Fractals/island wake

The biggest island of the p/q wake

  • for p/q <= 1/3
    • is located on the first branch ( reading from the left to right)
    • has period = q+1
    • external rays ?
  • for p/q > 1/3
    • is located on the last branch ( reading from the left to right)
    • has period = q+2

Examples edit

Wake 1/2 edit

period 3 island is the biggest island of wake 1/2


find angles of some child bulbs of period 3 component ( island) on main antenna with external rays (3/7,4/7)

Plane description :[1]

-1.76733 +0.00002 i @ 0.05

One can check it using program Mandel by Wolf Jung :

The angle  3/7  or  p011 has  preperiod = 0  and  period = 3.
The conjugate angle is  4/7  or  p100 .
The kneading sequence is  AB*  and the internal address is  1-2-3 .
The corresponding parameter rays are landing at the root of a primitive component of period 3.

See also:


1/3 edit

Period 4 island

  • Center X -0.15710375803
  • Center Y +1.03258348530
  • Pixel step +0.0000375

wake 1/7 edit

12/25 edit

wake 12/25

  • root c = -0.738203140939397 +0.124839088573366 i
  • The 12/25-wake of the main cardioid is bounded by the parameter rays with the angles
    • 11184809/33554431 or p0101010101010101010101001 and
    • 11184810/33554431 or p0101010101010101010101010 .
  • the center of the satellite component c = -0.739829393511579 +0.125072144080321 i period = 25

biggest island of 12/25 wake

  • cardioid
    • center c = -0.744245042107463 +0.127908444364520 i
    • period = 27
    • cusp


Size edit

Haskell program

size formula degree p a b = do
  -- z = x + i y = 0
  x <- real
  y <- real
  x .= 0
  y .= 0
  -- matrix L
  lxa <- real
  lxb <- real
  lya <- real
  lyb <- real
  -- L = identity
  lxa .= 1
  lxb .= 0
  lya .= 0
  lyb .= 1
  -- matrix B
  bxa <- real
  bxb <- real
  bya <- real
  byb <- real
  -- B = identity
  bxa .= 1
  bxb .= 0
  bya .= 0
  byb .= 1
  -- loop one period
  j <- int
  j .= 1
  while_ (j .< p) $ do
    -- allocate next z, l
    xn <- real
    yn <- real
    lxan <- real
    lxbn <- real
    lyan <- real
    lybn <- real
    -- calculate next z, l
    let R2 fx fy = formula (R2 a b) (R2 x y)
        -- calculate derivatives
        fdxa = ddx d x fx
        fdxb = ddx d y fx
        fdya = ddx d x fy
        fdyb = ddx d y fy
        d v u
              | u == x && v == x = lxa
              | u == x && v == y = lxb
              | u == y && v == x = lya
              | u == y && v == y = lyb
              | u == v = 1
              | otherwise = 0
    -- z = f(z, c)
    xn .= fx
    yn .= fy
    x .= xn
    y .= yn
    -- L = J_f(z, c)
    lxan .= fdxa
    lxbn .= fdxb
    lyan .= fdya
    lybn .= fdyb
    lxa .= lxan
    lxb .= lxbn
    lya .= lyan
    lyb .= lybn
    -- B = B + 1/L
    det <- real
    det .= lxa * lyb - lxb * lya
    bxa .= bxa + lyb / det
    bxb .= bxb - lxb / det
    bya .= bya - lya / det
    byb .= byb + lxa / det
    -- loop counter
    j .= j + 1
  -- l = sqrt (abs (det L))
  l <- real
  l .= sqrt (abs (lxa * lyb - lxb * lya))
  -- beta = sqrt (abs (det B))
  beta <- real
  beta .= sqrt (abs (bxa * byb - bxb * bya))
  -- compute l^d b
  d <- float
  d .= degree / (degree - 1)
  llb <- real
  llb .= l ** d * beta
  return_ (1 / llb)

References edit

  1. R2F(1/2B1)S by Robert P. Munafo, 2008 Feb 28.