This post was updated on Feb. 26, 2007 to fix the links.
Watch this animation [found via MeFi, link opens in a new window]. Intriguing, no? After watching it the first time, I thought it was pretty neat, but knew it wasn’t possible. So I did the math, and even showed my work.
It’s an optical illusion, powered by the outlines on the sections they move. First, they make two slices and rearrange things a bit. The blue slice creates two triangles, and the red slice creates two trapezoids.
Rearrange as desired, and without any outlines to cover things up, this is what you get:
See that white gap? It’s a parallelogram with an area of 1, which is why 64 seemingly equals 65. Warning: geometry and trig follow.
Start off noting that the two blue triangles are identical, as are the two red trapezoids.
For the blue triangle: looking at the smallest angle, you know the adjacent side is 8, and the opposite side is 3.
32 + 82 = x2, so the hypotenuse is 8.544
tan-1(3/8) = 20.556°
For the red trapezoid: look at the slice that happened. Basically, it took a triangle (green) out of a square to make the red trapezoid.
On that triangular piece, looking at the smallest angle, you know the adjacent side is 5, and the opposite side is 2.
22 + 52 = y2, so the hypotenuse is 5.385
tan-1(2/5) = 21.80°
Given that it’s the slice that was removed from the 5 x 5 square to result in the red trapezoid, you know the acute angle of the trapezoid and the length of the slanty side.
90° – 21.80° = 68.20°
Length of slanty side = removed section hypotenuse = 5.385
Now you know the acute angle of the parallelogram in the middle.
90° – 68.2° – 20.556° = 1.244°
So now you know everything you need to find the area of the parallelogram: opposite sides blue are 8.544, opposite sides red are 5.385, and the acute angles are 1.244°.
Area of a parallelogram is base times height, so you need to draw a right triangle inside. The following diagram is totally skewed for clarity, but the values are accurate.
You know the angle is 1.244°, and the hypotenuse of your newly drawn right triangle is 5.385. So you know the height of the parallelogram.
sin(1.244) x 5.385 = 0.1169
Knowing that the base is 8.544, you now know the area of the parallelogram.
8.544 x 0.1169 = 0.9988
… which is close enough to 1 to explain the illusion, because I rounded some things and didn’t follow the rule of significant digits.
QED, bitchez. :) Not that you care.