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.

Theorem:

A convex polygon with central symmetry and 2*n* 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

Dissection

# 2
Special cases

## 2.1
Equilateral polygons

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

Rhombi

## 2.2
Regular polygons

We
discuss now regular polygons with 2*n*
vertices. The dissection we can obtain by moving like above or by rotating the
2*n*-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

References

[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]

http://www.calpoly.edu/~glfisher/MAA_Dahlia_Paper3.pdf