# 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

## Wake 1/2

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.
```

## 1/3

Period 4 island

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

## 12/25

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

```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)
```