Aloha from lovely North Kohala on the wonderful Big Island of Hawaii!
This remote and stunningly beautiful location is ideally suited for performing original, paradigm-shifting research into the innermost workings of reality.
The club will convene electronically as well as in person (time and place TBD). All who share an interest in any level of mathematics and science are welcome to join. I personally offer my services as a mathematics educator, researcher and tutor to any members.
Inaugural Virtual Meeting: The Unreasonable Effectiveness of Fractal Mathematics, or, the “Why?” of it all…
Back in 2015, Hawaiian astronomer Paul Coleman delivered a thought-provoking lecture on fractals in astrophysics, at Maui’s Institute for Astronomy: The Fractal Nature of the Universe.
Toward the end of his talk, he asked (I paraphrase) “why does fractal geometry seem to show up everywhere (in nature)?” I what follows I will attempt to sketch an explanation.
In my PhD thesis: Scale Covariance of Fractal Sets and Measures: A Differential Approach to the Box Counting Function of a Fractal, with Applications, we show that a fractal can be represented by a Kronecker product of vectors or matrices. See also, Willi-Hans Steeb’s The Nonlinear Workbook.
Quantum states can be represented by vectors called “Bras” and “Kets”, and when many body quantum systems are constructed from the states of their components, the Kronecker product is used to combine the states. Also, quantum processes are represented by matrices, which can be combined with the Kronecker product as well.
The above results imply that quantum states have a natural interpretation in terms of fractal geometry, for which there is experimental evidence here. Also, some of the latest research into a quantum theory of gravity indicates that quantum entanglement is fundamental to the makeup of space-time. This means that fractals are a fundamental descriptor of reality at the most extreme scales of the universe! (When a fractal is represented as a Kronecker product, the removed portions, are not able to be represented as a power of vectors, and are hence described as entangled states [see my unpublished paper on this subject-email me for the pdf]).
But Really, Why?
Let’s back up from physics for a moment and just talk about fractals in general terms. Loosely speaking, fractals are objects made up of exact or approximate copies of themselves, constructed according to simple rules. There is a rigorous mathematical definition of “fractality” that can be found in my former PhD advisor’s book: Fractal Geometry, Complex Dimensions and Zeta Functions, Geometry and Spectra of Fractal Strings. The simplicity of the rules used to construct a fractal conceal the complexity of the end result, because the rules are iterated infinitely.
I call the philosophy of this act of construction (which I call “fractalization”), “If it feels good, do it! (and proceed via induction)”. While this is not meant as an advocacy for unchecked hedonism, our instinctual urges do serve a purpose in terms of our biological, mental, emotional and even spiritual evolution. To reign in our urge to do whatever feels good, we need a conscience, some way to evaluate what we should or should not be doing. (Personally, I try to arrange my habits so that what feels good to me is actually good for me and others. Sometimes, though, there is no substitute for self-discipline.) I will call this conscience “de-fractalization”. Fractals are tremendously complicated shapes that often boggle the imagination. So sometimes we boil them down to a number such as a fractal dimension to more easily understand them. This is one of many examples of processes that make fractals more accessible by reducing their complexity to that of the number line through this process of defractalization (other such processes include self-similar measures, entropies, other notions of dimension, even the Kronecker representation itself). Fractal geometry, utilizing techniques of both fractalization (creation of complexity) and defractalization (evaluation of complexity) forms a dialectic, or union of opposites.
It turns out that defractalization is a key process in machine learning, for example, in the problem of density estimation, an application to computing fractal dimension to many contexts, including machine vision and non-linear inference. (BTW fractal geometry has many more applications to machine learning, particularly to ANN’s, or Artificial Neural Nets.) Kronecker powers of states are ubiquitous in quantum computing as well, showing that fractals may play a role in quantum information. In the absence of a universally agreed-upon model for cognition, machine-learning and quantum computing likely serve as important models for these processes as well, hence fractals could serve as valuable models for cognitive processes.
The growing importance of perspectives from fractal geometry in the key realms of quantum physics, mind processes, and the structure and evolution of the cosmos, I think, is best exemplified by the interpretation of the space of quantum states (called Fock space), in terms of fractal geometry. And that, I submit, is why fractals “seem to turn up everywhere.”
