In 1967, a mathematician named Benoit Mandelbrot sat down to ponder a deceptively simple question: “How long is the coast of Britain?” You’d think the answer would be fairly straightforward — I mean, shit, we’ve unravelled the human genome and put a man on the moon, so surely we can figure this out, no?
But the problem, as Mandelbrot illustrated in the paper that took its name from the question he was trying to answer, is that the closer you look at a piece of coastline, the more complicated it becomes — look into every nook and cranny and bay and cove, and you’ll find ever smaller nooks and crannies. In other words, the shorter the yardstick you use, the larger the figure you’ll come up with for the distance you’re measuring. In fact, it’s nigh on impossible to get an accurate measurement, because the coastline is infinitely complicated.
The paper caused quite a stir at the time, and Mandelbrot expanded on the ideas therein in a series of subsequent papers. What he came to realize is that the coast of Britain is an example of what he’d eventually go on to call a “fractal” — basically, an infinitely complex system that replicates itself at smaller and smaller levels of magnification, in ways that are similar but never exactly the same.
Fractal equations can be used to create some remarkably beautiful patterns — such as the Mandelbrot set, which bears its creator’s name — and they also turn up in a remarkably broad variety of places in nature. Walk outside and look at a tree — the pattern of its branches is a fractal, with branches dividing from the trunk that subsequently divide into smaller branches, then twigs, then leaves, then the little twisty capillaries that come off the stems of those leaves, and so on. So too are the vascular and nervous systems inside the human body, and the synapses of the brain, and the patterns formed by mountain ranges and river systems.
The theory behind the study of fractals is a fairly recent innovation, but humanity has been recognizing and marveling at mathematical and geometric patterns in nature for millennia — as far back as the fourth century AD, for instance, one Pappus of Alexandria penned an essay called “The Sagacity of Bees”, wherein he pondered the perfectly hexagonal nature of the cells that made up a honeycomb, and why they might be the way they are. The world is full of such patterns, some bewilderingly complicated, some brilliantly simple. Some require no mathematical knowledge at all — even if you have no idea what a logarithm is, you can still pick up a nautilus shell and recognize that there’s some sort of pattern that governs the perfect logarithmic spiral of its shell.
People being people, the temptation has always been to ascribe some meaning or significance to the patterns we discover in the world, and to attempt to replicate or utilise those patterns for our own reasons. Religions the world over have ascribed spiritual import to mystical numbers or designs, and incorporated geometric and numerological ideas into the planning of their sacred buildings. There are plenty of examples: Hindu temples built to reflect the shape of mandalas and nautilus-esque logarithmic spirals in the capitals of Ionic columns to the use of the Golden Ratio in Renaissance architecture and aesthetics. Perhaps the most beautiful, though, are the geometric designs in the sacred art of Islam, which evolved because of the Koranic interdiction on creating images of living things — Umayyad and Abbasid mosques, in particular, featured some incredibly intricate tessellations that reflected mathematical ideas centuries ahead of their time.
The use of this sacred geometry is a physical embodiment of the idea that the beautiful patterns we see in nature have some sort of purpose or deeper meaning. It’s easy to see the appeal of this notion: after all, it’s human nature to want to find explanations for why things are the way they are, and the remarkable geometric symmetry of something like a beehive seems to demand some sort of explanation. You could argue that this is the whole reason that religions come to exist: to provide explanations for things that seem otherwise inexplicable. And, indeed, creationists have long used the simple yet intricate beauty of nature to support the contention that the universe must have some sort of intelligent design behind it (an idea apparently considered less controversial in the US education system these days than the ideas of that old crackpot Charles Darwin, who’d no doubt argue that bees have simply “discovered” through innumerable generations of evolution that a hexagonal matrix is the most efficient way of storing honey, in terms of using the least amount of wax without allowing any honey to escape.)
In any case, even setting aside the dangerous ground of organized religion, the idea that there is some meaning behind nature’s intriguing symmetries is a persistent one. In particular, the New Age movement has picked up the idea of “sacred geometry” and run like crazy with it. (We hesitate to speculate as to the reasons why this might be, although anyone who’s, erm, taken any sort of psychedelics will know that the brain can have itself a fine old time tracing all sorts of geometric patterns on the inside of your eyelids, a fact that we rather doubt escaped your average hippie during the 1960s.)
Anyway, visit any New Age-y shop or website and you’ll be bombarded with an assortment of theories about some sort of unifying geometric code that pervades every aspect of existence. Take this, for instance, from something called sacred-geometry.com: “The strands of our DNA, the cornea of our eyes, snowflakes, [Innumberable other examples — Brevity Ed.] … and all life forms as we know them emerge out of timeless geometric codes. Modern scholars ridiculed this idea until the 1980’s, when Professor Robert Moon at the University of Chicago demonstrated that the entire Periodic Table of Elements — literally everything in the physical world — truly is based on … five geometric forms. In fact, throughout modern physics, chemistry, and biology, the sacred geometric patterns of creation are today being rediscovered.”
If this all sounds a bit too Dan Brown for you, well, you’re not alone. As the good folk of sacred-geometry.com acknowledge, scientists generally have little time for such ideas — as such, the idea of sacred geometry has largely lurked around the fringes of the scientific world, being seen as pseudoscience at best and quackery at worst (and, considering that you can rarely navigate any sort of New Age text on the topic without quickly running across portentous rumblings about hidden ancient knowledge and Platonic solids and the energy of the Pyramids, you’ll understand the skepticism it attracts). Still, if you set aside the cosmic theorizing and ideas about ancient Egyptians, the notion that there’s some grand decipherable pattern behind the universe isn’t a hugely controversial one.
Indeed, whether there’s a grand unified theory that unites the macro-scale equations of gravity with the atomic-level theories of quantum mechanics is one of the great unanswered questions of modern physics. As Stephen Hawking famously wrote at the end of his bestselling A Brief History of Time, discovering a grand unified theory would be “the ultimate triumph of reason — for then we should know the mind of God”.
Hawking’s grand rhetorical flourish has been roundly misinterpreted over the years, both by the media and by outraged religious types — as he himself claimed, even if a grand unified theory could be found, trying to use it to explain anything beyond the first couple of seconds of the universe’s existence would be like trying to measure the coast of Britain with a microscope. The harder you look, the more complicated things get, and if you’re trying to look at the whole damn universe… well, you’re going to need an awfully big computer, put it that way.
In other words, even the most grand of grand unified theories would only explain how the universe works. The more vexed question is why it works the way it does, and this is a question that we fear that even the energy of the Pyramids can’t answer.
But ultimately, perhaps the significance of sacred geometry is this: nature has to be the way it is, because if the world was any other way, we probably wouldn’t be here to wonder about such things. This is what’s called the “anthropic principle” — that the universe is the way it is because it couldn’t be any other way for us to be observing it. If this seems self-evident, consider this: the universe could conceivably existed any number of ways. The smallest variation in any number of fundamental constants that govern the environment we live in could have precluded our existence in the first place. And yet, here we are. What does it all mean? Well, that’s something we all have to decide for ourselves, isn’t it?