Trust and the golden mean
When I got up this morning and tuned in to my day - I'm self-employed so don't have a schedule as such - two words came: 'educate yourself'. This surprised me. I've been writing every day for the past little while – I'm working on a book of my experiences – and was expecting to pick up where I left off. With no sense of what I was to educate myself about, I continued making preparations for the day.
As I left the house later in the morning for what has become my 'daily cappuccino' - I sit in a coffee shop and scribble in my notebook - my eyes fell on a set of books that I'd recently bought, but had not read. One, on the golden 'section' or golden mean, kind of 'sparkled' at me as I went past. I picked it up as I went out of the door.
Ensconced at the coffee shop, I soon found myself immersed in algebra and number, geometry, harmony and cosmology, discovering the universal laws of ratio and proportion. And finding what linked the one to the whole, architecture to music to nature: Nature's greatest 'secret' - the golden section or golden mean.
Not only does the whole of nature express itself through a very simple series of whole numbers (the Fibonacci number sequence), all related to the golden mean, but the golden mean is represented in much of the most popular classial music of all time, including the works of Bach, Mozart and Bartok and in the design of many of the most celebrated paintings of da Vinci and Botticelli.
It was Luca Pacioli, the 'monk drunk on beauty', who first wrote about the golden section in his treatise, Divina Proportione. I could relate. I remembered stumbling upon a building in my early twenties – the Duomo in Milan - and having nearly fainted at its beautiful symmetry. I later discovered it was founded on the principles of sacred geometry – again related to the golden section.
Then my eyes fell on one of a series of geometrical shapes on the page that I was reading. Called a 'claithrin', this perfectly symmetrical shape, it said, was a symbol of consciousness and, as the author noted, 'abuzz with golden ratios'. But the thing about it was that it brought to my mind one of the most memorable dreams I have ever had.
All of humanity – myself included – were standing round this giant multi-faceted crystal – identical to the claithrin that was represented. It was beautiful, harmonious and symmetrical in its shape. Yet as each one of its facets moved round in perfect symmetry, each person would flinch. As I stood there the words 'just trust' came to me. And what, I realise, this dream was telling me (and the others that stood around the giant crystal) was that we can trust. That everything is happening in perfect harmony and order, down to the movement of the tiniest particle. And that in surrendering as deeply as we can and trusting in each moment, we will find our way through, even should everything fall apart around us.
I suddenly remembered the words that started off my day. I had educated myself. I had inspired myself. I had had a realisation. A good day.
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Re: Trust and the golden mean
What a wonderful education Tonya. Such a beautiful invitation to trust that as all else falls away we will find our way through. Thank you for sharing. Keep shining
Trinity
x
Fibonacci
Yes. Holy Geometry is one of the fields that are most fascinating me. I did some research on it for an article and came into contact with some very interessting people.
There is a lot to learn and understand in that area for mankind and I belive this understanding will be a crucial part of science in our future.
Here is a nice collection of fibonacci spirals:
http://www.facebook.com/album.php?aid=122572&id=157719019022
David
"Sacred geometry", numbers and cosmic order
Here's a text that shows that an understanding of harmonious structures in creation and the interrelatedness of mathematics, architecture and music for example has been common knowledge for a long time. A little theoretical, but the message really comes through - the author is Anna Bofill Levi, a Jewish Catalan architect, musician, mathematician and feminist and the sister of an internationally renowned architect. (The translation is mine with a lot of help from a native speaker).
"In ancient times and during the Middle Ages, music, arithmetic, astronomy and geometry were part of the Quadrivium and thus of more advanced scientific studies. The artists of the Renaissance seized on the Pythagorean notion of "everything is number" and, in their reliance on Neoplatonic ideas, were convinced that the universe and the whole of creation had a harmonious structure. The architects, on the other hand, created a system whereby each part of a building, both inside and outside, had to be integrated into one and the same system of mathematical proportions. Furthermore, in accordance with the dictates of Vitruvius, the building was to reflect the proportions of the human body, since the human being was considered to be the image of God. Likewise, in compliance with the Platonic-Pythagorean tradition, the laws of the cosmic order were derived from systems of mathematical proportions. In the ninth book of De re aedificatoria (1450) Alberti gives a definition of beauty based on Vitruvius which is in accordance with the principles of the rational integration of the proportions of all parts of a building so that each part be commensurate with the whole, and the whole with each part. Alberti discusses similarities between musical intervals and architectural proportions. Referring to Pythagoras, he affirms: "The very same numbers, however, that cause sounds to have that concinnitas (harmony),pleasing to the ears, can also fill the eyes and mind with wondrous delight. Palladio, the last of the great humanistic architects, published a treatise in 1570 entitled Quattro Libri dell'Architettura, in which he also gives a mathematical definition of beauty. To sum up, it can be said that the history of Western music has been characterized by the search for the ideal of creating a language that was to be as perfect as possible: the so-called harmony of the spheres. As Wittkower stated, the symbolism and mysticism of numbers have been held in high esteem for 2000 years and have exerted a dominating influence on human thought."
