Type and grid - Fibonacci sequence and golden section

A task set by Phil was to go away and investigate the meaning of the fibonacci sequence and the golden section, and attempt to discover how it can be applied to art and design, below id the information if have found. 

 Fibonacci sequence

In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:

 (sequence A000045 in OEIS)

By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

In mathematical terms, the sequence F_n of Fibonacci numbers is defined by the recurrence relation

with seed values^[3]

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics. By modern convention, the sequence begins either withF₀ = 0 or with F₁ = 1. The Liber Abaci began the sequence with F₁ = 1, without an initial 0.

A tiling with squares whose side lengths are successive Fibonacci numbers

Fibonacci numbers are closely related to Lucas numbers in that they are a complementary pair of Lucas sequences. They are intimately connected with the golden ratio; for example, the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... . Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.

Golden ratio

An approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.

In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.

Applied to art

As the Golden Section is found in the design and beauty of nature, it can also be used to achieve beauty and balance in the design of art.  This is only a tool though, and not a rule, for composition.

The Golden Section was used extensively by Leonardo Da Vinci.  Note how all the key dimensions of the room and the table in Da Vinci’s “The Last Supper” were based on the Golden Ratio, which was known in the Renaissance period as The Divine Proportion.

A more detailed view of Da Vinci’s intricate use of the Divine proportion is available by using PhiMatrix

Sources:

http://en.wikipedia.org

http://www.goldennumber.net/art-composition-design/