Geometric vanish puzzles

Look at what happens here, when the pieces are rearranged! One square disappears. How can that be?

The video shows you another geometric vanish that looks even more amazing and puzzling. Where do those pieces go????? It looks like they vanish just by rearranging the pieces, yet the overall puzzle stays the same size.

Here's an explanation of the first puzzle. The one in the video follows the same principles.

Image from https://en.wikipedia.org/wiki/File:Missing_square_puzzle.svg

The overall triangle formed by the four pieces isn't really a triangle! Its longest side isn't a straight line but is actually "bent". Your eye just can't distinguish it easily.

Let's use some math. We will look at the ratio of the two perpendicular sides of each triangle, which also is the SLOPE of the hypotenuse (how steeply the hypotenuse "rises").

The two perpendicular sides of the overall "triangle" shape are 5 and 13, so the overall "triangle" should have a slope of 5/13 if it was a triangle (which it isn't).

The RED triangle has a slope of 3/8.
The BLUE triangle has a slope of 2/5.

NONE of those slopes or ratios are equal!!!! But, they are close:

5/13 ≈ 0.384615385

3/8 = 0.375

2/5 = 0.4

It is most interesting that the numbers involved (2, 3, 5, 8, and 13) in those ratios are consecutive numbers from the Fibonacci series:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 87, ...

In fact, by choosing some other consecutive Fibonacci numbers we can construct similar puzzles. If I use 1, 1, 2, 3, and 5, you can EASILY see how the longest side of the impostor "triangle" is actually bent:

The area between those two types of "bendings" accounts exactly for the area of the missing square.

And... if we take consecutive numbers from further on in the Fibonacci series, our eyes can NOT distinguish the difference in the slopes of the resulting triangles at all.

So... not everything is what it looks like to the human eye. We need to take a VERY close look — and then there is no paradox!

Further resources

See also

Missing Square puzzle - Wikipedia

Geometrical Vanishes from Mathematics Awareness Month
A geometrical vanish is a puzzle comprising several pieces whose area seems to change according to how the pieces are arranged. Common sense tells us this is impossible! But is it? Find out the explanation and how to take the topic further.