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Mathematics LibreTexts

15: Fractals

  • Page ID
    34274
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    Fractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties.

    • 15.1: Fractals
      Fractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties. We’ll explore what that sentence means through the rest of the chapter. For now, we can begin with the idea of self-similarity, a characteristic of most fractals.
    • 15.2: Iterated Fractals
      Fractal self-similar behavior can be replicated through recursion: repeating a process over and over.
    • 15.3: Fractal Dimension
      In addition to visual self-similarity, fractals exhibit other interesting properties. For example, notice that each step of the Sierpinski gasket iteration removes one quarter of the remaining area. If this process is continued indefinitely, we would end up essentially removing all the area, meaning we started with a 2-dimensional area, and somehow end up with something less than that, but seemingly more than just a 1-dimensional line.
    • 15.4: Complex Numbers
    • 15.5: Complex Recursive Sequences
      We will now explore recursively defined sequences of complex numbers.
    • 15.6: Exercises

    Thumbnail: Zoom in of the Mandelbrot set (Public Domain; Simpsons contributor via Wikipedia)


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