HPC Complex Fractal Generation

JP Appel

Prerequisite Knowledge

Complex Numbers

i2 =  − 1 z = x + iy

Addition

z1 + z2 = (x1+x2) + i(y1+y2)

Multiplication

z1z2 = (x1x2−y1y2) + i(x1y2+x2y1)

What is the Mandelbrot Set

zn = zn − 12 + z0

Fractals

  • infinite self-similar geometric shape
  • have “fractional dimension”
Sripenski Triangle

The Mandelbrot set is a fractal in the complex plane

Fractal in Nature

Romanesco Cauliflower

Escape Time Algorithm

  • Inputs
    • Maximum Number of iterations
    • Upper bound
  1. Create a grid of points to sample
  2. For each point in the sample space
    1. Compute the next term in the sequence
    2. if greater than the upper bound return the number of iterations
    3. else repeat until the maximum number of iterations and return

Implementation

Program Structure

Mandelbrot

Image

Tricorn

Image

Burning Ship

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Multibrot

Image

Multicorn

GIF

Julia

Image

Analysis

Interactive Plots