Strides Across the Immeasurable

Looking towards Moriches Inlet, photo by author

Early last spring I took a walk along Westhampton Beach, heading west from Cupsogue State Park towards the Moriches Bay Inlet. It was early in the year, the park was only sparsely populated, and as I got closer to the inlet I found myself totally alone, even though as the crow flies I was less than 60 miles from New York City. I was alone by chance — like the leading edge of the wave that rolls up the sand bank almost, but not quite to my feet — the population of the city, thinned by the distance to the Hamptons and the earliness of the season, and filtered by the need to take a long, slow drive back westward towards the park on Dune Road, left only a few people in the park, and none where I was, walking slowly west towards the inlet.

I enjoyed the solitude. My inner silence was punctuated by the slap of my bare feet in the cool, wet sand. Successive waves would approach and return. Occasionally one of them would make it all the way to my feet, submerging them in the final heave of the wave before it receded back to its ocean. I looked westward towards the inlet, across the meandering front of the waves, and wondered how far away the inlet was, exactly.

I wanted to reach the inlet before I turned back, and since it was taking longer than I expected I considered straightening my path to shorten the distance. I remembered something I leaned in school: A shoreline really has no length. The inlet may be just a half mile or so by line of sight, but when my eye follows the zigs and zags of this moment’s breakers, I can see that the distance I would need to walk will increase the more closely I try and follow the meanderings of the wave edges. And as I look downwards at the last few inches of wash, I can see that to try and follow even a moment’s arrangement of the waters edge would require more steps, further increasing the distance from the straight line of sight.

In my imagination I can shrink myself down to the size of one of the littered crab shells and note that the wave edge I try and skirt would require, at any particular moment, a jagged path with edges perhaps an inch in length. Finer still, if I were one of the sand gnats that flit along the drying edge of the last watery surge, my hops along the edge would be segments just a few millimeters long. Such fine tracings along this moment’s edge would increase the distance even more.

From my days studying fractal geometry, I know that for some boundaries, each increase in resolution, down to and beyond even the atomic scale, new levels of jaggedness are introduced in edges that seem straight at lower magnifications. The length of a perimeter increases with each increase in resolution. In the abstract model of something that has a fractal dimension, like a shoreline, there is no limit to the jaggedness, and hence the length, and so the infinite sum of segments is divergent. Each magnification of a fractal shoreline yields a new shoreline, just as rough, and so the length increases without limit.

The mathematical model of a shoreline has a fractal dimension, and so the abstract shoreline model always increases in length with finer and finer measures, where the smaller and smaller segments sum to larger and larger lengths. But the shoreline I walk along is not abstract, it is real. My feet are either in the water or just outside it. For a moment I feel comforted by the clarity, until I realize that the real shoreline has nuances that make its length just as indeterminate as the abstract fractal model.

The next wave surges past my feet and the cold surf curls around my ankles. When the wave retreats, my abandoned feet sink down as some of the sand beneath them is pulled back with the wave. I notice that the retreating wave leaves behind a darker, shinier region of lingering moisture on the shore, marking the farthest reach of the wave. My eyes follow along the jagged boundary of that wetness, a boundary that moves rapidly as the wetness evaporates in the hot sun. The edge of the darkened sand slides down towards the ocean, as if it were just the shadow of the wave that abandoned it. It never quite reaches its parent wave before a new one rushes in to bury it.

What is the true boundary between the earth and the sea at any particular moment? If I make a gross measurement with my footsteps, should I walk along the edge of the waterfront, or further inland along the line of wet sand holding the remnants of the preceding wave? The slower, but more accurate crab would face the same decision, especially when a new surge rushes up past the prior high-water mark in the sand.

So even though the shoreline is a real, tangible thing in the world, and not some theoretical, abstraction that scales to infinite detail, it doesn’t have the plainest of physical measurements: a length. And even if I could decide on a measurement policy — wave front or wet sand ?— and a scale, it would still be immeasurable.

Suppose I were as small as the sand gnats that flit along the wet sand. I might suspend my flight, and alight on the the very edge of the wet sand, my left legs wet, my right ones dry. I could mark my start and begin to count my steps as I race along the boundary, running the total steps into the hundreds. But then a newly arriving surge lifts me aloft and carries me up the sand bank. I would be cast upon a new region of wet sand. I could resume my count there as the sand begins to dry, but it would be pointless, because there was be no accounting of my time and distance when I was carried by the wave.

