Looking up through a skylight can open up new dimensions in geometry, especially when the straight lines framing the skylight contrast sharply with the fuzzy, irregular boundaries of puffy clouds visible through the glass.
Clouds seen through a skylight at Hood College in Frederick, Md.
As mathematician Benoit Mandelbrot famously noted in his book The Fractal Geometry of Nature, "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
Instead, a fragment of a rock may look like the mountain from which it was fractured. Clouds keep their distinctive wispiness whether viewed distantly from the ground or close up from an airplane window. A tree's twigs often have the same branching pattern seen near its trunk.
Rugged coastlines, ragged clouds, breaking waves, and fractured rocks are examples of natural forms that resemble self-similar mathematical objects described as fractals.
So clouds, mountains, and trees wear their irregularity in an unexpectedly orderly fashion. Indeed, nature is full of shapes that repeat themselves on different scales within the same object. In all these examples, zooming in for a closer view doesn’t smooth out the irregularities. Objects tend to show the same degree of roughness at different levels of magnification.
Mandelbrot coined the word “fractal” as a convenient label for these self-similar shapes—those structures that look the same on different scales.
The straight lines of the Washington Monument (background) contrast sharply with the branching geometry of a fir tree on the National Mall in Washington, D.C.
Mandelbrot, B.B. 1982. The Fractal Geometry of Nature. W.H. Freeman.
Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.
______. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. W.H. Freeman.
Photos by I. Peterson