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Types of Peano Curves

Perhaps best known as the founder of set theory and mathematical notation, Giuseppe Peano was an esteemed Italian mathematician at the University of Turin. In addition to contributing innovative new concepts to mathematics, he also worked to write many textbooks; during his life he even wrote a dictionary for mathematics. For his work, Peano was made a Knight in the Order of the Crown of Italy in 1905. Today, his name is remembered through its application to the title of mathematical concepts, including Peano axioms and Peano curves. The latter deals with concepts that can be easily applied to artistic expressions.

What are Peano Curves
- Peano curves are any type of fractal that has a dimension of two. This extremely basic definition can be stretched to cover a multitude of various appearances of said Peano curves, and because of this, you must look further into the meaning of a ̶0;two-dimensional fractal̶1; to fully understand what forms Peano curves may encompass. All Peano curves are made up of base-motifs, and the original ̶0;Peano curve̶1; was composed of a base-motif square form. Peano curves are then so full of twisted curves that the designs are two-dimensional in nature. While Peano curves include all fractals with a dimension of two by definition, the following are three examples of better known Peano curves.
Cesaro̵7;s Sweep
- The base for the Cesaro̵7;s Sweep fractal is a simple horizontal line, while the motif is a pair of lines that form an angle together. The particular degree of the angle between the motif lines will determine the appearance/form of the particular version of Cesaro̵7;s Sweep. Like all Peano curves, Cesaro̵7;s Sweep is a fractal that can be iterated an endless number of times, resulting in a fractal of indefinite length.
Polya's Sweep
- Very similar to Cesaro̵7;s Sweep in both its base and motif, Polya̵7;s Sweep is another type of Peano curve. This one is made by an alternation between the main and flipped versions of the base-motif pattern, creating a unique pattern that can also be iterated an indefinite number of times.
Paper Folding Fractals
- Paper Folding fractals is a larger overarching title for several forms of these fractals. The Dragon Fractal is just one type of such Paper Folding fractals that also fits within the definition of a Peano curve.

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