Penrose tiling, Golden Mean, and other stuff
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Alright, nothing to do with primes yet, my apparently dangling participle of a subject to get around to writing about, but I got onto some other thoughts after reading Mario Livio’s The Golden Ratio. I’d kinda like to do a mini-book report on a bit of it and try to tie what I’ve learned from it back into my thoughts on brain-structure, memory and possible connections to multi-dimensional (even possible prime number/signal fixation) brain modeling.
He touches on the stuff only briefly, but traces a path from Johannes Vermeer, Islamic art and architecture, and M.C. Escher to Roger Penrose and more modern experimentation about tiling or tessellations. From Vermeer he gets into talking about simple shapes that can be used to completely cover a plane that exhibit an ordered, periodic pattern such as a chessboard pattern of squares (as seen in several Vermeer paintings) or a hex-map. From Islamic art and Escher he talks about the use of odder and odder shapes that can still be used to form other periodic though more complicated patterns that are also capable of filling a plane. As the title of his book might suggest he then goes on to talk about patterns of tessellation that Penrose adapted from interior triangles of a pentagon, that bastion of Golden Mean-ness.
The pentagon alone cannot perform the feat of completely and periodically covering a plane that triangles, squares or hexagons can. Most everybody even thought that such five-fold, periodic symmetry was impossible. But in a feat of mental exercise, Livio suggests , Penrose found a way to at least combine parts of the pentagon into a new pair of shapes that, though they don’t cover it specifically periodically, do exhibit a definite sense of long-range order and pattern. Together with John Horton Conway from Princeton, he found that the ratio of the number of each shape of this new pair they used approached the Golden Mean the larger space they covered.
In 1984, Dany Schectman, an Israeli materials engineer, found aluminum manganese alloy crystals also exhibited both long-range periodicity and five-fold symmetry. All known crystals up until that time either were completely ordered, forming periodic “unit-cells” or repeating patterns or were totally amorphous. This in-between state of matter he discovered became known as quasi-crystals. Livio goes on to say that Penrose’s mental game was all fun and nifty and everything but then questions how do you go about explaining the connection to an actual physical structure?
For brevity, I’m now going to quickly leap-frog over huge details but cite names and overall effects for the next section. Building upon that work and the work of Sergei Burkov of the Landau Institute of Theoretical Physics in Moscow, Petra Gummelt of Ernst Moritz Arndt University in Greifswald, Germany proved that rather than using Penrose pairs, a single specially marked decagon could be used to cover the same effect provided that the same-marked sections could overlap. Paul Steinhardt of Princeton and Heyeong-Chai Jeong of Sejong University in Seoul used this theoretical model to perform X-ray and electron scans of quasi-crystalline alloys of aluminum, nickel and cobalt. They found remarkable similarity in their scans to the models that Burkov and Gummelt first proposed, finding further that such overlapping of the “decorated” sections of the decagons could actually be the result of the quasi-crystal “unit cells” sharing atoms from each of the constituent elements in a high density-low energy form that is incredibly stable.
The Steinhardt-Jeong (Burkov, Gummelt) model also shares an interesting general property with the Fibonnaci sequence, that of long-range, self-similar order. In S-J(B,G), that order is descriptive of a physical structure; in the Fib. Sequence, a mathematical one. They are both of the nature that Mandelbrot had his huge leap into: fractals.
Sheesh…2:30 a.m. day (of and) after Christmas… hella tired…
Note for myself : I wanna get to talk about my thoughts on how that S-J(B,G) model can be related to a possible fractal structuring of brain connectivity that deals with memory. The jist of it being that this super-dense, low-energy kind of structuring could be going on in the brain and could be responsible for our ability to overcome the entropic arrow of time as a limited form of time travel, at least in as much as our being able to recall and model past events. Much, much later I wanna tie this all into a fifth-dimensional model for how we can recall events and keep them fixed in what we perceive as a four-dimensional reality. But I gotta be patient… and I gotta sleep…