Take a solid metal plate (or any other rigid material) and sprinkle some particles on it. It can either be sand, salt, or even baking soda. Then, generate some vibration on the plate by using either a violin bow to excite the plate, or place it on a wave generator and set it to specific frequencies. What you will observe are the beautiful patterns generated as shown in the figure. This experiment sounds relatively simple, but as simple as it sounds, this experiment reveals some extremely complex and intensive underlying mathematical principles.

Ok, so how does this work? Well, it turns out these are examples of standing waves in the two-dimensional space. In one dimension, a “node” in a standing wave is a point along the wave where the amplitude is at minimum. Similarly, in two dimensions, there are “nodal lines,” which are static points or lines on the plate where the particles settle.

Given two algorithms that perform the same task, what is the best way to compare their performance? Timing them is the obvious answer and also the wrong one: a runtime depends far more on the hardware it was measured on than on the algorithm itself, and it varies from run to run even on the same machine. What is wanted is a measure of how an algorithm scales as its input grows, independent of the machine underneath it.

That measure is time complexity. Write f(n) for the worst-case cost of an algorithm on an input of size n, and the hardware becomes constant factors that the big-O notation then discards, leaving only the trend: constant, logarithmic, linear, quadratic, exponential, etc. From that definition come the properties of big O, big Omega, and big Theta, the Master Theorem, and the theoretical analysis of real algorithms.