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May 16th, 2013, 10:59 PM  #1  
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Skyscrapers Consisting of the Golden Ratio's Perfectly Proportional Standard
Over the past few days, ThatOneGuy and I have been discussing buildings that consist of the Golden Ratio of proportions. When numbers consist of the golden ratio, a/b=a+b/a, as shown below:
The Golden Ratio is precisely 1.618033988749..., or (1+√5)/2, and is represented by the Greek letter φ (phi). It can be generated by drawing a midsegment of a triangle inscribed inside a circle and extending that segment until it intersects with the circle's circumference, as shown below: A golden ratio triangle is where one of two leg segments on the triangle is equal to segment AB above, and the base is segment BC. Base angles are precisely 72 degrees, and the third angle is 36 degrees. These triangles appear on the outsides of pentagrams and the interior of decagons. The number is also the basis of the Fibonacci Sequence, where each square's perimeter is Phi x the smaller square's. It is also the solution to the problem y = x^3x^2x when y is equal to zero. This thread's purpose is to discuss buildings using the Golden Ratio, and should help us all learn much through looking into it. Eric, I'll give you the honors of posting the first building (I think we both know why ). Cheers, L.A.F.2.
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Last edited by L.A.F.2.; May 17th, 2013 at 03:11 AM. 

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May 16th, 2013, 11:42 PM  #2 
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Ah, I love the Golden Ratio  it's so simple, yet beautiful. One of the reasons I became addicted to math (BSc for now). Only few theorems like Euler's formula are more aesthetic / elegant.

May 17th, 2013, 12:03 AM  #3 
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Yup. Golden Ratio's an aspect of architecture that often goes ignored. Basically if the shorter side of a rectangle is multiplied by 1.62 and it reaches the length of the longer side, that's the golden ratio. The Fibonacci Sequence works with the golden ratio as well.
For example Ryugyong Hotel has a base width of around 204 meters. Multiplied by 1.62, it reaches around 330 meters, it's height. You can see where the setback starts from the Fibonnaci sequence spiral, and where the edge of the square meets with the edge of the sloped building side. The cone section, up to the antenna, is also in proportion to the golden ratio, as you can see from the diagram. [img]http://oi41.************/bgw9vn.jpg[/img] Some other examples [img]http://oi40.************/2h7f8y8.jpg[/img] [img]http://oi42.************/2jd16s8.jpg[/img] Last edited by ThatOneGuy; May 17th, 2013 at 12:08 AM. 
May 17th, 2013, 02:24 AM  #6 
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I know Gothic architects loved the Golden Ratio, but I had no idea the CN Tower used it!

May 17th, 2013, 03:08 AM  #7  
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Neither did I, but now that I think about it, it really does look like it! I'm sure there are many other OB Towers which do, but I'll have to do some more digging and math.
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May 17th, 2013, 03:08 AM  #8 
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May 17th, 2013, 03:15 AM  #9  
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That's a great one. If you look closely, you'll see that each of the rectangles of ornament above the Doric columns uses the Golden Ratio. The building's a gold (haha) mine for the number.
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May 17th, 2013, 12:37 PM  #10 
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Fantastic. Please keep them coming.

May 17th, 2013, 07:24 PM  #12 
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Very interesting! So the base to the pod is roughly 1.618 times the height of the pod to the pinnacle?
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World's 1st Baseball Game: June 4th, 1838, Beachville, Ontario, Canada North America's Oldest Pro Football Teams: Toronto Argonauts (1873) and Hamilton Tiger Cats (1869) I started my first photo thread documenting a recent trip to Halifax, Nova Scotia. Have a peek: http://www.skyscrapercity.com/showthread.php?t=724898 Last edited by isaidso; May 17th, 2013 at 07:31 PM. 
May 17th, 2013, 10:23 PM  #13  
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The pinnacle height (553.3 m) divided by 1.618 is 341.95 m, which is 1.95 m above the main OB deck's roof (~340 m).
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May 18th, 2013, 05:38 AM  #14  
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Well that's the inverse way of looking at the same ratio. Off by 1.95 m would suggest that they did look to the Golden Ratio. There's something to be said for the eye being able to recognize pleasing proportions though: it could be a coincidence.
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World's 1st Baseball Game: June 4th, 1838, Beachville, Ontario, Canada North America's Oldest Pro Football Teams: Toronto Argonauts (1873) and Hamilton Tiger Cats (1869) I started my first photo thread documenting a recent trip to Halifax, Nova Scotia. Have a peek: http://www.skyscrapercity.com/showthread.php?t=724898


May 18th, 2013, 06:42 AM  #15 
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A bit of background and added information which may be helpful: The images of the CNN Tower and Notre Dame in Paris are from http://www.goldennumber.net/architecture/, which also shows the golden ratio proportions found in the Taj Mahal, the Great Pyramid of Giza and the Parthenon. The golden ratio grid lines shown on Notre Dame above were created with PhiMatrix, a golden ratio design and analysis app which makes it easy to reveal and apply the golden ratio and other custom ratios with dozens of templates.

May 19th, 2013, 01:11 AM  #16 
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The golden ratio is greatly overrated overmystified if you ask me. Some people seem to consider it some kind of magic superharmonious division.
First of all, in architecture it is a very practical kind of measurement, if you have a ground plan with the golden ratio, one (golden ratio) room can be followed by a big square room and the ground plan keeps the same (golden) proportions. It is also not superharmonious because of some divine plan, it looks harmonious because it is somewhat between square and long, therefore buildings do not look semisquare and also not narrow/long. If you intuitively design a building that mustn't be too square and not too long it automatically falls somewhat in that range. Nothing mystical about that. That said, the same counts for [√2 by 1] which forms the basis for A4, A3, A... paper. And that has the advantage that a √2rectangle can be cut in half with both halves still √2 by 1. A4 doesn't sounds very sexy right? Perhaps that's the reason why there are no websites devote too it. A lot analysis of the golden ratio in ancient buildings are bad and greatly arbitrary. Just look at the icture of the Notre Dame above . Bad, because a lot of times the measurements don´t fit correctly. Arbitrary, because in every building can always an element be found to semialign your rectangle with. Isn't it the top of a dome, then you can pick the top of the lantern on top of the dome. If the windows don't fall within the right frame, they pick the door, and so on. A lot of the spirals found in nature do nót follow the golden ratio, even though it is claimed that they do, including the famed nautilus. Like spiral galaxies. There are an estimated 100bln galaxies in the observable universe. Among those are irregular galaxies, cluster galaxies, barred spiral galaxy and then there are the spiral galaxies of which just a fraction resembles the golden spiral. The same is done with all those plants that supposedly resemble the golden spiral. And above all. You can make ugly buildings with the golden ratio and beautiful building without. 
May 20th, 2013, 02:03 AM  #17 
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Re the post from Vincen1, it is true that the golden ratio is often claimed to exist where it does not. Most spirals in nature are equiangular or logarithmic, but that does not mean they have anything to do with the golden ratio.
While the properties of the golden ratio may not make it "magical" or "mystical", they go far beyond being simply a "practical" compromise between square and too long. The golden ratio has unique mathematical properties unlike any other number, as well as unique geometric properties. Its innate aesthetic appeal is likely due to its appearance in the human face and body. Perhaps its application to the image of the Notre Dame needs explanation:

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