(this article is a result of my study of fractals and their relationship to science and spirituality)
Buckminster Fuller and Benoit Mandelbrot (among others) show us that the world is connected fractally, over scales, meaning that when we look into something that appears smooth on the surface, we actually discover entire worlds within. These worlds often contain repeating, self–similar patterns. The universe itself seems to be arranged on a scale from smallest to largest (the quantum scale all the way up to the scale of galactic superclusters) but how far down does it go, and how far up? Are these micro and macro scales separated at the top and the bottom –– do they continue on to infinity –– or are they somehow connected?
How we answer these questions will formulate our worldview. For at least the past century, mankind has considered himself largely separate from the world around him, and these ideas have been reflected in the way we organize our societies and how we act toward each other.
Fractals and Paradoxes
The fractal geometry of nature requires that physical effects occur over scales, but these scaling effects are completely invisible to the human senses, or even to instrumentation.
Mandelbrot’s analysis of the coastline of Britain, for example, shows that it’s traditionally measured perimeter (using a measure ε, which gets smaller and smaller as we examine the coastline more carefully) is essentially infinite in length, unless we measure it fractally.
When one looks at the coastline, of course, one identifies more and more rocks, shallows and indentations the further one magnifies the area, increasing the size of the perimeter. However, when we talk about the coast of Britain in normal conversation, we are only talking about that which is visible to our perception. The complexity of the underlying scale of magnification is hidden from us.
The fractal nature of reality resolves apparent anomalies like the Paradox of Zeno, or Olbers Paradox.
The Paradox of Zeno says that it is really impossible to travel from point A to point B, because one must first traverse half that distance, then half of that distance, then half of that distance, and so on forever, so that one can never reach point B. What Zeno is describing is the fractal nature of reality, in which self-similar patterns reproduce themselves over micro- scales. Of course, we simply walk the distance between A and B, never thinking about the underlying fractal / scalar reality.
Olbers Paradox says that since the stars are distributed approximately evenly throughout the sky as visible from earth, as we look deeper into space we should see a greater and greater volume of stars, so that the night sky should be brilliantly lit. We know that the distribution of stars is not uniform, however, and so a line of sight into the celestial sphere will encounter many more barren areas of space than of stars. But we are not aware of this, because the galaxies are distributed fractally, which is not apparent to our sense of sight.
When one examines nature, one discovers more and more irregularity and broken-ness. The coastline of Britain looks smooth from a satellite photograph, but appears more and more fragmented, the closer we zoom in on it. One discovers shoals, and small bays, islands and rocks, all of which look more and more irregular and fragmented. When one looks deeply into something, as in the fractal demonstration above (see link) brokenness is evident everywhere: upon magnification, even a smooth curve is irregular. In other words, the apparent continuity of a surface disappears as we magnify into it (Buckminster Fuller claimed that a sphere was a series of finely broken lines from the construction of a polyhedron). And if we examine the surface of the earth, which looks so smooth from outer space, we see the broken-ness of mountains, seas, forests, and the irregular contours of ground. Look out the window and see if you can find a curve in nature. If you can, examine it more closely to find the irregularities.
These phenomena have lead mankind to conclude that nature herself is broken and fragmented, and that we are separate and individuated from nature. But this separation is an apparency.
Beauty and Continuity
What is beauty?
It is generally acknowledged that beauty comes from symmetry. And symmetry is regularity. Studies show that a face that is considered beautiful has symmetry and regularity: eyes evenly spaced, nose and lips set in the middle of the face; in general, features that are regularly spaced and balanced. In general, human beings find curves more appealing than angles. A beautifully curved automobile is much more attractive and stylish than a square, boxy one. A building with appropriate curves is more aesthetically appealing than utilitarian designs. One need only visit Moscow and examine Soviet architecture, then travel to St. Petersberg and examine the glorious buildings there.
Why is the Pieta so admired? Because of it’s lovely curves!
