Fractals: Cheat Sheet

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Essential Concepts

  • Fractals are special shapes in math that look similar to themselves when you zoom in or zoom out. They have interesting properties and are made using a process called recursion. When something is self-similar, it means that it looks the same from far away as it does when you look at it up close.
  • To create other fractals, we use a process called recursion. In this process, we have two important things: initiators and generators.
    • The initiator is the starting shape we begin with, and the generator is a group of smaller copies of the initiator arranged together.
    • To generate fractals, we follow a simple rule. In each step, we replace every copy of the initiator with a smaller copy of the generator. We may also rotate the copies if needed. This rule helps us create more complex and interesting fractals.
  • Dimension is a way to describe how many directions an object can move in. For example, a line is one-dimensional because it only has length. A rectangle or circle is two-dimensional because it has length and width, describing an area. A box or cylinder is three-dimensional because it has length, width, and height, describing a volume.
  • The relationship between scaling and the number of copies needed follows a pattern. In one dimension, the number of copies needed is equal to the scaling factor. In two dimensions, the number of copies needed is equal to the scaling factor squared. In three dimensions, it is equal to the scaling factor cubed. This pattern helps us understand how fractals behave and how they can have interesting properties that don’t fit into traditional dimensions.
  • An imaginary number is defined as [latex]i=\sqrt{-1}[/latex] and is represented by the symbol “[latex]i[/latex].” If we multiply “[latex]i[/latex]” by any real number, like [latex]5i[/latex], we get another imaginary number.
  • A complex number is a combination of a real number and an imaginary number. It is written as [latex]z = a + bi[/latex], where “[latex]a[/latex]” is the real part of the complex number and “[latex]b[/latex]” is the imaginary part of the complex number.
  • The complex plane has two axes: the horizontal axis is for the real numbers, and the vertical axis is for the imaginary numbers. This is used to plot complex numbers.
  • When adding or subtracting complex numbers, we just add or subtract the real parts together and the imaginary parts together. Adding complex numbers can be seen as shifting a point in the complex plane.
  • When can also multiply complex numbers by a real number or multiply two complex numbers. Multiplication by a complex number involves changing the distance from the origin and rotating around the origin.
  • A recursive sequence is a sequence of numbers where each value is determined by a formula that relates it to the previous value, starting with an initial value.
  • The Mandelbrot Set is a special set of numbers that is defined using a recursive sequence. We check if a number is part of the set by seeing if its corresponding sequence stays close to the origin or not. When we plot the numbers in the set on a graph, we get a unique and complex shape that exhibits self-similarity. By counting the number of iterations it takes for a point to move away from the origin, we can create colorful patterns. Zooming in on different parts of the set reveals intricate and mesmerizing shapes.

Glossary

complex number

[latex]z = a + b i[/latex]

complex plane

where the horizontal axis is the real axis and the vertical axis is the imaginary axis

generator

an arranged collection of scaled copies of the initiator

imaginary number

[latex]i=\sqrt{-1}[/latex]

initiator

a starting shape

recursive relationship

a formula which relates the next value, [latex]{{z}_{n+1}}[/latex], in a sequence to the previous value, [latex]{{z}_{n}}[/latex], with an initial value of [latex]{{z}_{0}}[/latex]

recursive sequence

the sequence of values from a recursive relationship

self-similar

when a shape looks essentially the same from a distance as it does closer up

Key Equations

scaling-dimension relation

[latex]\text{Copies}=\text{Scale}^{\text{Dimension}}[/latex], or [latex]C=S^{D}[/latex]

scaling-dimension relation to find dimension

[latex]D=\frac{\log\left(C\right)}{\log(S)}[/latex]