The Main Idea 

Self-similarity refers to a characteristic where a fractal pattern retains its shape, irrespective of the level of magnification. In other words, no matter how much you zoom in or out, you will encounter the same pattern repeating over and over. This recursive, infinite detailing and scaling is a fundamental property that distinguishes fractals from other geometric shapes, serving as a visual manifestation of certain mathematical concepts and phenomena in nature.

The Main Idea 

An initiator is a starting shape.

A generator is an arranged collection of scaled copies of the initiator.

To construct a fractal, at each step, replace every copy of the initiator with a scaled copy of the generator, rotating as necessary.

The Main Idea 

Fractal dimension is a statistical quantity that provides a measure of complexity in a fractal, representing how detail in a pattern changes with the scale at which it is measured. It goes beyond the traditional dimensions (1D, 2D, 3D) by incorporating the scaling properties of fractals. For instance, a fractal line could have a dimension between [latex]1[/latex] and [latex]2[/latex], depending on how much space it takes up as it twists and curves.

Scaling-dimension relation: To scale a [latex]D[/latex]-dimensional shape by a scaling factor [latex]S[/latex], the number of copies [latex]C[/latex] of the original shape needed will be given by:

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

Scaling-dimension relation to find dimension: To find the dimension [latex]D[/latex] of a fractal, determine the scaling factor [latex]S[/latex] and the number of copies [latex]C[/latex] of the original shape needed, then use the formula:

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