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 (501, 501) (x*y², x²*y²) x² + y² > 6.25 Pixel
above right Mandelbrot (501, 501) (x*y², x²*y²) x² + y² > 6.25 Point
cabbage Mandelbrot (801, 801) (x + y*sin(x*y² + y*x²), y + y*cos(y*x² + x*y²)) x² + y² > 12.0/k³ Point
fireflies #1 Mandelbrot (501, 501) (sin(y²*x - x - y), sin(x²*y - y + x)) x² + y² > 6.25/k Point
fireflies #2 Mandelbrot (501, 501) (sin(-x*y² - x), sin(x²*y - y)) x² + y² > 6.25/k Point
gadget #1 Julia (801, 801) (-(x² - y²)*sin(y²), -2.0*x*y*cos(x²), (0.0, 0.0)) x² + y² > 0.0 Point
gadget #2 Julia (801, 801) (-(x² - y²)*cos(x²), -2.0*x*y*sin(y²), (0.0, 0.0)) x² + y² > 0.0 Point
gadget #3 Julia (801, 801) (-(x² - y²)*cos(y²), -2.0*x*y*cos(x²), (0.0, 0.0)) x² + y² > 0.0 Point
horn Julia (801, 801) (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 (1401, 801) (x*y, x) |x| + |y| > 16.0 Pixel
pinwheel left Julia (501, 501) (-x*y², -y*x², (1.73725, 0.0)) x² + y² > 6.25 Pixel
pinwheel right Julia (501, 501) (-x*y², -y*x², (1.73725, 0.0)) x² + y² > 6.25 Point
spiral Julia (801, 801) (sin(x² + y²), cos(x² + y²), (0.0, 0.0)) x² + y² > m*2.0 / k³ Point
tile in blue and gray Julia (801, 801) (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