OpArt

OpArt is, by definition, a style of visual art based upon optical illusions. Let it be a painting, a photograph or any other mean, the objective of this style is to play with the interaction of what you see, and what it really is. A classical OpArt piece involves confusion by giving impression of movement, impossible solids, hidden images, conflicting patterns, warping, etc. And of course, Mathematics is a perfect vehicle to study—and even perform—this form of art.

In this post I would like to show an example of how to use trivial mathematics to implement a well-known example (shown above) in with the tikz package, and leave as homework another more interesting example.

Observe first the image above: the optical effect arises when conflicting concentric squares change the direction of their patterns. You may think that the color is the culprit of this effect but, as you will see below, it is only the relationship between the pure black-and-white patterns what produces the impression of movement.

The Cantor Pairing Function

The objective of this post is to construct a pairing function, that presents us with a bijection between the set of natural numbers, and the lattice of points in the plane with non-negative integer coordinates.

We will accomplish this by creating the corresponding map (and its inverse), that takes each natural number and drops it at a location in the lattice, as the following diagram suggests: