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This image is clearly a set of triangles embedded in a larger triangle in such a way that it can be continually cut into three identical pieces and still retain its internal structure.
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This idea is known as self-similarity {{"self-similar" | cite }}, and it is usually the first aspect of fractals to catch an audience's attention.
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In fact, there are plenty of uses of fractals and their mathematical underpinnings, such as estimating the coastline of Britain {{ "mandelbrot1967long" | cite}}, identifying fingerprints {{ "jampour2010new" | cite }}, and image compression {{ "fractal-compression" | cite }}{{ "saupe1994review" | cite }}.
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This idea is known as self-similarity {{"self-similar" | cite }}, and it is usually the first aspect of fractals to catch an audience's attention.
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In fact, there are plenty of uses of fractals and their mathematical underpinnings, such as estimating the coastline of Britain {{ "mandelbrot1967long" | cite}}, identifying fingerprints {{ "jampour2010new" | cite }}, and image compression {{ "fractal-compression" | cite }}{{ "saupe1994review" | cite }}.
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In many more rigorous definitions, a fractal can be described as any system that has a non-integer Hausdorff dimension {{ "3b1bfractal" | cite }}{{ "hausdorff" | cite }}{{ "gneiting2012estimators" | cite }}.
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Though this is an incredibly interesting concept, the discussion of this chapter will instead focus on methods to generate fractal patterns through iterated function systems.
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@@ -41,7 +41,7 @@ f_3(P) &= \frac{P + C}{2}\\
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$$
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Each function will read in a particular location in space (here, $$P \in \mathbb{R}^2$$) and output a new location that is the midpoint between the input location and $$A$$, $$B$$, or $$C$$ for $$f_1$$, $$f_2$$, and $$f_3$$ respectively.
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The union of all of these functions (the set of all possible functions available for use) is often notated as the _Hutchinson operator_ {{ "hutchinson-operator" | cite }}{{ "hutchinson1981fractals" | cite}}, and for this case it would look like this:
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The union of all of these functions (the set of all possible functions available for use) is often notated as the _Hutchinson operator_ {{ "hutchinson-operator" | cite }}{{ "hutchinson1981fractals" | cite}}, and for this case it would look like this:
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