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`                         MATHEMATICS OF THE FRACTAL TYPES`

Fractal Type(s) Formula(s) used ———————– ——————————————— Mandel, Julia Z(n+1) = Z(n)^2 + C Newton, Newtbasin (roots of) Z^n - 1, wnere n is an integer ComplexNewton, ComplexBasin (roots of) Z^a - b, where a,b are complex plasma (see the Plasma section for details) Mandelsine, Lambdasine Z(n+1) = Lambda * sine(Z(n)) + C Mandelcos, Lambdacos Z(n+1) = Lambda * cos(Z(n)) + C Mandelexp, Lambdaexp Z(n+1) = Lambda * exp(Z(n)) + C Mandelsinh, Lambdasinh Z(n+1) = Lambda * sinh(Z(n)) + C Mandelcosh, Lambdacosh Z(n+1) = Lambda * cosh(Z(n)) + C BarnsleyM1, BarnsleyJ1 Z(n+1) = (Z(n)-1) * C if Real(z) >= 0

`                           else    (Z(n)+1) * modulus(C)/C`

BarnsleyM2, BarnsleyJ2 Z(n+1) = (Z(n)-1) * C if Real(Z(n))*Imag(C)

```                                   +Real(C)*Imag(Z(n)) >= 0
else    (Z(n)+1) * C```

BarnsleyM3, BarnsleyJ3 Z(n+1) = (Real(Z(n))^2 - Imag(Z(n))^2 - 1)

```                                  + i * (2 * Real(Z((n)) * Imag(Z((n)))
if Real(Z(n) > 0
else    (Real(Z(n))^2 - Imag(Z(n))^2 - 1
+ lambda * Real(Z(n))
+ i * (2 * Real(Z((n)) * Imag(Z((n))
+ lambda * Real(Z(n))```

Sierpinski Z(n+1) = (2x, 2y - 1) if y > .5

```                           else    (2x - 1, 2y) if x > .5
else    (2X, 2y)```

MandelLambda, Lambda Z(n+1) = (C) * (Z(n)^2) + C MarksMandel, MarksJulia Z(n+1) = (C^(Period-1)) * (Z(n)^2) + C

`                          ("Period" is a parameter)`

Unity (see the Unity section for details) ifs, ifs3D (see the IFS section for details) Mandel4, Julia4 Z(n+1) = Z(n)^4 + C Test (as distributed, as simple Mandelbrot fractal) Mansinzsqrd, Julsinzsqrd Z(n+1) = Z(n)^2 + sin(Z(n)) + C Manzpower, Julzpower Z(n+1) = Z(n)^M + C (M is a parameter) Manzzpwr, Julzzpwr Z(n+1) = Z(n)^Z(n) + Z(n)^M + C Mansinexp, Julsinexp Z(n+1) = sin(Z(n)) + e^(Z(n)) + C popcorn Z(n+1) = x(n+1) + i * y(n+1), where

```                          x(n+1) = x(n) - 0.05*sin(y(n)) + tan(3*y(n))
y(n+1) = y(n) - 0.05*sin(x(n)) + tan(3*x(n))```

demm, demj (Mandelbrot, Julia fractals calculated and

`                          colored using the "Distance Estimator" method`

Bifurcation (see the Bifurcation section for details) Lorenz, Lorenz3d Lorenz Attractor - orbits of differential

```                          equation
x = x + (-a * x * dt) + (a * y * dt)
y = y + (b * x * dt) - (y * dt) - (z * x * dt)
z = z + (-c * z * dt) + (x * y * dt)
(defaults: dt = .02, a = 5, b = 15, c = 1)
(Lorenz3D is the Lorenz Attractor with the
as specified by the IFS <E>ditor's
transformation option)```

The following trig identities are invaluable for coding fractals that use complex-valued transcendental functions.

``` e^(x+iy) = (e^x)cos(y) + i(e^x)sin(y)
sin(x+iy) = sin(x)cosh(y) + icos(x)sinh(y)
cos(x+iy) = cos(x)cosh(y) - isin(x)sinh(y)
sinh(x+iy) = sinh(x)cos(y) + icosh(x)sin(y)
cosh(x+iy) = cosh(x)cos(y) + isinh(x)sin(y)
ln(x+iy) = (1/2)ln(x*x + y*y) + i(arctan(y/x) + 2kPi)
(k = 0, +-1, +-2, +-....)```
```                   sin(2x)               sinh(2y)
tan(x+iy) = ------------------  + i------------------
cos(2x) + cosh(2y)     cos(2x) + cosh(2y)```
```                  sinh(2x)                sin(2y)
tanh(x+iy) = ------------------ + i------------------
cosh(2x) + cos(2y)    cosh(2x) + cos(2y)```
```     z^z = e^(log(z)*z)
log(x + iy) = 1/2(log(x*x + y*y) + i(arc_tan(y/x))
e^(x + iy) = (cosh(x) + sinh(x)) * (cos(y) + isin(y))
= e^x * (cos(y) + isin(y))
= (e^x * cos(y)) + i(e^x * sin(y))```

Extract from FRACTINT.DOC 