tile.jpg FRACTAL FIND
Explore Fractal and Quantum Variations

Lynn Wienck

FRACTAL INFORMATION
Build Information for Mandelbrot, Julia, Julia Unit Circle, Sequence Unit Circle

Mandelbrot
  • Map (m, n) plane where m: pixels real axis and n: pixels imaginary axis
  • Build (f(x,y), g(x,y))
    The set is zn+1 = zn² + c and written:
    xn+1 = xn² - yn² + cx
    yn+1 = 2.0 * xn * yn + cy
    (cx, cy) is mapped to a pixel in the (m, n) plane.
    (x0, y0) = (0, 0)
    The computation may be written:
    xn+1 = f(xn, yn) + cx
    yn+1 = g(xn, yn) + cy
  • Escape h(x,y) > value, k > kmax where k exceeds limit or x² + y² > value
  • Plot Pixels
    The escape count, k, is assigned a color.
    Color plotted for each (cx, cy) in the (m, n) plane.
    Plot Points
    The final point (xf, yf) is scaled and plotted.
    There is one (xf, yf) for each (cx, cy)
Julia
  • Map (m, n) plane where m: pixels real axis and n: pixels imaginary axis
  • Build (f(x,y), g(x,y), c)
    The set is zn+1 = zn² + c and written:
    xn+1 = xn² - yn² + cx
    yn+1 = 2.0 * xn * yn + cy
    (cx, cy) is a fixed point over all iterations.
    (x0, y0) is mapped to a pixel in the (m, n) plane.
    The computation may be written:
    xn+1 = f(xn, yn) + cx
    yn+1 = g(xn, yn) + cy
  • Escape h(x,y) > value, k > kmax where k exceeds limit or x² + y² > value
  • Plot Pixels
    The escape count, k, is assigned a color.
    Color plotted for each (x0, y0) in the (m, n) plane.
    Plot Points
    The final point (xf, yf) is scaled and plotted.
    There is one (xf, yf) for each (x0, y0).

Julia Unit Circle
  • Map double (θ)
    1 * θ
    2 * θ
    4 * θ
    8 * θ
  • Build (f(θ,x,y), g(θ,x,y))
    The set is zn+1 = zn² + c and written:
    xn+1 = cos(2k * θ)
    yn+1 = sin(2k * θ)
    The computation may be written:
    xn+1 = f(θ, xn, yn)
    yn+1 = g(θ, xn, yn)
  • Escape k > kmax 0 ≤ kmax ≤ 16
  • Scale and plot final point (xf, yf)
Sequence Unit Circle
  • Map increment (θ)
    0°
    1°
    2°
    3°
  • Build (fa(θ), ga(θ)) (fb(θ), gb(θ))
    xa = fa(θ)
    ya = ga(θ)
    xb = fb(θ)
    yb = gb(θ)
  • Scale scalea, scaleb These values scale (xa, ya) and (xb, yb) respectively when plotting points.
  • Escape θ > θmax Escape value is arbitrarily set to any value. These figures repeat the same form.
  • Offset mmin, mmax The minimum value and maximum value of m, the offset count.
  • Scale and plot point where ((scalea*xa + scaleb*xb*m), (scalea*ya + scaleb*yb*m))

gallery.jpg

GALLERY Variation Map Build Escape Plot
above left Mandelbrot (500, 500) (x*y², x²*y²) x² + y² > 6.25 Pixel
above right Mandelbrot (500, 500) (x*y², x²*y²) x² + y² > 6.25 Point
cabbage Mandelbrot (800, 800) (x + y*sin(x*y² + y*x²), y + y*cos(y*x² + x*y²)) x² + y² > 12.0/k³ Point
fireflies #1 Mandelbrot (500, 500) (sin(y²*x - x - y), sin(x²*y - y + x)) x² + y² > 6.25/k Point
fireflies #2 Mandelbrot (500, 500) (sin(-x*y² - x), sin(x²*y - y)) x² + y² > 6.25/k Point
gadget #1 Julia (800, 800) (-(x² - y²)*sin(y²), -2.0*x*y*cos(x²), (0.0, 0.0)) x² + y² > 0.0 Point
gadget #2 Julia (800, 800) (-(x² - y²)*cos(x²), -2.0*x*y*sin(y²), (0.0, 0.0)) x² + y² > 0.0 Point
gadget #3 Julia (800, 800) (-(x² - y²)*cos(y²), -2.0*x*y*cos(x²), (0.0, 0.0)) x² + y² > 0.0 Point
horn Julia (800, 800) (x*y² - x*cos(y²), y*x²*sin(y²), (0.0, 0.0)) x² + y² > 0.0 Point
moebius-klein Julia Unit Circle θ (cos(2k * θ) ± 0.5*x, sin(2k * θ) ± 0.5*y) k > 16 Point
monochrome Mandelbrot (1400, 800) (x*y, x) |x| + |y| > 16.0 Pixel
pinwheel left Julia (500, 500) (-x*y², -y*x², (1.73725, 0.0)) x² + y² > 6.25 Pixel
pinwheel right Julia (500, 500) (-x*y², -y*x², (1.73725, 0.0)) x² + y² > 6.25 Point
spiral Julia (800, 800) (sin(x² + y²), cos(x² + y²), (0.0, 0.0)) x² + y² > m*2.0 / k³ Point
tile in blue and gray Julia (800, 800) (x*y² - x - y, y*x² - y + x), (0.0, 0.0)) x² + y² > 16.0 Pixel
wave Julia Unit Circle θ (cos(2k * θ+x) - x, sin(2k * θ+y) + y) k > 16 Point

GALLERY Variation Map Build Scale Offset Plot
round corners Sequence Unit Circle θ (cos(-0.75*θ), sin(0.5*θ)) (cos(θ), sin(θ)) 250.0, 4.0 0, 19 Point
shaded ball Sequence Unit Circle θ (cos(θ), sin(θ)) (cos(-0.5*θ), sin(0.55*θ)) 200.0, 4.0 0, 45 Point
up-down square Sequence Unit Circle θ (cos(0.501*θ)*sin(0.501*θ), cos(0.999*θ)*sin(0.999*θ)) (cos(0.501*θ)*sin(0.501*θ), cos(0.999*θ)*sin(0.999*θ)) 225.0, 3.0 0, 51 Point
universe Sequence Unit Circle θ (cos(0.301*θ)*sin(0.302*θ), sin(0.301*θ)) (cos(0.301*θ)*sin(0.302*θ), sin(0.301*θ)) 225.0, 3.0 0, 51 Point