![]() Weierstrass’ Monster was already one such curve, existing in some dimension greater than one but not filling the plane. These examples by Peano, Hilbert and Boltzmann inspired searches for continuous curves whose dimensionality similarly exceeded one dimension, yet without filling space. This ability of a one-dimensional trajectory to fill space mirrored the ergodic hypothesis that Boltzmann relied upon as he developed statistical mechanics. ![]() The space-filling curves of Peano and Hilbert have the extreme property that a one-dimensional curve approaches every point in a two-dimensional space. When the iterations are taken to infinity, the curves approach every point of two-dimensional space arbitrarily closely, giving them a dimension D H = D E = 2, although their topological dimensions are D T = 1. 3 Peano’s (1890) and Hilbert’s (1891) plane-filling curves. This was followed quickly by another construction, invented by David Hilbert in 1891, that divided the square into four instead of nine, simplifying the construction, but also showing that such constructions were easily generated.įig. Where Cantor had proven abstractly that the cardinality of the real numbers was the same as the points in n-dimensional space, Peano created a specific example. In this way, a line is made to fill a plane. ![]() This process is repeated infinitely many times, resulting in a curve that passes through every point of the original plane square. At this stage, the original pattern, repeated 9 times, is connected together by 8 links, forming a single curve. Then each sub square is divided again into 9 sub squares whose centers are all connected by lines. Lines connect the centers of each of the sub squares. The construction of Peano’s curve proceeds by taking a square and dividing it into 9 equal sub squares. Only two years after he had axiomatized linear vector spaces, Peano constructed a continuous curve that filled space. What does it mean to trace the path of a trajectory in an n-dimensional space, if all the points in n dimensions were just numbers on a line? What could such a trajectory look like? A graphic example of a plane-filling path was constructed in 1890 by Peano, who was a peripatetic mathematician with interests that wandered broadly across the landscape of the mathematical problems of his day-usually ahead of his time. He was so surprised by his own result that he wrote to Dedekind “I see it, but I don’t believe it.” The solid concepts of dimension and dimensionality were dissolving before his eyes. In 1878, in a letter to his friend Richard Dedekind, Cantor showed that there was a one-to-one correspondence between the real numbers and the points in any n-dimensional space. Cantor set (below) and the Cantor staircase (above, as the indefinite integral over the set). The fractal dimension of the ternary Cantor set is D H = ln(2)/ln(3) = 0.6309.įig. This clumpiness is an essential feature that distinguishes it from the one-dimensional number line, and it raised important questions about dimensionality. But whereas the real numbers are uniformly distributed, Cantor’s set is “clumped”. It is a striking example of a function that is not equal to the integral of its derivative! Cantor demonstrated that the size of his set is aleph0, which is the cardinality of the real numbers. The set generates a function (The Cantor Staircase) that has a derivative equal to zero almost everywhere, yet whose area integrates to unity. Partially inspired by Weierstrass’ discovery, George Cantor (1845 – 1918) published an example of an unusual ternary set in 1883 in “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (“Foundations of a General Theory of Aggregates”). It is a fractal with fractal dimension D = 2 + ln(0.5)/ln(5) = 1.5693. This continuous function is nowhere differentiable. Riemann had asked whether the functionįig. Karl Weierstrass (1815 – 1897) was studying convergence properties of infinite power series in 1872 when he began with a problem that Bernhard Riemann had given to his students some years earlier. This blog page presents the history through a set of publications that successively altered how mathematicians thought about curves in spaces, beginning with Karl Weierstrass in 1872. Here is a short history of fractal dimension, partially excerpted from my history of dynamics in Galileo Unbound (Oxford University Press, 2018) pg. From then onward the concept of dimension had to be rebuilt from the ground up, leading ultimately to fractals. Child’s play!īut how do you think of fractional dimensions? What is a fractional dimension? For that matter, what is a dimension? Even the integer dimensions began to unravel when George Cantor showed in 1877 that the line and the plane, which clearly had different “dimensionalities”, both had the same cardinality and could be put into a one-to-one correspondence. ![]()
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