Even supposing we could account for that ride in the surf, and add to the sum of tiny footsteps the distance the gnat was carried on the surf, we’d still have a long way to go to measure the wave edge exactly. Even the tiniest of creatures looms like a galaxy when pondering the edge of the wave on an atomic scale. You could take our gnat and shrink it down to molecular scale, like the tiny submarine in the 1966 movie Fantastic Voyage, and you’d still have problems making exact measurements of the wave. Because at this scale it’s just a foam, primarily consisting of water and salt molecules, and to measure it you have to decide whether a droplet flung by a wave as it smashes against the sand should be considered part of the wave or not. Should the tiny, tiny insect riding on the lip of the wave ride the droplet as it leaves and measure that distance, or stay with the parent body of water?

As we strive for more exactitude, and consider the surf at smaller and smaller scales, new problems arise. At the atomic scale we have the question of whether a water or salt molecule is part of one wave, or the evaporating remnant of a prior wave. Now that we’re at the atomic scale, quantum mechanical issues arise: where precisely is the molecule? And which way is it moving? Quantum uncertainty says you can’t determine both at the same time.

It’s worth stopping here and pointing out that there is a lot of distance accumulated by tiny turns back and forth at the atomic boundary of the wave. It’s estimated that one coiled strand of DNA — a single molecule — inside one of our cells would be about two full meters in length if it was uncoiled. And so atomic scale edges matter a lot when measuring the length of a shoreline.

Suppose we could shrink down to the atomic scale, and attempt a measurement. We’d find that quantum effects and the nature of time make that impossible. Measuring the distance between a segment anchored by two molecules that might be at the extreme edge of the wave would require fixing their position while the distance between them is measured. But this would make their velocity uncertain, and velocity is needed to determine whether a water molecule in the sand is part of one wave or the remnant of a prior one.

But even if we could discount quantum uncertainty, and make a precise measurement of the shore at the tiniest scales, we’d also need to stop time, letting the wave hang suspended in place for a moment while while we traverse atom by atom along its farthest edge. When time resumes, the dammed up energy inside the wave will uncoil, pushing the wave to rearrange itself on the shore and ocean, radically altering any measurement we just made.

But we forgot something! We can’t really stop time, because measuring the distance between two molecules on the edge of a wave requires time: time to spot one molecules, and somehow fix its exact position while we search for the next molecule along the very edge of the wave. Identifying molecules, and using their position and velocity to assign them as wave, or not-wave, is a mental process that requires time. So is the process of adding the segments between molecules into a running length total.

It feels disorientating, to realize that even as I step along the edge of the shoreline, feeling the sudsy coolness stream past my toes, I am feeling something that is as insubstantial as the nimbus clouds that drift slowly above. Something that never stops changing, the shoreline’s curling edges twist and meander along all the beaches of the world. Beaches equally immeasurable in their turn, for the same reasons.

So the shoreline is a real, physical thing I can walk along, yet it is immeasurable. It belongs in the world, but in a way that is more cloudy, less substantial than any single, hard thing with a clear boundary. Any attempt to precisely measure the shoreline, and fix its exact dimension, drops our minds deep into the timeless, foamy chaos: the loom we are drawn on; the ground of our being.

I reach my goal. I arrive at Moriches Inlet, and turn around and head back towards Cupsogue Park. I see that all but the last few footprints of my western path are eroded, having settled back into the wet sand left by the waves. I realize that I am every bit as immeasurable as the shoreline. I am no more substantial than the edge of the ocean. My body may seem hard and substantial, and far less volatile than the breakers that fling themselves again and again onto the shore. This seeming stability is just an illusion of time.

Accelerate time and it becomes clear that I am myself but the leading edge of a wave, not a wave of water but a wave of people. A molecule launched by the city to the west, having no more control, no more enduring result than a molecule of salt or water jostled about in the sea.

It is good, this commonality between myself and the surf I walk along. The very comparison I make between us is a measurement of common features, and like all measurements it is anchored to the pace of time. Time at my scale sees the named, individual person, whose feet feel the cool touch of the next nameless breaker. Reset the metronome, and time at the ocean’s pace reveals me as just some tiny thing, come alive for its instant in the melt of the great glacier.

Photo by Enelia Sandoval on Unsplash