Deep Pockets: Zeno, Coastlines, and the Dimension of Scale

All of these things were present–the natural history of the land, my ancestry in relationship to it, California, my ex-boyfriend, mono, fox finding. They were all watching and waiting, to see the outcome of this one insignificantly significant moment of choice on the sunny banks of Deep Eddy: book or boy? In fact is not even just my entire personal and ancestral history that is present, waiting and watching to see if I am going to go for it or not, but the entire cosmos. No pressure.

The universe created itself so it could have experiences just like this one, tiny moments of complicated, stomach twisting, simultaneously exciting and scary. The universe hides from itself, just so it can sit on the edge of its seat in suspense, “Ooh! Ooh! How I am going to deal with this one!?!?” The timeless, eternal omnipresent universe, covers its eyes squealing with delight, “Don’t show me! Don’t show me! Don’t ruin the ending! I can’t wait! Don’t show me!”

It carefully wraps its own presents in the wrapping paper of finite space-time late at night. Then in the morning, it wakes up one tiny part of itself and says, “Let’s go see what the Tooth Fairy/ Easter Bunny/ Santa Claus, brought for you! Oooooh, what is that? What can you do with this one? Do you like it?” The universe delights in delighting itself.

“How about a sun that comes up in the morning and makes pretty colors when it goes to bed at night? And then a moon that glows and changes shape? Ummmm… twinkly stars? How about pine trees and snowy mountains? And squirrels? And Eagles? How about poop? That’s funny, right? And grasshoppers? And parrots? How about kangaroos?!?! Bears? Whoa! Mosquitos? Cockroaches? Humming birds. Ooh oooh – thunderstorms!” When you’re the universe, to continually delight one’s self like this, it helps to keep a few secrets.

But every once in a while, one of those little pieces of the universe that woke up, says, “Wait a second… What is going on here? What is really under all that wrapping paper of finite space-time, of waterfalls, snowflakes, fur, feathers, scales and skin? Where did all this stuff come from? What is it made out of? Who made it? How did they make it? Universe/ Mommy/ Daddy, where did you come from? Why do you do all this for me? How is it that finite space-time is infinitely full? How does time do that? How does consciousness do that? How do time and consciousness, together, fit the infinite expanse of time into a split second?”

The universe desperately wants to answer all these questions for itself. It desperately wants to reveal itself to itself, entirely, completely, with total abandon, wrapping paper torn and crumpled in a corner. But the only way it can communicate to itself is through the wrapping paper of finite space-time. Wrapping paper of words, bodies, leaves, homes, adornments–these layers of beauty, meaning, and remembrance judiciously protect and enshrine the universe’s tender secrets. Because, of course, if all was revealed everything would cease to be.

Every once in a while the fabric rips and adventurous souls peer through. Some fall into the infinite layers behind the wrapping paper. Some stand guard at these portals, showing others the way in and helping them back out. Some of these guards we know as shamans.

Every time the universe turns itself inside out trying to understand itself, trying to get beyond the wrapping paper, all it can do is create more wrapping paper in infinitely more complex patterns and colors, trying to remember how meaningful and non-trivial the wrapping paper is. The wrapping paper of finite space-time, after all, is not trivial. It is an expression the protection, the care, the tenderness, the attentiveness with which we relate to one another, the universe to the different expressions of itself.

How does infinity fit into finite space-time? How does the trivialness of wrapping paper become imbued with infinite meaning? Mostly its just because moments have super deep pockets. I’m talking clown-car deep pockets. I’m talking Mama says, “You better empty your pockets before you set foot in this house!” because she knows there is no telling what is coming out of those bottomless pits–crushed flowers, a red striped rock, two rusty screws, three handfuls of dirt, with worms, five loquats for later, a frog, only three from the freshly hatched nest of tiny, unidentified snakes, etc. At least that’s how it was before video games and concrete took over anyhow.

The universe fits infinity into finitude by nesting, by fitting many small things into and in between a few large thing–pebbles in pockets, in pants, on a child, held in mama’s arms, held in daddy’s arms, held in granny’s house, on ancestral land, amidst an ecosystem, nourished by a watershed, on a goldilocks planet, in the radiance of a generous sun, hugged by an outer spiral arm of the milky way, in a family of galaxies, in a vast, mysterious universe. The smaller something is, the more of them can fit.

Paradoxes: Zeno and Coastlines

Zeno wrestled with this paradox of bottomless pockets, or infinity within finitude, back in 400 BCE. He claimed that it was impossible for Achilles to ever catch a tortoise since as soon as Achilles reached the place where the tortoise was, the tortoise will have moved on ahead, no matter how slowly. The distance between them will grow ever smaller and smaller without ever actually closing. He knew, of course, that people do actually come from behind to win races all the time. This was the paradox, that logical impossibilities played out all the time.

