Exploring The Grand Canyon’s Fractal Dimensions

Once upon a time, there was a mathematician named Benoit Mandelbrot, who was concerned with figuring out how to calculate the length of Britain’s coastline. Though this may sound straightforward, it’s really not.

Zooming In

The trouble stems from the fact that the boundary between sea and land, like all boundaries, is blurry. A satellite view of a given landmass shows crisp lines, but the more you zoom in, the more texture is revealed: bays, coves, deltas, stream outlets, the saturated spaces between grains of beach sand, and so on.

Indeed, the closer you look at a coastline — the smaller your ‘yardstick’ is — the more distance you seem to find. It’s almost like, contrary to common sense, every island and continent on Earth has a circumference of infinite (or indefinite) length.

Mandelbrot’s solution to this measurement problem is technical, and I won’t delve into it at the moment. Suffice it to say it involves dimension-like textures that hover weirdly between between line and shape.

Added Depth

I couldn’t help but think of Mandelbrot and his mathematics of roughness during a pilgrimage to the Grand Canyon last weekend. Each of the four or five trips I’ve made to the park over the years have felt like a closer zoom-in. As a result, each visit has seemed to add a new measure of depth to the place and a new measure of surface area.

This time around, not only did we hike off the rim and intersect a side canyon or two, we also got a permit to enter Cave of the Domes — one of the countless natural voids vesiculating the Grand Canyon’s Redwall and Muav limestone layers.

No Eye Hath Seen

What caving has done to my perception of the Canyon has proven exceedingly difficult for me to verbalize — perhaps in the same way my visceral understanding of a dimension between one and two is nearly impossible to put into words.

There’s something about knowing there are hallowed halls, vaulted copulas and apses, ceilings dripping with stalactites that no human eyes have seen. There’s something about knowing the universe is by-and-large unfolding without us — opening, ornamenting and then filling in dark and silent spaces according to its own alien timeframe and its own mysterious motives.

There’s something about all this that makes my brain feel as if it’s circuits are being overloaded by electrical current. You feel the immensity of that which you don’t know, that which you couldn’t possibly know, even if you had a million lives to live. While I am resolved to zoom in as far as I can, I am also resigned to the fact that I’ll barely scratch the surface of the surface of the surface.

Learn More:

  • PDF: “How Long Is The Coast Of Britain?” By Benoit Mandelbrot (Science)
  • VIDEO: Hunting The Hidden Dimension (Nova)
  • Caving In El Malpais National Monument, Part 1: Xenolith Cave (Farther Still)

(Today’s photo, which was taken from Horseshoe Mesa, is courtesy of yours truly.) 

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