190Tutorial4

2. (Mindscape 15, Chapter 2.2 of the textbook). The enlarging area paradox. The square drawn on the next page has sides equal to 8 (a Fibonacci number) and so has area 8x8=64. However, it can be cut up into four pieces whereby the short sides have lengths 3 and 5, as illustrated on the separate page. Cut it up as indicated along the dotted lines. Now use those pieces to construct a rectangle having base 13 and height 5. The area of this rectangle is 13x5=65. So by moving around the four pieces, we increased the area by 1. What's going on? How can you explain this impossible thing?

• Discussed with Stephen and Nick.

The area of the pieces has not increased as the new rectangle is not actually whole.

The new rectangle is made up of two right angle triangles. The hypotenuse of these triangles are not straight however. They are slightly concaved. This can be shown by taking the angles of the hypotenuse at both ends. The whole triangle can be broken up into a section of a 3x5 rectangle with a 2x5 rectangle on the top. And a second triangle with dimensions of 8x3.

The first part of the hypotenuse has a gradient of tan-1(2/5) = 21.8 degrees and the second section has a gradient of tan-1(3/8) = 20.6 degrees. As these are so similar, it looks like the line is straight. This is also helped by the general dimensions. The ratios of the sides of the triangles and the rectangle itself are all made up using Fibonacci numbers. 3/8 and3/8 are both very close to the golden ratio. As a consequence, the final rectangle looks 'pretty' which makes it harder to see an obvious distortion in the hypotenuse.