While is there is no consensus among historical scholars as to the point Zeno was trying to make. It seems that perhaps the paradox was his point. Paradox is a fundamental reality demarcating the limits of logic. Zeno points to the infinite depths of smaller and smaller divisions within any continuum. Between any two discrete numbers like 3 and 4, there are infinitely many places to stop in between, 3.5, 3.75, 3.375, 3.1875, etc.

Along the same lines of Zeno’s paradox, 20th century French American mathematician Benoit Mandelbrot (1967) presented a challenge to measure the coastline of Britain . Though this may seem fairly straightforward, his challenge brought attention to the fact that the measurement will depend on the length of the measuring stick used. If one uses a mile-long measuring stick, the measurement will gloss over certain nooks and crannies along the coast. If one uses a yard-long stick, the measurement will take more of these irregularities into account, resulting in a longer overall measurement than the measurement taken with the mile-long stick. The length of the coastline will increase further still with the employment of an inch-long measuring device, incorporating even more irregularities.

This reasoning process then logically concludes that the length of Britain’s coastline is actually infinite. As the measuring stick gets ever smaller, the length gets ever longer. While the length of the coastline may approach infinity, it is actually contained within a finite space, herein lays the apparent paradox.
How, then, might one measure a lifetime? Measurement in years, months, or minutes may all add up to the same length of time, but does that really include the textured contours of each moment? Are our temporal measures simply smoothing over the infinite depths present in each moment? Infinity is not so easy to work with, so physicists and mathematicians had long chosen to ignore the true depths of such mathematical monsters.

Topographic Textures

Similarly the a landscape’s surface contains infinitely many scales of texture–the jaggedness of a mountain vista slopes enfolded with the hills and ravines, with boulders facilitating stream crossings, the contours of a rocky trail underfoot, the points and flatness of each individual rock.

A map may not reveal all these textures, but a topographic map begins to hint at a landscapes internal complexity by offering lines of elevation. A regular map may suggest that it is one mile between two points, only to discover that in real life that mile also includes 1000 feet of elevation gain. San Francisco reveals the importance of topography, where gridded streets march laboriously straight up and down steep sloped hills. The city layout was planned without the benefit of a topographic map.

To get the idea of how a topographic map works make a fist and use a marker to trace circles around each knuckle—a small circle at the top of each knuckle then a wider concentric circle a little further down until the circles merge into one large circle around all your knuckles. Then flatten your hand out. The circles then illustrate on the flat map of your hand, where the peak of your knuckles emerge when your fist in in three dimensions. Similarly, a topographic map reveals where a landscape’s peaks and valleys lay. Elevations lines stacked on top of one another reveal sheer cliffs and gentle slopes roll out from between more broadly spaced lines. In this way a topographic map includes an additional dimension of description beyond that of a normal map.

Beyond the landscape of the earth, we also traverse the landscape of time, conveniently flattened for ease of navigation until we came to believe that that was all it was. Early in the 20th century however, General Relativity revealed that space and time both have contours more akin to high mountains and deep valleys than the flatness of a black and white checked floor.

Like a topographic map, general relativity shifted our conception of space-time from flat to textured, by gravitating matter. Just as topographic maps can offer a more accurate map of the terrain, relativity might also hold some wisdom for more accurately describing our textured experiences of time, a topology of time, which we navigate and co-create. The mathematical study of topology has to do with boundaries and continuity. Topography specifically deals with the earth boundaries. The topology of time, as I propose, deals with the boundary between matter and energy, time as it emerges from the frontier of timelessness. The fractal, multi-scaled, continuous, and non-differentiable, finite and infinite nature of this boundary is the topological issue that concerns me.

Imagine the flat map as a timeline and the topographic contours as the subjective experience of time. Sometimes one experiences time as laboriously slow, like moving uphill, and sometimes it proceeds at a breakneck pace, flying on the down hills. The flat map indicates the steady progression through time, but gives no indication of how it feels to traverse that particular stretch of time, whether arduous and slow going or pleasantly flowing. These contours describe the topology, or the texture of subjective time.

A topographic map however mostly still only describes just one or two scales of textures. The scale described depends on the size of the measuring scale used, as in the coastline paradox. Consider a map of the roughness across one square mile of land, with elevation measured in hundreds of feet. Then consider a map of the roughness of one square foot of land within that square mile, with elevation measured in inches. The roughness of these two maps might look quite similar, suggesting that at any scale, nature offers a similar degree of roughness. Our body size simply limits the ranges of our participation.

