dissectable into m triangles of equal areas iff m is a multiple of n. For the proof the author uses Sperner’s lemma, non- Archimedean valuations and dissectability-techniques.
Reviewer: J.M.Wills
MSC:
52C17Packing and covering in n dimensions (discrete geometry)
51M20Polyhedra and polytopes; regular figures, division of spaces
52A10Convex sets in 2 dimensions (including convex curves)
05B45Tessellation and tiling problems
References:
[1]G. Bachman,Introduction to p-adic Numbers and Valuation Theory, Academic Press, New York, 1964.
[2]H. Hasse,Number Theory, Springer-Verlag, Berlin, 1980.
[3]K. Ireland and M. Rosen,A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1982.
[4]D. G. Mead, Dissection of the hypercube into simplexes,Proc. Amer. Math. Soc.,76 (1979), 302–304. · doi:10.1090/S0002-9939-1979-0537093-6
[5]P. Monsky, On dividing a square into triangles,Amer. Math. Monthly,77 (1970), 161–164. · Zbl 0187.19701 · doi:10.2307/2317329
[6]F. Richman and J. Thomas, Problem 5479,Amer. Math. Monthly,74 (1967), 329. · doi:10.2307/2316056
[7]J. Thomas, A dissection problem,Math. Mag.,41 (1968), 187–190. · Zbl 0164.51502 · doi:10.2307/2689143