In Part 1, I talked about the overlapping ideas of eternity, perfection, and immortality and the seeming paradox of making something perfect:
“…making perfection a reality seems to evoke the mathematical concept of the asymptote— ‘a line that continually approaches a given curve but does not meet it.’ One can approach perfection, but the closer one gets, the deeper the gulf between ideal and real becomes…
But if perfection is unattainable, then it would seem that so is the infinite and the eternal. How could something transcend entropy, decay, and death if it wasn’t perfect? How can one craft something perfect using imperfect tools, or make something perfect out of imperfect materials?
It seems like a paradox.”
You can read the full text in Part 1.
In Part 2, I want to talk about fractals, which seem to be a bridge between the abstract world of eternity and the tangible world of nature, and how we measure the infinite.
Along the way, perhaps it will become clearer how paradoxes may not represent dead ends, but routes to transcendence.
Fractals & Infinite Resolution
Fractals are geometric patterns that defy normal standards of measurement and perspective. Each fractal is made of a single pattern that is repeated again and again without end. Every discrete part of a fractal contains the whole design in miniature, and the more you ‘zoom in’, the more iterations of the design you see. Thus, the level of magnification one uses to view a fractal is mostly irrelevant, because the result will always be identical.
You can’t make a fractal by just copy-pasting any arbitrary pattern—the pattern has to have a kind of symmetry that allows it to nest into itself infinitely. Already, fractals touch on something strange and profound: the idea that a finite pattern can have the hidden potential to be infinite.
Another strange thing about fractals is the way we measure them—they don’t fit into the normal classifications of one, two, or three dimensions, because all of them end up veering into the infinite.
Like a lot of bizarre, abstract mathematical concepts, fractals have a strange relationship with reality. One example is the Coastline Paradox. The idea is that using smaller units of measurement (ie, inches instead of feet) should get a cartographer closer to figuring out the true length of a jagged coastline.
For example, measuring a coastline in inches might produce a length of 216.54 miles, while measuring in millimeters might produce a total measurement of 218.13 miles. The length of the coastline seems to be longer when measured in millimeters, because millimeters are able to account for lengths that are too small to be measured by inches, and thus are not recorded.
By switching to smaller and smaller sets of units, one would expect the closer the measurements get to an upper limit, like 220.43 miles.
Instead, it turns out that the smaller those units of measurement become, the closer their sum gets to infinity. According to Wikipedia:
…if one were to measure a coastline with infinite or near-infinite resolution, the length of the infinitely short kinks in the coastline would add up to infinity.
This paradox became one of the cornerstones of study into fractals, since measuring the dimensions of a fractal required finding a way to measure the eternally repeating patterns as they produced smaller and smaller iterations.
There’s a lot of fascinating aspects to fractals, but I want to focus on two ideas in particular.
Seeing the Infinite Everywhere
First, fractals suggest that the closer one’s perspective gets to an ‘infinite resolution’, the more one is able to see the infinite in all things, even a coastline.
Fractal (ie, repeating, self-similar) patterns are found in all different facets of nature, from the growth of plants to frost crystals to the structure of cells. Blood vessels, river systems, and tree roots all share self-similar, branching structures, despite operating on vastly different scales.
Meanwhile, fractals are closely related to chaos theory, a mathematical field that seeks to uncover the patterns that underlie complex systems that may seem random or unpredictable, such as weather patterns or the population growth of certain animals.
Graphical models of chaotic systems sometimes show data points orbiting around certain areas on the graph, which are called ‘strange attractors’. These strange attractors often have a fractal shape, revealing that there are strongly structured patterns amidst apparent randomness.
If you’ve ever read Jurassic Park, you may have seen iterations of a Dragon Curve (a fractal) accompanied by quotes from the story’s mathematician and chaos theorist character Ian Malcolm, which serve as openings for each section of the book. Malcolm’s quotes (and the fractal itself) demonstrate how relatively simple patterns can quickly develop in seemingly unpredictable ways, mirroring the way Jurassic Park itself devolves from order to disorder by introducing a chaotic, unpredictable force—life.
In short, fractals provide a link between the abstract, geometric world of the infinite and the complex, tangible world of nature, life, and growing things, revealing that the two are linked in a myriad of hidden ways.
Macrocosm and Microcosm
The second interesting idea that fractals touch on is that the microcosm can be identical to the macrocosm.
Fractals may be magnified 1x or 100x without a meaningful difference in their appearance, making ‘big’ (macro) and ‘small’ (micro) interchangeable. This might be demonstrated with a miles-long river system or inches-long blood vessels—despite their vastly disparate size, both might have basically the same structure and may even appear identical when viewed in side-by-side images (see the photos at the right).
Indeed, the very nature of fractals is that each individual piece (the microcosm) is identical to the whole (the macrocosm). There is an inversion of scale, as well as a vertigo-inducing sense of being unmoored from an objective viewpoint—if a fractal is the same at 1x, 10x, 100x, and 1000x magnification, 1x and 1000x become interchangeable, despite apparently being different scales (a seeming paradox).
Normally, one’s perception of reality is colored by a static perspective, which usually focuses on a narrow scale. Humans, for example, are used to a scale of time measured in minutes, hours, and days. Trying to imagine a geologic timescale, measured in thousands or millions of years, ends up straining a normal person’s imagination in the same way trying to imagine the timescale of a nuclear fission or chemical reactions, which may be measured in milliseconds or microseconds.
Human perception prioritizes our own timescale, but the macro-level and micro-level timescales are just as real and important in the big picture of reality. As I’ll discuss later, eternity makes the difference between micro and macro-level time scales almost meaningless. However, the concept of macrocosm and microcosm will also return in a different context—the relationship between the cosmos and the individual.
But before we get into that, I want to talk about something a bit more familiar—swordsmanship.
Leave a Reply