Peano Curve | Research & Encyclopedia Articles

Peano Curve

The Italian mathematician Giuseppe Peano (1858-1932) is best known for creating an axiom system for arithmetic that today remains the starting point for most rigorous developments of modern mathematics, but he is also famous for his construction of a curve that fills an entire planar region. This seems counterintuitive since our usual notion of a curve is that it is one-dimensional. So how could a one-dimensional object fill a two-dimensional region? Peano's curve was something of a curiosity when it appeared in an 1890 article, but a year later the German mathematician, David Hilbert (1862-1943), produced another "plane-filling" curve. At this point some mathematicians began to wring their hands and proclaim such curves to be "non-intuitive," "monstrous," and "pathological"; and they feared that these things threatened to undermine some of the most cherished concepts in mathematics. Another problem for the mathematicians, in addition to the quandary about the meaning of "dimension," was that these curves possessed the bizarre property of being everywhere continuous but nowhere differentiable. Karl Weierstrass (1815-1897), sometimes called the father of mathematical analysis, had produced such a curve, although his was not plane-filling, in 1861, to the dismay of mathematical analysts who relied on their intuition to guide their mathematical research. Continuity and differentiability are concepts at the heart of calculus. In non-technical terms, a curve is continuous if there are no "breaks" or "gaps" in it. A function is differentiable if it is "smooth" with no "sharp" corners or cusps. One of the most important concepts in calculus is that differentiability of a curve at a point implies continuity of the curve at that point, but the converse is not true. A curve may be continuous at a point without being differentiable at that point. Mathematicians were well aware of this--the absolute value function is continuous at the origin, but it has a sharp corner there so that it is not differentiable there. The famous cycloid curve has infinitely many cusps, and so infinitely many points at which it is not differentiable; but the cycloid curve is continuous at all of those points. Mathematicians were used to seeing curves that were continuous but not differentiable at isolated points, but the Peano and Hilbert curves, like the Weierstrass curve, were continuous at all points but not differentiable at any point. Largely as a result of the non-intuitive nature of these curves, they were regarded as exceptions and oddities. They were ignored and kept safely out of sight for about 70 years until Benoit Mandelbrot (1924-) showed them to be members of a class of curves now called "fractals," a term coined by Mandelbrot in his 1975 work Les Objets Fractals. In this work, and in his 1982 book, The Fractal Geometry of Nature, Mandelbrot defined the "fractal dimension" of an object and showed that by this definition the Peano curve had fractal dimension 2.

It is now customary to call all plane-filling curves Peano curves, although Peano's original plane-filler was merely the first of many such curves to be discovered in the 19th and 20th centuries. In The Fractal Geometry of Nature, Mandelbrot shows how to construct fractal curves using an infinitely repeated iterative process. He starts with an initial figure, called the initiator, and then shows a second construction called the generator that produces the next stage of the curve. After that the process is repeated creating the curve stage by stage. At each new stage the figure will look more and more like the fractal being constructed, but the true fractal curve is complete only after an infinite number of iterations. For example, Peano's original curve has a line segment as its initiator. The generator is formed by shrinking the initiator (a) by 1/3 (b) and placing 9 copies of the shrunken piece in the configuration shown below.

Now the process is repeated, shrinking each of the generator segments by 1/3 and placing 9 copies on each segment of the generator. Repeating this process to infinity produces the Peano curve.