OT: Fibonacci Numbers Video

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When working to get proportions on things, I usually try Fibonacci numbers first. Here's a nice short video showing some of where we run into them in nature.
http://www.wimp.com/fibonaccisequence /
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On 7/25/11 11:33 AM, kimosabe wrote:

cool video! I've often referred to the "Golden Ratio," but have never seen the Fibonacci number string. It's a lot easier to remember.
This is going on facebook. :-)
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Those numbers closely resemble those that determine ideal internal dimensions of acoustic loudspeaker enclosures. If the baffle board is 12 " wide than the depth of the box be will 7.5" and the height 19.2" or any ratio close to 0.625:1:1.6. That is for a basic box.
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wrote:

I wonder what the basis for that is in physics. Daughter have opinions?
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Han
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Reminds me of the concept and math of fractals.
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Sonny wrote:

They appear to share a pattern and a limit.
The number which is the limit of the successive ratios of the values in "Fibonacci Sequence" is called the "Golden Ratio" is just a number (approximately 1.618). You can play the same game (defining a "recurrence relation") starting with values besides 1 and 1, and by the same process you'll surely end up with a different limit, i.e. number.
The limit in the fractal case often lies in 2-D. A well-known one is called a "snowflake".
The numbers e and Pi share a pattern and a limit in some of their definitions too, maybe comparable to nature and evolution....
To get back on topic, some have suggested that a picture frame whose length and width are proportion alto the Golden Ratio, 1: 1.618, will look the most natural to the most people.
If this is so true, I wonder why the tv industry appears to have settled on the aspect ratio 16:9 ~ 1.77 instead of 16:10 (which would be closer to the Golden Ratio)?
Bill
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On 7/25/2011 3:48 PM, Bill wrote:

Their programming is so bad that they have calculated the most aggravating ratio to keep you awake so you don't miss the commercials?
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On 7/25/11 3:48 PM, Bill wrote:

The camera ads ten pounds?
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Not true. Using "the same process", i.e., the same recurrence relation, you will end up with the golden ratio no matter what the starting values (as long as at least one of the two is non-zero). e.g.
8, -6, 2, -4, -2, -6, -8, -14, -22, -36, -58, -94, ...
At this point, the ratio of successive values has already converged to 1.62, and it keeps getting closer. Try to find a pair that will NOT get you to within .01 of the golden ratio after ten iterations, and you will quickly convince yourself of this.
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On 7/26/2011 7:55 AM, alexy wrote:

...
Well, he specifically said "defining a recurrence relation" which would imply some other than the same one... :)
But, it isn't true that _any_ recurrence relation will generate a limit as is implied, either...
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alexy wrote:

I don't doubt it. IIRC, the trickiest part is showing the limit exists, then the quadratic formula decides the value of the limit. Probably, the more general result can be obtained by using the original one: If you multiply the Fibonacci sequence by some any non-zero value c, then when you take the ratio of successive elements the c cancels out. So the sequuence 0, c, c, 2c, ... will yield the Golden Ratio too. Beginning with -1, 0 instead we get -1, 0, -1, -1, -2,.. and this also yields the Golden Ratio (discard the first term to see this). Taking the sum of the 2 sequences 0, 1, 1, 2, 3, ... and 1, 0, 1, 1, 2, 3, ..., or equivalently any zon-zero multiples of them 0, a, a, 2a, 3a, ... b, 0, b, 2b, 3b ... basically helps establish your result. Minor technicalities omitted. If a or b is zero, but not both, the conclusion is unaffected. So yes, I agree. Score another one for the Golden Ratio!
Bill
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There is a complex set of trade-offs involved.
Peripheral vision is more effective to the sides than up/down. This is in part because, 'historically', threats were more likely to appear from the sides. The 'range of vision', _vertically_, is typically about +/- 60 degrees from the horizontal. However, 'to the sides', it is typically 80+ degrees from 'straight ahead', and in a significant number of people it can range to 90-95 degrees _and_more.
"Portrait" orientation (the long dimension vertical) is optimal -- in terms of 'visually pleasing', that is -- at 1.618:1. This ratio occurs 'naturally' in a bunch of aspects in the human body -- See da Vinci's figure studies.
"Landscape" is more natural, and 'panoramic', at a ratio that is 'wider' and 'flatter'. You don't get much from the extra 'sky' in an exterior shot. Similarly, for interiors, the floor-ceiling dimension tends to limit the usefulness of a greater display height.
Also, realize that 'wide-screen' in the movie theater is typically _1.88_:1.
and the famous 'Cinerama' process from the 1950s, 1960s, and 1970s, was *really* wide -- at _2.66_:1.
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http://en.wikipedia.org/wiki/Fibonacci_number
More in Mathematics but applications in Physics and the Bio plants and animals. I've watched complex presentations on the mathematics of Shell shapes.
Most interesting line of mathematics. Martin
On 7/25/2011 12:36 PM, Han wrote:

