# New version of an interactive program f

New version of an interactive program for exploring Möbius transformations. Now with properly curved grid lines and arrows.

Question: is the resemblance of the image below to vortices you might see in your cup of coffee just a pleasing coincidence, or is there something about smooth transforms of sufficient complexity that leads to what looks like turbulence?

Question: is the resemblance of the image below to vortices you might see in your cup of coffee just a pleasing coincidence, or is there something about smooth transforms of sufficient complexity that leads to what looks like turbulence?

### timhutton/mobius-transforms

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I bet the latter chapters of

Visual Complex Analysiswould help answer your question, because they tie Möbius transformations to physical systems.I do know the general case has particle paths that are

loxodromic. The following video is a cool example that I think exhibits the behavior ofellipticMöbius transformations with two fixed points, perhaps similar to patterns you've seen in coffee swirls.https://m.youtube.com/watch?v=pnbJEg9r1o8

These will look as vortices, because these are also solutions to the Laplace equation. I guess your arrows would be a complex function, ie the arrow (dx +idy) are f(x+iy).

Just thinking:

Maybe instead of lines orthogonal to a potential flow, you could construct a viscous flow directly that emulates the arrows. Instead of pumping fluid in and out at the poles and zero’s, you put shafts (axis orthogonal to the complex plane) rotating counterclockwise at the poles, and clockwise at the zero’s, and solve the viscous flow.

That might be fun..