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.

(In reply to re(2): spoiler by Bractals)

The link in the Bractals' comment  nicely  demonstrates how to transform a rectangle into a square. However, we have to find the side of that square first . This graph shows it for the rectangles with the side ratio between 0.5 and 2. For prolonged rectangles, we can cut in half and stack the halves on top of each other.

 

ABCD is the initial rectangle. Continue AB so that BE=BC, then draw a circle on AE as the diameter. Continue BC to obtain F. BF is the side of the required square since AB*BE=BF^2 (AFE is a right triangle and FB is the altitude). Now, in order to transform ABCD into HBFG, we cut it along AK (DK=GF), slide triangle ADK up and move the small triangle from a to b.

 

P.S. Is it possible to insert an image directly into the comment?

Edited on September 27, 2006, 2:17 pm

Posted by Art M on 2006-09-27 02:46:00