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Fractals and Self-Similarity
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Cantor Set
Characteristic: Constructed by repeatedly deleting the middle third of a line segment. Dimension: Approximately 0.630.
Mandelbrot Set
Characteristic: Complex structure at any scale. Dimension: Fractal (non-integer) dimension, approximately 2.0.
Sierpinski Triangle
Characteristic: Triangular shape with smaller triangles removed. Dimension: Approximately 1.585.
Julia Set
Characteristic: Each set is associated with a specific complex parameter. Dimension: Fractal (non-integer) dimension, varies with parameter.
Koch Snowflake
Characteristic: Starts with an equilateral triangle; each iteration adds smaller triangles to each side. Dimension: Approximately 1.261.
Dragon Curve (also known as the Harter-Heighway Dragon)
Characteristic: Folded paper dragon curve. Dimension: Exactly 2.
Menger Sponge
Characteristic: Three-dimensional fractal formed by removing cubes. Dimension: Approximately 2.726.
Barnsley Fern
Characteristic: Resembles a natural fern with self-similar structure. Dimension: Approximately 1.85.
Sierpinski Carpet
Characteristic: Square shape with central square removed, repeated recursively. Dimension: Approximately 1.892.
Apollonian Gasket
Characteristic: Composed of mutually tangent circles with curvilinear triangular gaps. Dimension: Varies, but often around 1.3.
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