Is there a koch circle?

Question

Is there some fractal like the koch snowflake, but only with many circles around a bigger initial circle, each of them surrounded by smaller circles and so on (but all of them kissing one bigger circle)? So circles instead of the triangles in a koch snowflake... If not, why?

Answer 1

Is this any good?

It's the last frame in the animation for this answer.

Answer 2

Parts of the Mandelbrot set look like circles with smaller circles attached recursively:

They aren't quite exact circles (except for the one centered at $-1+0i$), but the property that the radius of the smaller circle attached at rational angle $\frac{p}{q}$ measured in turns is approximately $q^2$ times smaller provides a way to construct a similar fractal from exact circles:

Haskell source code using the Diagrams library:

import Diagrams.Prelude
import Diagrams.Backend.SVG.CmdLine (defaultMain)

main
  = defaultMain
  $ diagram 1
  # rotateBy (-0.25)
  # pad 1.1
  # lw thin
  # bg white

power = 2

minimumRadius = 0.001

diagram radius
  | radius < minimumRadius = mempty
  | otherwise = circle radius <> mconcat
     [ diagram r
     # rotateBy (s - 0.5)
     # translate (r2 (rr * cos t, rr * sin t))
     | den <- [ 2 .. ceiling (sqrt (radius / minimumRadius)) ]
     , num <- [ 1 .. den - 1 ]
     , num `gcd` den == 1
     , let s = fi num / fi den
     , let t = 2 * pi * s
     , let r = radius / fi den ** power
     , let rr = radius + r
     ]
  where
    fi = fromInteger

Reducing the power makes the circles larger, but too low and they eventually overlap - in any case the power must be larger than one to ensure the circles actually do get smaller.

Answer 3

Maybe the "Pharaoh's Breastplate" described by Mandelbrot.

Here plotted by Ken Monks