The golden ratio has long been known in nature, and long used in art and architecture the world over. I personally like it in photography as an aspect ratio for prints. There's something about it that's just beautiful, and... right.
Now there's new evidence that the golden ratio might be a fundamental principle -- at the quantum level. Thought there were people in this crowd who might find it interesting.
Bruce Watson
"The first two notes show a perfect relationship with each other. Their frequencies (pitch) are in the ratio of 1.618…, which is the golden ratio famous from art and architecture."
What about the rest in the series. If it's a simple fibonacci series, the others should converge on the same ratio. Too bad they don't tell us more in the article.
Unless you're a professional skeptic, this stuff shouldn't be too surprising, since it's all just based on a simple arithmetic series, and ends up being the most convenient design.
The Fibonacci series creates a different spiral, compared to the Golden Ratio Spiral, especially when you consider the absolute values in the Fibonacci sequence. The Fibonacci series creates a different spiral than the Golden Ratio Spiral too, even when you calculate the Fibonacci Ratios that eventually approach the Golden Ratio. If this sounds confusing, it is, since most folks simply think of the Golden Ratio as a fixed value, and periodically identify the Fibonacci series as the source of the Golden Ratio. The Fibonacci Series happens to be one source that appraches the Golden Ratio.
That said, the Golden Rectangle produces the Golden Ratio value quickly, whereas the Fibonacci sequence eventually approaches the Golden Ratio value when you travel deeper into the Fibonacci sequence, while using the require mathematical ratio calculations. The Golden Rectangle, which produces the Golden Ratio value, is very prominent in Architecture and Art, whereas the Fibonacci Sequence which approaches the Golden Ratio value, happens to be more prominently displayed and more commonly found in nature.
While studying Architecture, the Golden Ratio was prominently displayed, but many students could not feel or see this ratio in three dimensional space, because they continuously saw the value in two dimensions only. This ratio happens to be a natural fit for any viewer, since the Golden Rectangle produces a pleasing shape. If you truly want to use the Golden Ratio within your artistic images, and to use it beyond your simple physical print size boundary, then you should also consider placing the Golden Spiral within your image and place your subject matter within the spiral; accordingly, too.
For those of you that may be interested, I attached a PDF that demonstrates how you can physically create the Golden Rectangle and, or calculate the Golden Ratio. I also included a PDF that demonstrates the different spirals, and I also included the Golden Ratio Number to a few decimal places, which possibly does not terminate, but who knows...
jim k
Thanks for the link, Bruce. (You probably knew who some of the interested parties would be - no surprise to see Ken jumping on this one!) It's always fun to see another place the Golden ratio springs up. Here's another one: My sister just sent me a clipping about a 20 year study of 166,000 Austrians. The average ratio of systolic to diastolic blood pressure was 1.6235 overall. For those who died during the study the average was 1.7459 and for those who survived the entire 20 years the average was 1.6180! I haven't seen the study itself, so I don't have any more details.
I have to (respectfully) take issue with you on one point Jim, the implication that the Fibonacci sequence lolligags around in getting to the Golden ratio. There is an explicit formula for the terms of the Fibonacci sequence that contains the Golden ratio. Let a_1, a_2, a_3, a_4, ... be the numbers in the Fibonacci sequence 1, 1, 2, 3, 4, ... (I'm using the underscore to indicate a subscript.)
Also, let A be the golden ratio (1 + sqrt(5))/2 and let B be its "conjugate" (1 - sqrt(5))/2
Then for n = 1, 2, 3, 4, ...
a_n = (A^n - B^n)/sqrt(5)
You may well be aware of this Jim. It can be derived by solving the difference equation initial value problem
a_(n+2) = a_(n+1) + a_n, a_1 = 1, a_2 = 1
Fun stuff, at least to the nerds amongst us!
Dear h2oman,
Lolligag is a good term. I should have thought of that...
I cannot argue your point, since it is correct too, but if you look at another basic Golden Ratio calculation, which happens to be the premise for my "lallygagging" intent while migrating through the Fibonacci sequence, you will recognize that most non nerdy types are taught to take the preceding sequence number and divide this value into the following sequence number. The resultant values from each subsequent calculation produce a value that eventually bumps into the Golden Ratio (Phi), but not immediately as your equation suggests. Somewhere around the fortieth calculation you should see the Fibonacci sequence converge to the Golden Ratio, and it is considered to be accurate to fifteen decimal places.
If you want to try a totally different approach that can keep anyone's youngsters busy with a calculator, while sitting in the backseat of your SUV during a family trip, you could try the following:
1. enter "1" into the calculator
2. take the reciprocal of the displayed number, using the obvious (1/x button) and add "1"
3. repeat step two (2) until the display is constant...
Your approach was first determined by Euclid circa 300 BC, where he defined Phi using the quadratic equation Phi(squared) - Phi - 1 = 0, and where he defined Phi being positive, and greater than 1. His equation of Phi = 1/2*[1+sqrt(5)] = 1.618033988749 etc, etc, etc, happens to illustrate the Golden ratio very effectively.
You could also exact trigonometric formulas for Phi, such as:
Phi = 2 * cos (Pi/5)
You could also illustrate Phi as a nested radical, an infinite series, or even as an irrational number because it has a continued fraction.
Such an interesting number...
jim k
If it only takes 40 calculations to be accurate to fifteen decimal places then it gets "close enough" much earlier than that. I generally don't care about one part in a quadrillion and would usually give up after 5 or 6 significant figures.Originally Posted by jim kitchen
(sqrt5+1)/2 ≈ 42
+1 slartibartfast
It's good to have a sense of humor, and to be skeptical whenever the hyperbole starts to arise. At the same time, it's good to keep an eye open for fundamental design principles. We don't want to throw out the baby with the bath water, as they say.
Gentlemen,
I think your calculators need early retirement...
Jack, I would have quit after the second inverse.
jim k
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