A Faulty Dissection:
What Is Wrong?

The applet presents a famous dissection "paradox": take a 8×8 square and cut it as shown in the left portion of the applet. The four pieces thus obtained can be dragged into the right portion of the applet where they appear to form a 5×13 rectangle. Comparison of the areas of the two shapes creates a paradox: 64 = 65. How is this possible?

(In the applet you can either drag the pieces manually or make them move automatically by pressing the "Animate" button.)

When the pieces put together to form a rectangle, the inner edges of the polygons do not fit together. The applet blurs the effect by adjusting the pieces ever so little. In an "honest" version of the applet the effect is clearly noticeable. It becomes obvious there how small an adjustment is really required. The explanation involves the famous Fibonacci numbers.

The dissection pieces can be rearranged in a different manner with a no less remarkable result.

Dissection Paradoxes

• Curry's Paradox
• Dissection of a 10×13 Rectangle into Two Chessboards
• A Faulty Dissection
• Hooper's Paradox
• John Sharp's Paradox
• Langman's Paradox
• Popping A Square
• Sam Loyd's Son's Dissection

Copyright © 1996-2008 Alexander Bogomolny