We will in this post showcase a method for generating psychedelic animations using the basic trigonometry functions *sin(x), cos(x) *and *tan(x)*.

By animation, we mean a smoothly blending series of images of a given size. Suppose we just want to create a single image, we could define a function of two dependant variables *f(x, y)*, where *x* and *y *are the coordinates of a given point on the image. This means that the function depends on the *x* and *y *coordinates, giving a smooth variation across the image.

Here is the image representing *f(x, y) = x + y:*

The brightness of a pixel is* 256 – f(x) mod 256*

This is nice. It’s a smooth image, however, it’s not very “psychedelic”. To accomplish this, we construct not one, but *three *functions *R(x, y), G(x, y) *and *B(x, y)* – one for each color channel. This produces the following image instead:

*R(x, y) = sin(x+y) * **–** *G(x, y) = cos(x+y) * **–** *B(x, y) = sin(x) + cos(y)*

This is perfect! This is trippy enough for our purposes, however, it’s not very “animated”. This is a single image, so it doesn’t really count as an animation. To make this an animation, we introduce another independant variable *t* (for time), so we get three functions *R, G, B*, dependant on three variables *x, y, t*. Introducing a time variable, produces animations like the follow:

*Fewer colors in this than the actual version, due to .GIF file format*

This is perfect! We can now construct animated psychedelic animations using math!

Here’s a link to a YouTube channel I made, which showcases a bunch of these: http://www.youtube.com/mathematicart

**Future ideas: **Create the same program, but using HSB instead of RGB. My hypothesis is that this will create better, more trippy images.