Design Principles - Fibonacci

Liber Abaci also introduces the Hindu-Arabic numeral system and compares the system with other systems, such as Roman numerals, and provides methods to convert the other numeral systems into Hindu-Arabic numerals. Replacing the Roman numeral system and its ancient Egyptian multiplication method, with a Hindu-Arabic numeral system using an abacus for calculations, was an advance in making business calculations easier and faster. This thus led to the growth of banking and accounting in Europe.

Back to the Fibonacci Sequence. We can see examples of the sequence in nature most clearly in plants. For example, most flowers can be placed into petal categories, notably 3,5,8,13 or 21. Seeds in plants will usually add to a number in a Fibonacci Sequence, as well as sections in plants, such as an apple, which when cut in half will have 5 seeds, encompassed in 5, equally-spaced, containers. Rows of seeds and seed pods in nature will almost always add up to Fibonacci Numbers. Most fruit and vegetables you use regularly will have a Fibonacci Number somewhere on it, from sections, to seeds, or leaves.

For those without flowers or fruit at their disposal, your hand will suffice. You have 2 hands, 5 fingers divided in to 3 parts, by 2 knuckles.

These examples, by no means prove that the Fibonacci Sequence is prevalent in all of nature. For example, the reason there are X rows holding X seeds, is because it is mathematically the most efficient way to store that many seeds, giving the plant a higher chance of reproduction. You could easily argue that there are actually more examples in nature disproving it. However, it does crop up an awful lot, and even more so when linked to the Golden Ratio.

They’re two separate ideas, however they do coincide very neatly. The Golden Ratio can be devised if we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ...) and we divide each by the number before it.

So, for example,

1/1=1       2/1=2          3/2=1.5        5/3=1.6666…          8/5=1.6          13/8=1.625          21/13=1.6153

As you can see the first numbers are irrelevant, but once the numbers get higher, each answer is very similar. As they get higher, they get closer and closer until they match. However, as an average value, it is approximately 1.618034.

This was originally discovered by the ancient Greeks which they named ‘PHI’. The Golden Ratio has been used for countless uses over the years from Greek sculpture, to current-day architecture. It is still debated as to whether it is aesthetically pleasing or whether or not we have only convinced ourselves that it is and used them repetitively. It can also be argued whether we have manifested this perfect form or whether it is a subconscious desire, something primitive that we hold within us, a more ‘natural’ choice.

The ‘Golden Rectangle’ can be used to accurately describe how both the Golden Ratio and the Fibonacci Sequence can combine. A Golden Rectangle is one that’s sides are successive Fibonacci Numbers (1x1, 2x3, 3x5, 5x8, 8x13 etc). These can then be divided up into other rectangles that also have sides with successive sides, albeit including 1x1 (square). Once divided, an arc from each corner of an inner rectangle to the opposite will create a large spiral, reminiscent of classic natural shapes such as the Nautilus or a snail’s shell.

Examples of the Golden Ratio can be seen in the Parthenon, Stradivarius Violins, and in human anatomy. If we take the ratio as 0.618 and think of a segment divided in two, labelled as such.

a-------------------------b------------c

The ratio, or ‘relationship’, between the two is the sum of the two segments. i.e. If you swing your mind back to when you did algebra – bc/ab = ab/ac = 0.618. Using anatomy as the example, the ratio can be seen by looking at your leg and your arm. Both can be divided into two segments; the arm from fingertip to elbow is represented here by a/b, from elbow to shoulder by b/c. In all the examples, the ratio between the segments is approximately the Golden Ratio.

The Golden Ratio and the Fibonacci Sequence continue to be extremely prevalent in all areas of design. They can be found in the strangest places, from Beethoven and Mozart’s symphonies, to Corbusier’s ‘Modular’, which is incredibly useful. They can be used from compositions, measurements and patterns. Sometimes you will do it without even noticing, as we are constantly surrounded by it. Do consider, however, that forcing the Ratio or Sequence into a piece of work or design can be perceived as forced or contrived.

References and Links:

https://en.wikipedia.org/wiki/Fibonacci_number#History

https://en.wikipedia.org/wiki/Golden_ratio#History

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

https://en.wikipedia.org/wiki/Fibonacci

https://www.livescience.com/37470-fibonacci-sequence.html