This image depicts the interaction of nine plane waves—expanding sets of ripples, like the waves you would see if you simultaneously dropped nine stones into a still pond. The pattern is called a quasicrystal because it has an ordered structure, but the structure never repeats exactly. The waves produced by dropping four or more stones into a pond always form a quasicrystal.
Because of the wavelike properties of matter at subatomic scales, this pattern could also be seen in the waveform that describes the location of an electron. Harvard physicist Eric Heller created this computer rendering and added color to make the pattern’s structure easier to see. See: What Is This? A Psychedelic Place Mat?
See Also: 59. Medieval Mosque Shows Amazing Math Discovery
A CG movie inspired by the Persian Architecture, by Cristóbal Vila. Go to http://www.etereaestudios.com for more info.
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| Circle Limit III, 1959 |
In 1941, Escher wrote his first paper, now publicly recognized, called Regular Division of the Plane with Asymmetric Congruent Polygons, which detailed his mathematical approach to artwork creation. His intention in writing this was to aid himself in integrating mathematics into art. Escher is considered a research mathematician of his time because of his documentation with this paper. In it, he studied color based division, and developed a system of categorizing combinations of shape, color and symmetrical properties. By studying these areas, he explored an area that later mathematicians labeled crystallography.
Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher’s interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher’s works Circle Limit I–IV demonstrate this concept. In 1995, Coxeter verified that Escher had achieved mathematical perfection in his etchings in a published paper. Coxeter wrote, “Escher got it absolutely right to the millimeter.”
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| Snow Crystal Photo Gallery I |
If you have never studied the structure of Mandala origins of the Tibetan Buddhist you might never of recognize the structure given to this 2 dimensional surface? Rotate the 2d surface to the side view. It becomes a recognition of some Persian temple perhaps? I mean, the video really helps one to see this, and to understand the structural integrity is built upon.
So too, do we recognize this “snow flake” as some symmetrical realization of it’s individuality as some mathematical form constructed in nature?
I previous post I gave some inclination to the idea of time travel and how this is done within the scope of consciousness. In the same vein, I want you to realize that such journeys to our actualized past can bring us in contact with a book of Mandalas that helped me to realize and reveals a key of symmetrical expressions of the lifetime, or lifetimes.
Again in relation how science sees subjectivity I see that this is weak in expression in terms of how it can be useful in an objective sense as to be repeatable. But it helps too, to trace this beginning back to a source that while perceived as mathematical , shows the the mathematical relation embedded in nature.
Dialogos of Eide 
Nice CG movie. Persian architecture seems to evoke some kind of wafting all-most memory when I see it. It's been responsible for my superficial attraction to Gurdjieff.
Has nothing to do with Gurdjieff.The most important archetype of all is the self. The self is the ultimate unity of the personality and is symbolized by the circle, the cross, and the mandala figures that Jung was fond of painting. A mandala is a drawing that is used in meditation because it tends to draw your focus back to the center, and it can be as simple as a geometric figure or as complicated as a stained glass window. The personifications that best represent self are Christ and Buddha, two people who many believe achieved perfection. But Jung felt that perfection of the personality is only truly achieved in deathI think you missed the point about the Intent in Actualization. Thatwhether it's theobjectie reality where one finds these symbols metaphrs is talking about structures much wider o the platform Tom showed. Looking at the diagram of the tiling at first glance….is to understand this represents something I discovered in the dream world. Lives are built upon it. Tibetans build them and then take them apart. Really not superficial at all but have enormous depth.
"…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups. You know, those things the non-professionals call . . . kaleidoscopes! * (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism'– then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry (A polytope is an n-dimensional analog of a polygon or polyhedron. Chapter V of this book is entitled 'The Kaleidoscope'….)"
To clarify, the Gurdjieff, is my subjective correlation – didn't mean to suggest it was objective. It's a correlation i have yet to sort out.Since using this blog/Disqus I'm noting more inconsistencies. I use Firefox, so I've yet to confirm that is the 'cause' or not. In my experience IE may/may not provide solution but I stay away from it as much as I can, due to it's non-standardization.
"I think you missed the point about the Intent in Actualization."Very likely, as I tend to actually ignore such observations at that level unless they connect to me deeper than the intellectual level. Mandala's, in general, don't get me at the deeper level. They, however, are aesthetically pleasing in the intellectual mode, for me.
Yes thanks RBM for clarification. Still experimenting within Diqus format parameters, and I understand.
Again thanks for your clarification. I just wanted to point out that this sits in the same realm and aside from the intellectual. An internal exploration does in dreaming not deviate and represent the intellectual but fully empowers the observer about discovery.