The Golden Ratio is a special irrational constant that can be observed across all levels of the universe, and so it has been nicknamed "God's Fingerprint". The ratio itself is about 1.618 or -0.382 based upon its definition: Phi = (1 + sqrt(5))/2 and phi = (1 - sqrt(5))/2 There is also the Golden Angle which is most commonly seen in plants, but can also be seen in the spiraling arms in galaxies. The Golden Angle is 137.5 degrees. The Golden Ratio is most associated with the Fibonacci Numbers.
Leonard Bonacci aka The Fibonacci
Fibonacci was a mathematician in Pisa, Italy (the same place with the tower that people try to prevent from leaning) during the 12th century. He tried to develop a model for the population growth in rabbits by starting with 2, and seeing where things went from there (if you know what I mean). This produced the definition for the Fibonacci Numbers, which is as follows: F1 = 1, F2 = 1, Fn = F(n-1) + F_(n-2) This definition generated the following sequence of numbers, formally known as the Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The interesting thing about this is that as n approaches infinity, the ratio between these number approaches the Golden Ratio.
Golden Ratio Calipers
This is a device used to measure the Golden Ratio in things. You can get some on EBay for about $20 if you want to do some recreational mathematics.
So you like pinapples?
This sort of pattern is rather common with Fibonacci Numbers appearing in the developmental patterns of plants. The reason for this is yes unexplained, however some have conjectured that it is simply the most efficient way of doing things.
Leaves and the Golden Ratio
So here we can see the golden angle at work, in the development of leaves, and this is caused by the hormone used to grow leaves and how it is displaced and stuff. I can do a post on this alone if y'all want.
La fleur de l'amour
Here we can see how the golden spiral is represented in the shape of the rose, which is likely why it is so appealing to look at (I'll get there).
Dandelion Seeds: good for more than just blowing
As can be seen here, the seeds of a dandelion form interweaving Golden Spirals which form a pretty neat pattern.
Sunflower Seeds: good for more than just munching
As can be seen here, the seeds of a sunflower form the same pattern as the dandelion ... coincidence?
The branches and leaves of a fern form Golden Spirals in their development, as it allows for easy expansion, but just as easy storage.
The fractal patterns form layers of Golden Spirals all throughout, which again is due to efficiency.
Similar to the pineapple, the pine cone forms many spirals, where each type sums to a Fibonacci Number.
Aloe: good for more than treating solar radiation damage
Note the similar fractal patterns to the Romanesco Broccoli and that there are 5 spirals, which 5 is a Fibonacci Number
Here is another example of a Golden Spiral however this one is a geological formation rather than a floral formation.
Yes, even Hurricanes have the Golden Spiral in all its destructive magnitude!
It turns out that even objects obeying the laws of physics produce the Golden Spiral! Who knew?
As can be seen in this picture, the spiral shapes of galaxies also form Golden Spirals (I'm thinking the Illuminati are involved somehow ...).
So, ants also have the golden ratio, which is why they are such bad ass nopes! In fact most arthropods (fancy word for nopes) have the golden ratio in their body's length!
Yup, most seashells also produce the Golden ratio. This type of seashell is particularly special, because on top of having as similar pattern to the Romanesco Brocolli, it exhibits the golden ratio in its height v width, as well as in the height of each adjacent layer!
The Golden Ratio even exists within DNA
The Shape of the DNA also follows the Golden Spiral!
Even out teeth exhibit the Golden Ratio.
Our ears have the Golden Spiral in them, which is pretty cool, particularly because now when you see someone listening to a shell, you can exclaim "spiral-ception"
So, even the shape of our faces, mouths, nose, eyes, and pretty much every other part of our faces exhibit the Golden Ratio and Golden Spiral, which again is likely why we collectively consider the face to be aesthetically pleasing. For all the scientists out there(****): http://i.imgur.com/YCtXiRu.jpg Note: that this is likely why we consider bosoms to be so aesthetically pleasing. So the next time a girl catches you staring, just say you're observing the Golden Ratio in action. If anything, it'll at least be a conversation starter ;)
So because the Golden Ratio is engrained in our DNA and is pretty much everywhere in the universe (damned Illuminati), it would only make sense for us to replicate it in both architecture and art.
The CN Tower: Golden Ratio approved
Notre Dame: Golden Ratio approved
The Pyramids of Gyza: Illuminati Confirmed
The CN Tower: Golden Ratio approved
The Panthenon: Golden Ratio Approved