Visualizing mathematics

I love visual proofs of mathematical theorems. One visual proof I use quite often in my design courses (CEP817 or CEP917) is a visual proof of the fact that the sum of consecutive odd numbers is a square number.

In other words:

1 + 3 = 4 = 22

1 + 3 + 5 = 9 = 32

1 + 3 + 5 + 7 = 16 = 42

… and so on.

Mathematically speaking that translates to

1 + 3 + … + (2N-1) = N2

Now one can go about proving this fact this through algebra OR one could just do it visually. Check out this image below,

… or better still go to this applet

Pretty cool huh?

Looking for the above applet led me to this other cool visual proof for

(a + 1)2 = a2 + 2a + 1

Check it out.

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