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The mathematical waterfall March 31, 2010

Posted by Jenny in hiking, nature, Southern Appalachians.
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Falls on Grogan Creek, Pisgah National Forest

If you happen to visit a waterfall in the company of a pair of mathematicians, it may occur to you to see things in a different way.  I think it has something to do with the link between complexity and beauty.

Last Saturday I went on a hike with my friend Gary and his 16-year-old son Zach.  I’ve known Gary since college.  He is a professional mathematician who likes to tinker with algorithms, and Zach (and his sister Noura) grew up playing with math.  Zach could certainly follow in the footsteps of his dad, if he decides to do that.

We did a variation of a hike that I tried out back in January to Cedar Rock Mountain. We started at the Fish Hatchery trailhead in Pisgah National Forest, went up John Rock, and had lunch at Cat Gap on the Loeb trail.  Then we followed the Loeb trail to the unmaintained manway that goes up to the top of Cedar Rock, finding a lot of blowdown still lingering from the winter’s bad ice storms.  We met someone on top who told us the blowdowns had been at least partially cleared on the section of the Loeb trail west of there, so we retraced our steps to the trail and followed it to Butter Gap.  Then we went down the Butter Gap trail to the Fish Hatchery.

View from Cedar Rock Mtn. (taken in January)

We could have shortened our route considerably by descending via Cedar Rock’s other manway, the one on the northwest side that goes directly to Butter Gap.  But since that is said to involve tricky scrambling over steep ledges, I would first like to scout that from the bottom up.  For future reference, I noted where the path starts at the gap.

By the time we descended the Butter Gap trail, it was getting into the phase of the afternoon when the sunlight takes on a golden color.  Suddenly we heard the deep roar of water on rock.  There we were at the top of a waterfall.  It was hard to see very much, because the water simply disappeared over the edge.  But just down the trail we found a steep little side path that led to the bottom.

The water as it fell looked creamy white.  It cascaded over a series of ledges until it reached a final flat stairstep of rock, where the water somehow seemed to turn clear again (see top photo).  From this perspective, the water at the top glowed bright.  It came out of nowhere, as if out of a slot cut into the side of the mountain.  All around us, the pillars of hardwood trees gleamed in the afternoon light.

The falls does not have a name, and in the “Waterfalls of North Carolina” map used by western NC waterfall aficionados, it gets a “beauty rating” of only 4 out of 10.  I guess I can understand the relatively low rating.  After all, people who take waterfall aesthetics seriously have to maintain their standards! This one isn’t all that high (about 20′), and it doesn’t have any unusual bumps or bounces or dramatic, photogenic features.  But something about the simplicity of the waterfall’s overall shape made it easier to see how incredibly complicated it really was, in what was going on with the flow of the water.

It was Zach who pointed out that in a particular spot he was watching, a strand of water was dividing into two strands, then merging, then dividing.  I went over and looked at it.  The path of the water changed back and forth in a kind of pulsation.  It seemed that some sort of limit was reached in each half of the cycle, and then the pattern flipped over to the other half.  I probably wouldn’t have seen it this way, or seen it at all, except that I was in the company of mathematicians.

Zach and Jenny

The thing of it was, the tiny little portion of the water we were looking at represented—of course!—only an infinitesimal part of everything that was going on with the water flow.  Before our very eyes, an astronomical number of droplets were tumbling, gliding, bouncing, merging, separating, flowing in an unending sequence.  It was absolutely and ridiculously complicated!

We continued on our way, admiring a couple of well-crafted beaver dams in the lower section of the stream, looking for early spring wildflowers (we saw none) and for songbirds (none of those either), eventually wending our way back to the Fish Hatchery.

Gary found the best seat for viewing the waterfall