The Fourier Series
With this application you can see how a sum of enough sinusoidal functions may lead to a very different periodical function. The Fourier theorem states that any (non pathological) periodic function can be written as an infinite sum of sinusoidal functions.

Change the value of , representing the number of sinusoidal waves to add up. The maximum value for , the index of the sum, is shown and the sum of the waves is shown in blue, too. For small values of you can also see the single terms of the sum.