Topographic maps offer a fairly detailed picture of the contours of the landscape with quantified lines of elevation, but there is another level of detail they can only hint at. Only a fractal topology might begin to describe the infinite textures and scales that constitute our actual experiences of space and time.
While vegetated areas are often indicated in swatches of green on the map, the map does not describe the denseness or height of the vegetation in various areas. A hiker might find themself lost in the thick of the undergrowth, unable to see two steps ahead or behind, or in open old growth forest making quick time at the feet of giants, or perched on a peak high above tree line surveying the vast expanse of past and future journeys.

One might find similar textures in their experience of time. There are times in one’s life where one’s memories or plans seem muddled and unclear, in the thick of the undergrowth, and peak experiences where it all seems to make sense and fit together as part of a larger plan revealing a clear path forward. The present moment, like the landscape, is infinitely dense with information, stored both in our own bodies, the body of the land and the life that inhabits it.

This nesting of scales within scales, memories within moments, the infinite within the finite, and the finite within the infinite is exactly what makes time so mysterious. It is also what makes fractals the perfect mathematical candidate for describing the mysteries of time. Like time fractals describe degrees of roughness across multiple layers of scale, offering a description of the texture of the mutual enfolding of infinity and finitude.

Fractal Monsters

nature fractals

Mandelbrot’s coastline challenge revived the fundamental mystery of Zeno’s paradox, held safe for centuries in the bookshelves of philosophy until mathematics was ready to seriously consider it. He pulled the paradoxical monsters off the bookshelf, dusted them off, and gave them a proper introduction to mathematics. He called these little monsters fractals. Once mathematicians learned to recognize them, it soon became obvious that we were surrounded, embedded, and even made up of these bizarre little creatures.

Fractals appear everywhere in nature–the smaller and smaller leaflets of unfurling ferns, branching river deltas and culminating watersheds, jagged mountain horizons, and the successive subdivisions of the veins and arteries that circulate our blood. If these patterns show up everywhere in nature, might they be revealing a common underlying structure of space-time? A possible mathematical description of fractal time is what attracted me to Nottale’s theory of fractal space-time.

For a long time mathematics did not have the language to describe the detail and complexity of fractal structures. Yet since the 1800’s mathematicians occasionally produced geometrical objects whose structure and complexity equaled that of naturally occurring fractals’, failing to obey the laws of smoothness. Smatterings of these beings litter mathematical history, bearing their discoverer’s names: Weierstrass, Cantor, Klein, Poincare, Koch, Serpenski, Julia, Hausendorff, Levy, and Mandelbrot. The ability of mathematics to create, with just a few simple equations, structures as detailed and complex as those found in nature revealed a whole new layer of mathematical magic which Mandelbrot revealed to the world.

Mandelbrot began collecting and cataloging these unpredictable mathematical creatures into a book called “The Fractal Geometry of Nature.” He refers to his collection as a “scientific casebook,” a collection of like-themed cases such as a physician or a lawyer might have (1977, 2). Perhaps it also mimics a biological field notebook or taxonomy, following the inclination to organize and classify when confronted with the infinite diversity of the natural world.

Through collecting fractals, Mandelbrot identified several common characteristics. Under their unpredictable, monstrous wildness, lurked a desire to relate, organized by a mirroring of small patterns within larger patterns. The monsters had babies. This revealed their soft approachable side, through which we could understand, relate to them, and be reminded of our own softer sides.

For a long time mathematicians and physicists ignored these rough objects because their irregularities make them seem unpredictable, unmanageable, and wild. Mathematicians call them non-linear and non-differentiable. They preferred to work with smoother objects and equations, because pretending that the real world is smooth makes it seem more predictable and easier to work with. Smoothness is differentiable, and thus predictable.
Likewise, I wanted my life to obey the smooth lines of my intellect. However, as Mandelbrot pointed out, “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,” (Mandelbrot 1977, 1) To which I would add, the heart is not logical and time is not a line. The depths of the heart build time’s non-linearity, bringing infinite unpredictability to each finite, predictable moment through a desire to connect.

As 17th century French physicist and philosopher Blaise Pascal offered, “The heart has its reasons, which reason does not know. We feel it in a thousand things. It is the heart that experiences God, and not reason. This, then, is faith: God felt by the heart, not by reason.”

My heart and the heart of the world led me deeper into the mystery of my larger Self, through my break up and my fox finding. What these moments did for me, Mandelbrot did for mathematics. He brought the unpredictable heart alive within the cool exterior of mathematical predictability. It is the very desire to connect, to relate, to reflect one another that creates the unpredictable wildness that we desperately need and that tears apart our carefully constructed, carefully defended, domesticated personas.

This is an excerpt from my upcoming book “The Texture of Time”


One thought on “Deep Pockets: Zeno, Coastlines, and the Dimension of Scale

  1. It would appear my landing on this page was fated. Of all the pages my unrelated google search could have taken me here I arrived at something deeply relevant to whats been on my mind as of late. A sign that I’ve been looking in the right direction or that my programming will soon come to an end that I cannot know yet.


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