Describe how every simple polygon can be tranformed into any other simple polygon with the same area by dissecting it with straight cuts and rearranging the pieces.

I suspect that something of a more general nature is required rather than what I am offering here.

Essentially what I am doing is building upwards in side length, ie n-gon to (n+1) or (n+2)-gon.

1. a) Equilateral triangle is cut in half. The longest sides of the ensuing triangles are placed together [rotate one around the split vertex] forming a diagonal of a rectangle.
    b) If my triangle had been scalene and I split to the midpoint of the opposite side and then performed a similar rotation I would generate a simple concave 5-gon.
2. a) A rectangle, split from a vertex to the midpoint of one of the opposite sides, and rotation around the split vertex generates a simple concave 5-gon.
    b) Had I started with a square this process would have resulted in a quadrilateral with sides of 3, 1, √5 and √5
    c) I have a quadrilateral where two sides forming the split angle are of equal length.  After the split and rotation around the vertex I generate a 5-gon.
    d) With my quadrilateral being completely irregular I generate a concave 6-gon.

In general, if I have an n-gon where the sides adjacent to the vertex which I intend to split to a point on a different side are equal, then after the rotation around the split vertex I have an (n+1)-gon.

If the split occurs between two sides of unequal length, the an (n+2)-gon is formed.

Posted by brianjn on 2006-09-24 22:56:56