Today, the Peano Curve is recognized as just one of an infinite class of familiar objects known as fractals. But at the end of the 19th century it was an extravagant, completely counterintuitive thing; indeed, it was something that had been believed impossible. Writing of Peano's result in GrundzŁge der Mengenlehre in 1914, Felix Hausdorff said: "This is one of the most remarkable facts of set theory."
Originally, the Peano Curve was derived purely analytically, without any kind of drawing or attempt at visualization. But the first few steps in drawing it, as shown in the diagrams, are easy enough, even though the finished product is unattainable in this way – and totally unimaginable. To fill the unit square, as the Peano Curve does, without leaving any holes, the curve has to be both continuous and self-intersecting.
Related category FRACTALS AND PATHOLOGICAL CURVES
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