Why is nature broken, yet our concept of beauty revolves around curved, continuous surfaces? Is nature messed up, or are we?
The problem arises simply because it is impossible to physically construct a perfect curve. A curve is an ideal, never to be actually attained, and so it appeals to us aesthetically.
The chair I am sitting in, for example, is composed almost exclusively of smooth, curved surfaces, but a close examination shows tiny “rough spots.” No matter how finely you construct a curve, it requires a change of direction at every single point. It is at that point of change that broken-ness occurs.
To go from B to A in a perfect curve seems trivial. However, every point along the curve requires a change of direction, which generates an irregularity.
The perfect circle actually looks like this:
Every point along a circle, no matter how finely you look, has an “edge.”
As Fuller and Mandelbrot point out, the examination of anything in nature leads to the discovery of discontinuities.
Connected–ness and Singularities
All physically constructed forms have an error tolerance built-in, for that is the nature of physicality. There are always trade-offs, due to the complications of scale. When you machine a part for a device, that part has to comply with specifications, which means that it has to be manufactured to a certain level of detail and quality. Beyond that, no one cares, because the errors are not visible.
However, all physical systems are affected by forces and fields, down to the micro- and upward to the macro- levels. When one heats a contained gas, for example, that heat becomes distributed throughout the entire system, even though it is impossible to describe how this occurs on all scale levels within the container. For example, scientists who attempt to describe fluid dynamics mathematically often have a very difficult time, because it is simply impossible to track the movement of individual molecules, even though it is obvious that the fluid as a whole is responding in a certain way to forces placed upon it.
All scale levels in the universe are connected. For every action, there is a reaction at all scale levels, even if those reactions are not visible at the scale level of perception.
Science has determined, for example, that the quantum level is the lowest scale that we need consider. This level is so tiny, approximately 1.6 x10 ^-33 meter, and it is called the Planck length. Everything in nature, it is supposed, is composed of quantities no smaller than this quanta. By comparison, the nucleus of an atom is approximately 10^-14 meter. So the atomic nucleus is more than a million trillion trillion times bigger than the Planck length! But of course, the assumption of the Planck length is arbitrary. There is no reason not to suppose that something smaller - a smaller scale - exists.
If we continue further and further down the scale, zooming in as it were, we theoretically reach infinitely small fractal levels. Which is to say, perfect continuity. At an infinitely small scale, broken-ness disappears, because the particles get infinitely small and are packed so tightly that the distance between them approaches zero. The denser an object is, the more mass per unit volume, because more and more “stuff” gets packed into it. This tells us that a perfect, continuous surface or volume would have infinite density. Perfect density could only occur at an infinitely small granularity. Therefore, perfect continuity and density describes a singularity.
Broken-ness disappears, then, when there are no individuated particles to cause separation.
However, a material with infinitely small particles means that the radius of those particles is zero, which is the same as saying that there are no particles! Again, we have a singularity, but there is nothing discrete. And if the radius of particles in this continuous surface or volume is zero, the mass of each particle is also zero. Which means that this continuous entity is not infinitely dense, but is actually perfectly etheric, without substance. That is to say, it is non-physical. A singularity is itself an impossibility, so it must also be non–physical, or “virtual,” a product of our creative imagination. In Eastern philosophy, the universe emerges from something creative and latent, and is sometimes described as the dream of Brahma, the creative principle.
A singularity is actually a gigantic contradiction: something that is both infinitely dense, and massless; combining an infinitely great quantity and an infinitely small quantity. A singularity is the connection between the bottom (subatomic) and the top (macrocosmic) of the scale! In other words, physicality does not progress from small to large with small at the bottom separated from large at the top: the two ends of the spectrum are connected. Any physical process affects the whole, from the smallest scale to the largest, because of this invisible connection, even though it is only possible to observe the effect at the scale of observation.
The Shape of Space
To better understand this connection, consider the surface of a sphere.