Additional fractal applications:
The Role of Dimension in Understanding Cognition: According to mathematical physicist Douglas Hofstadter, analogy is the core of cognition. Analogies in language are akin to ratios in mathematics. A fundamental ratio that concerns us in fractal geometry is the box-counting dimension, a measure of the geometric complexity of an object. Thus it should be no surprise that density estimation is a key technique in machine learning.
A Toy Model of Fractal Quantum Gravity: Speaking of analogies, analogies between quantum and classical physics have been predicted by theory and confirmed by experiment. Of importance to this discussion is the analogy of a quantum Lagrange point, which allows us to consider the effect of quantum and classical Lagrange points in a heuristic, toy model of gravitation.
Two different features come to mind. They can be described as anti-gravity and quasi-crystal space time, respectively. However they really shouldn’t be as exotic as all that.
First, anti-gravity. It is not a controversial statement that an object orbiting at an L1 Lagrange point (between two gravitating bodies) orbits more slowly than an object on a different orbit, the same distance away from the greater mass M. Now imagine the less massive body in the limit of the very small. Let us ignore noise in space-time that will obscure our Lagrange points. This L1 point will have to get closer and closer to the mass m, yet still has this gravity-cancelling effect. Also, in this limit, quantum effects like uncertainty and charge separation may become important, perhaps opening up the possibility of harnessing these “negative gravitational charges” (presumably for relativistic-speed travel or some-such purpose). This could still work in the entanglement gravity paradigm, in which a quantum state carrying such a charge would have no, or reduced entanglement.
To get into the notion of quasi-crystal space-time (not entirely original to me), we first simply define a quasi-crystal as a structure carrying an unusual or “forbidden” symmetry. Since the metric of space-time changes with changes in coordinates, for us, this symmetry will be topological, rather than geometric. Observing that a gravitational field of a larger mass M acting on a smaller mass m has a fixed point at each of its 5 Lagrange points, and that in some neighborhood of each of these points the gravitational field as a mapping is contractive, so we can regard this field as the attractor of the maps induced by these fields. When the gravitating system under consideration interacts with another such, we perhaps combine the fields through taking their tensor product as we noted in the parallels between quantum mechanics and fractal geometry above (this is where the “quantum” comes in to quantum gravity). The quasi-crystal part is just because there are five mappings which induce a five-fold symmetry in the gravitational field.
The implication is that space time is fractal as well as quasi-crystal (since the number of interacting components is large), but how can any of this ever be tested? A simple prediction of this view is that Lagrange points should have their own Lagrange points, as use of the tensor product would suggest. I am not sure what conditions would have to prevail in order for them to be detectable. Given the zero net mass of the L point, perhaps all these points would serve to give the local, quantum structure to the gravitational potential in the neighborhood of a Lagrange point, since these points would likely be infinitesimally close to the original one. At least these would be quantum effects at low energies, so one might hope to detect them with the technologies of the foreseeable future.
So far we have considered the fixed points of the field only. What of the singularities corresponding to masses M and m (assuming their radii vanish)? Projective fractal geometry can help us here. As the magnitude of the gravitational force goes to infinity let us consider the field to be contractive about infinity, and work in the one point compactification of real space. Then the first 5 contractions about finite coordinates describe the local field about a gravitating object, while the contractions about M and m describe the large-scale fields at macroscopic distances. Then our full set of symmetries includes 7 mappings, another forbidden symmetry.
Properties of quasicrystals include brittleness: this describes the vulnerability of space-time to the decay of the false vacuum, yet these objects are stable at short distances as one should expect from the fractal local structure. Mapping this coordinate space and potential field into a phase space, we see that the resulting state space is fractal, indicating a chaotic space-time.
Special thanks to the Kohala community:
I have now had the honor and pleasure of working with the Big Island community in mathematics for over 3 years. I wish to extend my heartfelt thanks and gratitude to the people of this island and specifically to the community of North Kohala for welcoming myself and my family to this blessed place. It is my intention to serve this community for many more years.