Hans Walser, [20090720a]
Dissection of a convex polygon with central symmetry
A polygon with central symmetry has an even number of vertices.
Dissection of a 12-gon
Proof by induction according to the colors [Walser 1983].
We can easily find the dissection by moving copies of the polygon such that every vertex comes once to a particular vertex of the original polygon.
First two steps
An equilateral polygon with central symmetry can be dissected into rhombi.
We discuss now regular polygons with 2n vertices. The dissection we can obtain by moving like above or by rotating the 2n-gon by angles of multiples of around a particular vertex ([Lindgren 1972], p. 109, and cover).
Dissection of a regular 14-gon
If we are just interested in the vertices of the rhombi, we can work with circles.
Circles will work
In this case we get sets, each consisting of n congruent rhombi. The acute angles of the rhombi are .
n = 7
Solution of M. N. DeshpandeŐs Problem 1650 in Mathematics Magazine (2002), p. 227 ([Fischer 2003], with beautiful examples).
We get sets, each consisting of n congruent rhombi, and a set of squares.
n = 6
[Lindgren 1972] Lindgren, H.: Geometric Dissections. Revised and enlarged by Greg Frederickson. New York: Dover 1972.
[Walser 1983] Walser, Hans: Ein Zerlegungssatz fźr punktsymmetrische konvexe Vielecke. Elemente der Mathematik (38), 1983, p. 159-160.
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