If you start in Ann Arbor Michigan and walk in a straight line, theoretically you will wind up right back in Ann Arbor at the end of your walk (well, OK, you would have to have a boat, but you get the idea).
That’s because the surface of a sphere has no boundaries.
Now consider what topologists call a three–torus.
A three–torus is like a room that has the walls, ceiling and floor glued together –– not physically glued, but in the sense that if you walked through the left wall you’d find yourself emerging from the right wall. If you walked through the front wall you’d be at the back wall, and the same for the floor and ceiling.
This occurs because a three–torus has no edges or boundaries, just as the surface of a sphere has no boundaries. As soon as you reach one wall, you come through on the other side. If the universe itself has the properties of a three–torus, then it is connected in precisely this way, and all physical phenomena, from lowest to highest scale, are contained within it.
Here is what the Geometry Center says about this: “What seems to be a star in a distant galaxy could be our own sun. The light we receive from it could be light which left the sun billions of years ago, travelled around the universe, and is just now completing its trip. If we can someday find a pattern in the arrangement of the galaxies, then we will know the true shape of space.” This would be possible in a universe that was connected, and had no boundaries.
The point is that a boundary–less universe does NOT have to be infinite, if it has the properties of a three–torus. In other words, the universe could appear to be infinite, but actually be smaller than we think! We tend to think of the cosmos as so vast that it just spreads out forever; leading to a sense of separateness, and a view that we are mere specks in a rambling, limitless infinity. But if all scale levels are connected, as in the singularity, then we are actually living in a finite, holistic, and interconnected universe that is intimately responsive to us at all levels.
Oneness and Scale
The singularity is the perfect unity of polarity into Oneness. The infinitely large in perfect coexistence with the infinitely small! Such a thing is physically impossible, just as perfect continuity is physically impossible; so it cannot exist.
Oneness is physically impossible! You need at least two particles for a universe; for between two particles is a potential; a space where energy can flow, and thus duality, or polarity, is born. Oneness is outside of space and time. It describes something non-physical, or virtual.
A perfect curve (like a circle or a sphere) would contain no broken-ness at any scale. It would be perfectly smooth. So it too cannot physically exist, except in our minds. It is “virtual.”
Beauty is also an idealized perfection of form. Curves appeal to us, I think, because a curve reminds us that, from a spiritual perspective, we are inherently perfect ourselves. And that we are also, like the curve, non-physical.
Cognitive scientists cannot agree on definitions for consciousness, or the mind, or where these qualities or attributes might exist physically. It is really an impossible task, because a quality is virtual. In observing Jill play the piano, for example, we might say that she exhibits the qualities of joy and liveliness. But to physically describe these characteristics rigorously is impossible. We might say, “Jill is joyful because she appears to enjoy what she is doing, and is very animated,” but this is simply a trivial restatement of “Jill is joyful and lively.” Defining qualities rigorously is impossible, for they are non-physical characteristics, and language deals with the physical.
Qualities, like something perfectly continuous, exist at the virtual or non-physical level, combining the duality into Oneness. Qualities are a characteristic of consciousness itself.
Interestingly, to describe precisely how Jill plays the piano (which is what science is all about) is also unattainable, for a completely accurate analysis must encompass Jill’s every movement. This requires us to understand and describe how her hands move across the keyboard, how each finger touches the keys, how muscles and tendons work to generate movement, how the atoms within each tendon and bone come together to form the finger, how the subatomic particles come together to form the atom… a comprehensive physical analysis requires us to encompass all scales, which is physically impossible.
Haramein, Rauscher, and Hyson describe the underlying unity of matter and energy under all scales in a fascinating paper called, “A Scale Unification -- A Universal Scaling Law for Organized Matter,“ which details a fundamental relationship between the radius of anything––including the cell––and its rate of vibration. This shows us another glimpse into the underlying unity of everything in the universe.
