The famous Fibonacci sequence has captivated mathematicians, artists, designers, and scientists for centuries. Also known as the Golden Ratio, its ubiquity and astounding functionality in nature suggests its importance as a fundamental characteristic of the Universe.

Here’s more about the Golden Ratio from a mathematical and architectural standpoint:

Here’s more about the Golden ratio found in nature:

And another one for good luck:


 Constructing some phyllotaxic surfaces from Sunflower pattern:

It’s really magical that this works at all, since the spatial relationship of each seed to its neighbors is unique, changing constantly as the pattern expands outwardly—unlike, say, the cells in a honeycomb, which are all equivalent. If so, We wondered the same magic could be applied to surfaces that are not flat, like spheres, toruses,….. It’s an interesting question from an aesthetic point of view, but also a practical one: the answer has applications in space exploration and modern architecture.

Here is three secrets of the arrangement to reproduce the flat sunflower pattern mathematically:

  1. Seeds spiral outward from the center, each positioned at a fixed angle relative to its predecessor.
  2. The fixed angle is the golden angle, γ = 2π(1 – 1/Φ), where Φ is the golden ratio.
  3. The ith seed in the pattern is placed at a distance from the center proportional to the square root of i.

See more about technical analysis : How I Made Phyllotaxic Surfaces & Wine Glasses from Sunflowers by the author Christopher Carlson on

Some Images : Sunflower (gif)  & Sunflowers on Flickr



Anyone who has ever claimed there is no validity in the profoundness of “Sacred Geometry” needs to think again… This is completely mapped out using phi rectangles as a grid forming a perfect 64 star tetrahedron, which is a foundation to the building block of everything that creates this plane of existence. This is a multi part study so expect more variations based off of this model. ‪#‎sacredgeometry‬ ‪#‎geometry‬‪ #phi #fibonacci #goldenratio #mathematics #fineart #digital ‪#‎samuelfarrand‬

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The Golden Ratio φ can be derived from the Fibonacci Sequence:

Two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. Phi has many interesting and unique mathematical properties.The Golden Ratio φ can be approximated by process of successively dividing at term into the Fibonacci Sequence by the revious term. With each successive divition, the ration comes closer and closer to a value of 1, 618033987…….

2 : 1      = 2.0000                3 : 2      = 1.5000…

5: 3       = 1.6666…             8 : 5      = 1.6000…

13: 8     = 1. 6250…            21 : 13  = 1. 6154…

34 : 21  = 1. 6190…            55 : 34  = 1. 6176…

89 : 55  = 1. 6182…            144 : 89 = 1.6179…

233 : 144 = 1.61805…


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Here is an old post - Explain φ = (1 + √5)/2.

Image by flora-file on Tumblr.


Pincones - Fibonacci numbers in Nature.

On each pinecone , a double set of spirals – one going in a clockwise direction and one in the opposite direction. When these spirals are counted, the two sets are found to be adjacent Fibonacci numbers. In other words, the number of spirals in either direction are two consecutive Fibonacci numbers.

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Images: [1] - [2] - [3,4]