Only Echoes, Endlessly Repeating — Fractal
A complex geometric pattern exhibiting self-similarity in that small details of its structure viewed at any scale repeat elements of the overall pattern.
[This observation is unrelated to the manuscript, but it refers to an event so unusual for the climate of this region I thought it would be worth recording. Today, July 24, at four in the afternoon, it snowed.
The large and very cold snowflakes signaled to us that the wind currents that made this strange weather event possible drew precipitation from the clouds in the higher atmosphere faster than they could adjust.
Snow in summer is rare, but it happened before in this area, as the weather records from 1816 indicate.
The snowflakes melted quickly, because of the heat embodied in the earth, but not before we had a chance to immortalize their delicate array of intricate shapes, no two alike.
Because this event was as short lived as it was unexpected, it didn’t affect the library grounds, which have been a haven for many rare species of historical plants for the last two centuries.
All the vegetation, from mosses to succulents and from herbaceous perennials to trees shrugged off the late chill, all but the ferns, whose fiddlehead growth will probably be delayed by a month.
The cool and humid weather encouraged the proliferation of snails, to the great distress of the groundskeeper and to our unexpected amusement.
Their shells dazzled in a broad variety of colors, patterns and shapes, which, for some strange reason, remind me of the illuminations of the manuscript.]
Look around you and you will see the patterns of reality, the interlocking pieces and parts that are it and evolve it endlessly, in infinite variations.
We spoke about scale last time, and I purposefully left out of that discussion the kind of scale we will look at now, a very different kind that doesn’t fit inside reality but between it.
The concept of the space that lays between being is one we have great difficulty grasping, but if you ever doubted it exists, here is the proof: the building blocks of all dynamic systems, of everything that grows and adapts, the blueprints of reality itself, exist between dimensions, in a weird kind of scale that measures the sameness and consistency of an assembly, but not its size.
How do you measure sameness?
And yet reality is made of it, you can’t avoid it, it’s everywhere, large, small, in things that are living and things that are not, in your recurring thoughts, in the cycles of history, even in the workings of chaos.
Infinitely complex patterns of infinitely varying scale.
At their core they are infinitely simple too, a recursive game of outcomes derived from a simple equation that is being solved over and over in different contexts.
These patterns are the protagonists of a very old story, one which, once set in motion, can run indefinitely, and whose elements evolve and multiply, getting more and more complicated and dancing between sizes, inducing a strange sensation that the world is unreal, similar to the one you experience in a dream.
Look at the limbs of a tree. They are all the same. Their branching happens when the length of the limb reaches a certain proportion to its diameter, and that proportion repeats all the way to the top, no matter how large or small the limbs are. Look at the repetitive shapes of a mountain, at the self-similar nature of clouds, at the spiral growth of a seashell, at the circular distribution of seeds in the middle of a sunflower. Wherever you see sameness that ignores the big and small you are watching the patterns that lay in between reality model its shape.
After you noticed them, you can’t evade them, because they are everywhere. They run the functions of your body; they model the arrangement of petals on a flower; they guide the waves on the ocean and the wavy sand dunes on the coast; they are made evident in the flight of a flock of birds, in the flow of a river delta, in the organization of an anthill, in the flourishes of music.
This scale that lives between reality doesn’t answer to the rules of logic, it ruffles the fabric of being into smaller and smaller frills, in a way that never has to end, and which can fold infinity inside itself and make it fit in the palm of your hand.
This sounds like a story we tell our children by the fire on a summer night, but it only takes one look at a simple fern frond to know that it’s all real.
Left to its own devices the fern frond will continue to complicate itself around smaller and smaller indentations, making its boundary grow incredibly long, and it is only because of limitations to its size and nutrient availability that it slows down this incredible work of embroidery and eventually stops.
Learned people affirm that this pattern, that exists between dimensions, is more useful in modeling chaos, but what is chaotic about the growth of a flower, about the movements of the stars, about the shape of a starfish?
I dare say that this, if you want to call it chaos, is the nature of perfection, the only way a system as gigantic and complex as reality can organize itself around a unifying principle, the only way in which the sound of a million grains of sand and that of a million different stars can belong to the same harmonious symphony of matter, energy and motion.
I can’t imagine a simpler, more beautiful way to establish hierarchy!
[We isolated the rose drawings from the rest of the manuscript and placed them together in a field pattern, to see whether there are any commonalities between them.
The pattern is eerily consistent, most of the roses are the same size and level of detail, but there is something very unsettling that happens when one looks at it with an unfocused gaze.
Temporary swirls seem to develop, which draw the eye farther inside the field, in a constant motion that draws you into its depth, although the drawing obviously can’t have a depth, and this motion gets faster and faster the more you look at the field, and it spans longer and longer between repeating details.
It feels as if one is drawn into the essence of a rose ad infinitum, and one’s perception of time and length gets stretched to adjust to this increase in speed.
It is highly unlikely that the master conceived the drawings so that that their placement in the field pattern would yield this optical illusion.
We can only conclude that this is a quality intrinsic to the structure of the roses themselves, a hidden potential that only becomes active when the flowers are seen together.]