We’ve examined scales at the micro level, but what happens when we go the other way?
Let’s take our analysis up to the infinitely large. We have the planet earth rotating about its axis, the solar system, and the Milky Way galaxy. Our galaxy is part of a local group of galaxies, which in turn belongs to a supercluster, which in turn belongs to a larger group. In order to visualize these aggregations, however, we have moved our window of perception “out” farther and farther. In effect, we have minimized the prior physical structures and “rescaled” them. At the universal level, the planet earth is at approximately the scale that the Planck distance is for us at the human level. As we go further “out,” we require that our previous window of perception be forced further into smaller scales. Zooming in forces everything above the target point up the scale of perception, and zooming out forces everything below the target point down the scale of perception. If you think of the movie, “Powers of Ten,” you will understand what I’m talking about. (If you haven’t seen it, look it up on You Tube). In other words, the existence of anything is independent of how large or small it is, but rather on our ability to access that scale level.
The two ends of the scale spectrum – micro and macro - meet at the singularity. That singularity is non-physical; it is the source of the physical, and the physics, that surround it. It is the source of quality, and beauty.
Oneness is impossible for a physical universe, for the universe contains, practically, an infinity of particles. But what if the universe itself mirrors the properties of a singularity? What if the universe is a singularity? Remember that at different scale levels, things shrink or expand. That which appears to be infinitely small at one scale level, may be infinitely large at another! This tells us that a physical universe may simply be an extension of Oneness.
The hidden dimensions in string theory, for example, must exist at these tiny, hidden scale levels.
What is the singularity, ultimately? It is something physically impossible, yet it combines all aspects of materiality. The singularity is the creation point, the source point, from which all materiality arises and evolves. Quintessentially, it is the creative principle, the animating principle, chi, prana, Ka, life force, consciousness, whatever you’d like to call it. The source of beauty, of physicality itself, resides in something that itself cannot be quantified, yet is the origin point of all things. That source is joy, love, well-being, and the powerful energy of creation itself.
It is the Still Point, from which thought and creation, emanates.
It is YOU!
Fractals do not have to be self-similar, but many of them are.
Francesco Sylos Labini of the Enrico Fermi Centre in Rome and Luciano Pietronero of the University of Rome argue that the data from the latest Sloan Digital Sky Survey shows that the universe looks fractal as far out as our telescopes can see. According to their latest paper, which has been submitted to Nature Physics, Sylos Labini and Pietronero, along with physicists Nikolay Vasilyev and Yurij Baryshev of St Petersburg State University in Russia, argue that the new data shows that the galaxies exhibit an explicitly fractal pattern up to a scale of about 100 million light years. And they say if the universe does become homogeneous at some point, it has to be on a scale larger than 300 million light years across. There are opposite points of view, of course, but the fractal nature of scales below our level of perception suggests (to me, at least) that the same pattern continues at macroscopic scale levels.
–––Amanda Gefter, ”Galaxy Map Hints at Fractal Universe,” New Scientist, 25 June 2008, http://www.newscientist.com/article/dn14200
Heisenberg called this tradeoff the Uncertainty Principle, after it became obvious that physicists could not accurately determine both the momentum and location of an electron at the atomic scale level. Tradeoffs are an inherent property of the duality of a physical universe.
It seems absurd to refer to anything with zero radius, but a singularity is defined as something with infinite density and zero volume.
Preprint: N. Haramein, M. Hyson, E. A. Rauscher, Proceedings of The Unified Theories Conference (2008), Budapest, Hungary, “Scale Unification: A Universal Scaling Law for Organized Matter”, in Cs Varga, I. Dienes & R.L. Amoroso (eds.). Paper available at Haramein's website, http://www.theresonanceproject.org/research.html.
Assuming a universe 13.5 billion light years in diameter, this comes out to approximately 1.5 x 10^26 meters. The human body is approximately 2 meters in length.