I recently found a little-known piece of software with the astonishing ability to draw graphs of arbitrary implicit equations. It’s called GrafEq. I spent some time playing around with it looking for cool graphs, and let me tell you, when you’re able to enter arbitrary implicit equations, you can get some astonishing graphs! Here are a few of the ones I came up with:
This “Checkerboard” is quite interesting! It is probably misleading. There’s no way the graph of that equation “has interior”, in other words, there’s no way the checkerboard in the picture actually has solid squares in it. Probably what’s going on it we have some very rapidly oscillating curves which end up looking solid because they oscillate so fast.
What I love about this graph is it looks more like something from biology than from math. With math, generally things tend to be very well-behaved, very symmetric. I love how we have seemingly “random” behavior in the graph of this equation.
Interesting “bands” appear in this graph. Bands further away from the origin seem to be better behaved; the band which goes through the origin has some weird bubbles in it.
This looks like it would make an interesting sort of chain mesh. Or maybe some sort of level field for some current, or something. Again, if you saw this out of context, you’d have trouble believing it came from math, and not from some engineering or physical science.
I love the patterns in this one!! If I ever get around to building that giant math cathedral, I’ll use patterns like these for the stained glass windows 
Oh my gosh, is God trying to say something to us through mathematics? Actually, longtime readers will recognize this right away: it comes from my Inverse Graphing Calculator, where you can enter whatever phrase you like and get an equation whose graph is that phrase. The equations which result tend to be enormously complicated. Even just a two-letter phrase generates a pretty complicated equation, as you see above. I’m just pleased that I finally found graphing software which is actually sophisticated enough to handle the equations produced by the IGC. Wolfram Alpha, for example, shatters into a million pieces.
This one almost makes me feel dizzy! All I can say is, cool… it’s way too badly behaved for me to offer any sort of mathematical analysis
And again, more bands. If you squint, you can kind of see “glint on metal” in the parts where the bands are less thick.
FURTHER READING
The Inverse Graphing Calculator
Proving 1=0 Using Wolfram Alpha
What Color is an Electron?
Levels of Infinity