Hans Walser, [20090720a]

Dissection of a convex polygon with central symmetry

1        General case

A polygon with central symmetry has an even number of vertices.


A convex polygon with central symmetry and 2n vertices can be dissected into  parallelograms.


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




2        Special cases

2.1      Equilateral polygons

An equilateral polygon with central symmetry can be dissected into rhombi.




2.2      Regular polygons

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


2.2.1    n odd

In this case we get  sets, each consisting of n congruent rhombi. The acute angles of the rhombi are .


n = 7


2.2.2    n even

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.

Links (2009 / 7 / 20)

[Fischer 2003]