Order and disorder
Even though I just love ordered systems, and I'm also like HA!!! when I see how proportions, numbers, symmetry, and in general, that's why I went to study chemistry - I just love to see how formulas, numbers, mathematics can describe this crazy world outside. But also, I felt I just have to speak here for the 'Holy disorder' a little!!!
'Harmony' and beauty in my view don't need any special numbers, even though we may see some regularity in what looks/sounds harmoneous.
I reconsidered my attitude to traffic, industrial (which are beautifully used in electronic music by the way) and many other noises, after I saw a documentary with Lisa Gerrard, where she was saying that there are no not beautiful sounds for her, and that she finds highway very inspiring for her, and as she reproduced doppler effect of the driving away truck, I immediately got it, and since then...no rigidity about what I would define as 'harmoneous', maybe not resonating at specific moment, but...unable to define what's beautiful and what's not, and loosing my rigidity about this topic forever.
And here I will touch how the 'order-disorder' is seen in science...
In science these questions arise constantly - how do we define and work with disordered systems. For example, in previous year there was a very interesting lecture on the chemical conference about the nature of entropy (disorder).
The lecture was actually one big question - what entropy is?
Entropy is something that until today, nobody knows what it is, despite working with it and using it in calculations (by the way, just as NOBODY! really understands quantum physics, calculate, work with it, get used to its laws - yes, but...this is just too far from intuitive perception of this world, and we all admit it). To see how equations describe our physical reality on its smallest level is pure delight, but the mind is all smoking when working with these things.
Even in my work, motion in viscous medium, we see that it is NOT really perceivable how it woks. Intuition fails there. For example, in my calculations I got that when there is infinite number of small objects move in single file in a channel, in water, their behavior will be as if there are no objects and no channel, but simply a cylinder moving freely in water. Isn't it just beautiful? But... try to analyse this. Impossible.
(All these are like Zen-koans for the mind)
So what order is? If we see a crystal - then it's ordered, but how 'much' is it ordered? And then what is quazi-crystal? Ordered, but not periodic they say, but here we see that the word ordered begin to sort of loose it's 'sharpness'.
And just in the same way I ask - what is harmony? While in music it's very precise, but in art generally - here are some of them:
"harmony - Agreement; accord. A union or blend of aesthetically compatible components. A composition is harmonious when the interrelationships between its parts fulfill aesthetic requisites or are mutually beneficial.
"harmony. The principle of design that combines elements in a work of art to emphasize the similarities of separate but related parts."
"harmony The related qualities of the visual elements of a composition. Harmony is achieved by repetition of characteristics that are the same or similar."
The latter just reminds me of the same connundrum in sciense about order.
And here on the example of order/disorder we can see how really 'complicated' it all is, until it gets totally disordered
First of all, the entropy - it was slightly recognised (still nameless) by Carnot, when working with heat-engines, where he saw that the ends just won't converge in terms of energy, i.e. "we loose energy", don't know where... Then it was seen that the first thermodynamic law doesn't work, since, hey we have energy loss. After Carnot, Clausius defined it as a dissipative energy, then more scientists worked with it via statistical mechanics (thermodynamics), i.e. showing how microstates affect macrostates, and since then...mathematicians, physicists, chemists...more more work...
BUT, the point is still the same - entropy is something unmeasurable directly, you measure everything, but not it. And therefore define it however you want, and it WILL work, we use it, but! it's really some 'hole' where the energy goes. And if it's so, then define order/disorder. Entropy increases -> disorder increases, okaaaaay... (scratching head)
It's also funny to know what chaos is scientifically, I think. Chaos is supposed to be something 'chaotic'
but, actually it is very well defined, and can be calculated (measured), as follows:
In general, if you change just a little the initial conditions and then look at the final results - if the system is chaotic, the first feature will be that in phase-space you'll get that the results will be 'far away' from each other - the so-called butterfly effect. Now this can happen in totally deterministic system! how do we know we have chaos? In a very simple form - we measure time-evolutions of the states, and if the difference between them increases exponentially - it's chaos.
And if it's not enough, absolutely random systems also can be and are characterized by "what kind of diffusion we have here", for example? And once it can be classified, we already have a 'box' for it...so...
With all this being said, I'm a big fan of harmony in its conventional sense and enjoy the symmetry and the 'order' in it, but just take the most formless piece of anything to STM and look at these atoms there, HOLY it IS!!
And who would say that the turbulent river flow is not beautiful? Look at your teabag when you prepare yourself a cup of tea, look how the color is spreading slowly into the water around. Magnificent!!! Divine! And absolutely disordered
Love,
Yulia
Fibonacci
Thanks David for putting Fibonacci spirals pictures they are awesome.
Bliss.