Roughness, Self-Similarity and Chaos

Karl Weierstrass presented this function to the Prussian Academy of Sciences on July 18, 1872, as the first published example of a function continuous at every point yet differentiable at none. It overturned the assumption, held by mathematicians including Gauss, that such failures could only be isolated, and Charles Hermite called such functions a lamentable scourge.

Georg Cantor described this set in 1883 by repeatedly removing the open middle third of every remaining segment of the interval from 0 to 1. What survives is uncountably infinite yet has Lebesgue measure zero, and it consists of two scaled copies of itself, a foundational fractal before the word existed.

David Hilbert described this continuous curve in 1891, refining a construction Giuseppe Peano had published the year before. Though a one-dimensional curve, its iterations come arbitrarily close to every point of a square, and its locality-preserving property makes it useful in database indexing, image processing and computer graphics.

Wacław Sierpiński described this fractal in 1915, built by removing the central triangle from each remaining triangle, forever. Its Hausdorff dimension is log 3 divided by log 2, about 1.585, between a line and a plane, and marking the odd entries of Pascal's triangle approximates it.

Gaston Julia and Pierre Fatou published their foundational papers on complex iteration in 1918 and 1917. The Julia set holds the points where iteration behaves chaotically, so that an arbitrarily small change in the starting point gives drastically different outcomes, while the Fatou set holds the points where nearby starts behave alike.

Edward Lorenz developed the three equations behind this attractor in 1963 while modeling atmospheric convection. Fully deterministic, the system is so sensitive to initial conditions that nearly identical starts diverge, the idea behind the butterfly effect, and its butterfly-shaped trajectories made it a founding example of deterministic chaos.

Robert May popularized this simple feedback equation in a widely read 1976 paper, building on Verhulst's logistic equation for population growth. As the parameter r rises, behavior passes through period doublings governed by the Feigenbaum constant and becomes chaotic around r equal to 3.56995.

Benoit Mandelbrot obtained the first high-quality computer visualizations of this set in 1980 at IBM, though Fatou and Julia had studied the iteration decades earlier. Its boundary is a fractal curve revealing new detail at every magnification, and a value c belongs to it exactly when the matching Julia set is connected, so it maps every connected Julia set at once.