Lab 11. Dissections

Part A. Proving that a rectangle can be decomposed into other rectangles or a square

We want to understand how to rearrange this picture rigorously into a rectangle with one side = AF

Step 1. Shearing the first rectangle.

Construct a rectangle ABCD and a point F on BC. Translate F by vector AD to get F'. Explain why DAFF' has the same area as DABC. Construct the interior of ABF and translate it to a triangle that fits with DAFC to fill DAFF'.

 

 

Step 2. Shearing the parallelogram.

Construct a new rectangle AFGFH as in the figure. First construct the line DF' (on top of the segment) and the lines through A and F perpendicular to AF. Why does this new rectangle have the same area as the parallelogram DAFF'?

 

Step 3. Reassembling the rectangle ABCD

Let I be the intersection of FG and CD. Then construct polygon interiors in the original rectangle ABCD; color them differently Then translate each polygon so that the imaged fill up the rectangle AFGH. This should hold together when you slide F.

 

Step 4. Animating

Let I be the intersection of FG and CD. Then construct polygon interiors in the original rectangle ABCD and translations that move these polygons and reassemble them as the rectangle AFGH.

 

Part B. Animating a Dissection – sliders using dilation

On the side, construct a segment AB and a point C on the segment. Select A, B, C in that order and hold down the Shift Key and Measure Ratio. Then select the ratio measurement on the screen and choose Mark Scale Factor. Now choose two new points D and E and Mark D as center. Dilate E by the marked scale factor to get E'. Now mark vector DE'. Draw a shape S and translate S by vector DE'. Drag point C and see how the shape S' moves from its original position on S to the new one. You can hide S so that only one shape is visible. This technique can be applied to dissection figures to get moving pieces.