know anything about the GOLDEN RECTANGLE?

hi i have a math project to do and its really stupid but i have to research things about the golden rectangle/the golden mean/ratio/spiral-whatever you want to call it- and i need to: "study its appearance in art, architeture, biology, and geometry, and its connection with continued fractions, Fibonacci numbers, etc..." If you could find anything that would help that would be greatly appreciated!! thank you!! =)

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  • 1 decade ago
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    A link to the image I will be refering to is provided below.

    http://www.clintnmary.org/images/GoldenRec2.jpg

    The golden rectangle isn't the same thing as the golden ratio (this is a common mistake). In the diagram, phi is the golden ratio, and the large rectangle is the golden rectangle. Note that you could also use the Greek letter φ to represent the golden ratio, as they really are the same symbols (i.e., both are pronounced "phi").

    Notice that the outer rectangle has length phi and width 1. Also, the inner rectangle has length 1 and width phi-1.

    Since, for each rectangle, the ratio of width to length (or vise versa) is the same, you can write

    1/phi = (phi-1)/1

    which is the same as

    1/phi = phi - 1

    This means that you can obtain the reciprocal of phi, the golden ratio, simply by subtracting the number 1. How many numbers have that property?

    Now notice what happens when you multiply both sides by phi. You get the quadratic equation

    phi² - phi - 1 = 0

    which has the solution

    phi = (1+/- √(5))/2

    The positive root of this solution is the golden ratio:

    golden ratio = phi = (1+√(5))/2

    Using as many decimal places as I can, this number is

    1.6180339887498948482045868343656

    Thus, the golden rectangle can be used to obtain the golden ratio.

    As for this number's connection with continued fractions, consider the following sequence of calculations, each of which approximates the golden ratio with increasing accuracy.

    1+1/(1+1/(1)) = 1.5

    1+1/(1+1/(1+1/(1))) = 1.666...

    1+1/(1+1/(1+1/(1+1/(1)))) = 1.6

    1+1/(1+1/(1+1/(1+1/(1+1/(1))))) = 1.625

    1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1)))))) = 1.615384615384615

    1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1))))))))

    = 1.617647058823529

    and so on.

    See the second equation in

    http://www.phys.ualberta.ca/physicsupdate/images/e...

    for the general case.

    This sequence has an extremely interesting geometric interpretation, which is illustrated in the folloing links:

    http://images.google.com/imgres?imgurl=http://www....

    http://web.mit.edu/cjoye/www/art/img/Fibonacci1.jp...

    As for art and architecture, I would suggest starting with the ancient Greeks, as they were the first to discover both the golden ratio and the golden triangle.

    One thing I would point out, however, is that it is important not get too carried away. Much of the mystique surrounding the golden ratio is just that: mystique. Here is an excellent article which explains what I mean:

    http://www.maa.org/devlin/devlin_06_04.html

    It also discusses the golden ratio's relationship with Fibonacci numbers.

    "The oft repeated claim (actually, all claims about GR are oft repeated) that the ratios of successive terms of the Fibonacci sequence tend to GR is also correct. The Fibonacci sequence, you may recall, is generated by starting with 0, 1 and repeatedly applying the rule that each new number is equal to the sum of the two previous numbers. So 0+1 = 1, 1+1 = 2, 1+2 = 3, 2+3 = 5, etc., giving the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... The sequence of successive ratios of the numbers in this sequence, namely 1/1 = 1; 2/1 = 2; 3/2 = 1.5; 5/3 = 1.666... ; 8/5 = 1.6; 13/8 = 1.625; 21/13 = 1.615...; 34/21 = 1.619...; 55/34 = 1.6176...; 89/55 = 1.6181; ..., does indeed tend to GR. As I'll explain momentarily, this is a key part of the explanation of why the Fibonacci numbers keep appearing in flowers and plants - which they do."

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