About three years ago I ran across a strange coordinate system in which familiar functions lead to interesting plots. The system is called “circular coordinates” but it is not polar coordinates.

This morning I was playing around with this again.

Here’s a plot of *f*(*x*) = *x*.

And here’s a plot of *f*(*x*) = cos(8*x*).

See this post for details of circular coordinates.

Here is Python code to make the plots. You can experiment with your own plots by changing the definition of f.

# See Mathematics Magazine, v 52 no 3, p175 from numpy import cos from numpy import linspace import matplotlib.pyplot as plt plt.style.use('seaborn-v0_8-muted') def g(u, c, f): t = f(u) + c return 2*u*t**2 / (u**2 + t**2) def h(u, c, f): t = f(u) + c return 2*u*u*t / (u**2 + t**2) t = linspace(-7, 7, 10000) fig, ax = plt.subplots() f = lambda x: cos(8*x) for c in range(-10, 11): ax.plot(g(t, c, f), h(t, c, f)) plt.axis("off") plt.show()