64/65 paradox

[shivvy]

Could anyone explain how the 64=65 geometric triangle paradox works? I've read some explainations but it was all Greek to me.

[kempie]

When the pieces cut from the square are put together to form a rectangle, the inner edges of the polygons do not fit exactly together (the area left uncovered being equal to 1 unit). This is clearly noticeable for small values of the control parameter. For larger values the discrepancy is virtually imperceptible.

http://www.cut-the-knot.org/Curriculum/Geometr y/FibonacciBamboozlement.shtml

[Loosehead]

See here: http://www.mathematik.uni-bielefeld.de/~sillke /PUZZLES/jigsaw-paradox.html

The hypotenuse of the triangle is not straight, in the lower picture it bows out slightly, in the top picture it bows in slightly. If you use our old friend Trig you can see that the angle at the point of the green triangle is 21.8 degrees and the angle at the point of the red triangle is 20.56 degrees so the hypotenuse of the whole thing is not a straight line. The drawing gives the impression that it is but it is not. The area occupied by the bowing in/out accounts for the "Extra" unit.

[Loosehead]

when I said hypotenuse above, meant of course the line that looks like the hypotenuse.