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'Smoother' response --- no reinforcing resonances on two axes.
The 'ideal' ratio is .618:1:1.618.
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I'm with Mike. Put it on FB too ...
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Han
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On Mon, 25 Jul 2011 09:33:33 -0700 (PDT), kimosabe

Very interesting. Now I know where the term "Golden Rectangle" originated.
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Mathematician Mario Livio in his book: The Golden Ratio, examines the so-called Golden Ratio as it applied to the arts.
While it is true that the ratio shows up in most unexpected places (chambered nautilus being one), it is not so certain that artists have consciously employed the ratio in their art. The Parthenon is supposed to follow it, but it's not clear where to actually start the measurement to coincide with the ratio.
Here's a quote from a web page:
"Furthermore, I should note that the literature is bursting with false claims and misconceptions about the appearance of the Golden Ratio in the arts (e.g. in the works of Giotto, Seurat, Mondrian). The history of art has nevertheless shown that artists who have produced works of truly lasting value are precisely those who have departed from any formal canon for aesthetics. In spite of the Golden Ratio's truly amazing mathematical properties, and its propensity to pop up where least expected in natural phenomena, I believe that we should abandon its application as some sort of universal standard for "beauty," either in the human face or in the arts."
http://plus.maths.org/content/os/issue22/features/golden/index
The take away message is, don't get so trapped in following the formula that you overlook the art you are doing as a whole. Sometimes deviation from a "norm" is more enticing, thrilling and original.
MJ
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On 7/25/11 3:47 PM, MJ wrote:

I never took the fact that the golden ratio is present in so many great works of art to mean that the artists purposely employed the ratio in the design of their art.
I've always seen it as a *description* of aesthetic pleasure, not a *prescription.* Meaning, we have perceived certain works of art to be beautiful or pleasing because, for whatever reason, our brains are wired to like things that have this ratio.
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Ah, but suppose many, many artists avoided the trap you postulate, or more likely, never knew it existed. But yet their work product still emerges so proportioned. THAT speaks volumes. I doubt that sunflowers, ammonites, and myriad other works of nature got "trapped". I'd put this tendancy right in there with Gibb's free energy.
-Zz
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On 7/25/11 8:32 PM, Zz Yzx wrote:

That's what I contend. The great art isn't great because the artist followed a formula. It's great because it has beautiful proportions. The formula describes those proportions.
But the author at that link has a point worth discussing. Architecture these days seems to be at one extreme or another; either absurd grotesque or absurdly plain. The grotesque can be seen in Dubai where everything is to a ridiculous excess... simply because it *can* be done and someone will pay for it. The plain can be seen by the work that comes from several generations of architectural teaching that says the golden ratio is the perfect proportion. As the author implies, these architects seems to just shoot for that formula without any forethought for *art,* producing generic, homogenized structures that look almost attractive.
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