Figure 2. The 2D Peano curve is constructed using an iterative procedure. In this procedure, each line segment is folded into eight segments, which preserve the start- and end-point while sweeping out an adjacent 2D area. In the mathematical limit - after the procedure is repeated infinitely many times - the curve touches every point inside the square. This raises the question of whether the curve, first constructed by Giuseppe Peano in 1890 (Peano, 1890), is one- or two-dimensional, or whether this notion of dimensionality is even meaningful. The original report introducing space-filling curves by Peano introduces the concept using mathematical language, but does not include a visualization of the curve. It is challenging to make this seminal concept intelligible and intuitive to the mind and senses. Motivated by Peano’s study, Hilbert’s 1891 contribution included a new, similar curve, complete with a diagram to assist the reader’s intuition.
Five square images in a horizontal row. In the leftmost one, a multicolored line folds at right angles. In each of the subsequent images, each segment of the line folds into equal segments, but the space occupied by the total line stays the same. As a result, the images to the right are increasingly dense and complex.
Aiden, E.L. / - (2009)