I have a hobby of graphing equations on the Cartesian coordinate plane, not in order to solve any particular problem, but just to notice how they look. I’ve developed an intuitive understanding of what functions make what shapes. I’ve had the most luck with relations and trigonometric functions. Trigonometrics repeat themselves and give rise to complex repeating patterns, like those found in nature. Relations make liberal use of x and y on both sides of the equation, outputting a set of points that makes full use of the 2D space. Combining the two, trigs and relations, has allowed me to discover some astoundingly beautiful graphs. Here are some of my favorites and what I see in them:
“Tribal Patterns”
1 = cos(x) + cos(x*sin(x) + y*sin(y))
Looking at the graph of this equation, it’s hard to believe that it isn’t a hand drawn piece. With rounded shapes of triangles, wavy lines, and crescent moons, you get the feeling it is an ancient people’s depiction of the dreamland.
“Alligator Skin”
1 = cos(x+x*sin(y)) + cos(y+x*sin(x))
Most of the equations I come across tend to feel artificial and geometric, but this one curves in some of just the right ways to feel organic. It has big, broad shapes at the centerline and many more smaller ones at the peripheries, which is reminiscent of the scaly skin of an alligator.
“Globby Secant”
y + sin(8y) = sec(x)
This graph has a vibe like the Jamba Juice logo and might be able to pass for a funky wallpaper. I think it is a good example of how you can take an easily recognizable function like the secant and build complexity within it.