What is so interesting about these patterns? Do you remember how we discussed last time that the infinite can not fit into a bounded space? These patterns are infinite, and they do. Not only that, but some of them defy another rule, that of relative proportion. In the world of matter, when things organize themselves in structures, the smaller the components of the structure, the closer together they must be, the sun is farther away from the earth than the earth is from the moon, and so on. The structures that fold inside themselves don’t obey this rule. Quite the opposite: the deeper you go into their intricacy, the smaller their components and the farther away they get from each other. Imagine the jagged edge of a coastline. A bigger yardstick will yield a shorter coastline, a smaller yardstick will yield a longer one.
As measuring instruments become smaller and smaller, they get into all its bits and crevices, and, as they become incredibly small, the length of the coastline approaches infinity.
The world is a contradiction, its rules are there just to confound us and they work for a while, deceitfully lulling us into a false sense of security, only to fail us at the most inopportune moment.
These patterns have no dimensions, although they look normal and they are everywhere.
We can’t measure them the way we measure the world, although they make up the world. We can’t measure the rate of their change at any point in the structure either.
They exist outside the way we understand reality, and yet they are everywhere — the waves, the clouds, the growth of plants and animals, the shapes of the dunes, the ocean shores, the arrangement of planets, the births and deaths of stars, our own beings, the probabilities of events, the ebb and flow of wealth, the growth of a city and that of a giant mushroom circle, they govern everything, and yet they slip between dimensions in a world that can not exist in between.
We are used to the fact that in the real world a thing exists as a focus, as a direction, as a surface, or as a space. These patterns do not. A coast line is not really a line. Because it can grow infinitely long, it becomes something between a line and a plane, an in between dimension, right under our noses.
The world is much weirder than you can even think it is.
We make conventions for ourselves to make our lives easier. We find comfort in the things we can count, that we can see in their entirety, that we can break into components. We like perfect geometric shapes and logical consistency, but reality is nothing of the sort. It is an eminently recursive process which breaks through dimensions and doesn’t yield to any of the rules we made for ourselves.
How can things we can see and touch live between dimensions?
The things we see and touch don’t. They approximate these patterns well, but they are finite fragments of their endlessly repeating nature.
The patterns themselves are not real in the sense we attribute to the word, they are mathematical abstractions, organizing diagrams of the physical world.
These patterns are to existence as meaning is to words: a primal structure that can only express itself in local fragments.
If the intrinsic structure of being spans between dimensions, why do we limit ourselves to dimensions at all?
You are right to ask that question, dimensions are unnatural. Everything that exists in our world is an object in space, no matter how linear or flat we perceive it to be. We simplify reality by removing some of its aspects for our own convenience, and in the process we forget it doesn’t really look that way.
Why did reality evolve as a recursive game? How should I know? Why did reality evolve at all? Why does there have to be a reason for it?
As all self-organizing systems, reality doesn’t have a why, it has rules. It’s a giant game, a three-dimensional puzzle of infinite scale. It takes a seed, sprouts it and sets it in motion and after that its outgrowth keeps expanding and complicating into infinity.
Can I go over the measure of sameness again?
The essence of these patterns is that no matter how large or how small they are, taken outside of context they are nearly identical. Not only that, but the ways in which the smaller designs derive from the larger ones are nearly identical. Sameness measures the extent of this quality.
The fronds of a fern are very much like each other, and the indentations that define every one of them follow the same pattern, to smaller and smaller scale, but no two of them are ever alike.
Nature follows its own laws in spirit, not in rigid fashion.
Like a good tailor, it makes bespoke outfits. It cuts patterns according to the same rules, but uniquely designed to fit each individual instance.
What an embarrassment of riches it is for it to create every wave, every cloud, every living thing, every grain of sand, every snowflake, every heavenly body in a way that never repeats!
Can I give an example of a folded pattern that is infinite and yet can fit inside a closed boundary?
Our physical limitations don’t allow us to continue a pattern into infinity, but we have good approximations of this, which imply the pattern could go on forever: a cathedral rose window implies unending variations at the edge, all bound inside a circle, a tree has the potential of infinite growth, even if it never grows to the sky, you can divide a square into four parts, and each of them again, and again, until the grid becomes so fine you can’t even see it anymore, but no matter how close together the lines are, you can always fit another one in between.
Is it possible that if we were to look inside reality, no matter how deep and at how small a scale, we would find the same patterns repeating? Yes, but not in the same way I described above.
As I mentioned, some of these patterns defy the rule of relative proportion, they behave according to the rule of inverse proportion.
It is possible that the more we look inside the workings of matter, the more its repeating elements get spaced out, leaving us with the intermediate states, strangely elongated in ways that make them look very different, when in fact they are the same.
The smaller things get, the more their shapes get distorted and stretched, like gourds in the fall. Some look like giant mushrooms of vivid color, some are ghostly pale and their skin hangs around the bottom like melted wax on a candle, some turn out flat and pointy rosettes that look more like starfish and sea urchins than vegetables, but they are nothing but squashes, bent and twisted out of shape.
As things get smaller and smaller, they get farther and farther apart, and it gets harder to find the seeds of structure inside them, hidden in the vast spaces in between, but they are there, outside our limited perception, going much deeper into the nature of being than we are accustomed to believe.
Even for us humans the desire for recursive variations comes naturally, as if guided by instinct.
This desire gets reflected in art, in music, in architecture, in the embellishment of day-to-day items, in the shaping of our cities. The patterns of reality are written in our blood too.
