The Fibonacci Sequence: Mathematical patterns in nature and projects
Consider the following sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. And so on, into infinity. What do you notice?
You get the next number in the sequence by adding up the previous two numbers.
This is the Fibonacci series.
Life
Leonardo of Pisa, known as Fibonacci, was born in 1170. He accompanied his father, Guglielmo dei Bonacci, a wealthy merchant, on his travels. Leonardo studied with a Muslim teacher, who guided him in calculation techniques with Indo-Arabic numbers, which were not yet used in Europe.
Fibonacci was educated along the trade routes in Egypt, Syria and Greece, before returning to Pisa in 1200. He spent the next 25 years writing mathematical manuscripts.
His book ‘Liber Abaci’ was written for merchants, and covered product prices, calculating business profit and converting Mediterranean currencies.
He discovered his famous sequence when studying rabbits and the number and frequency of children that they, and their offspring, have.
Golden ratio
The Fibonacci series has a ratio between any number and the previous one that tends towards a well-defined value, 1.618. This is known as the ‘Golden ratio’, φ (Phi), that frequently occurs in nature. It’s believed to be the most aesthetically pleasing ratio – faces often approximate this 1.6:1 aspect.
Observing geometry in plants, flowers or fruit, we recognise recurring structures. The Fibonacci sequence plays a role in phyllotaxis, which studies leaves, branches or seeds, highlighting patterns. Curiously, the plant kingdom prefers spiral symmetries and follows surprising mathematical regularities.
We find Fibonacci numbers in the circular shapes of daisies, sunflowers, cauliflowers and broccoli. Leaves are aligned in patterns that include two Fibonacci numbers. Starting from any leaf, after one, two, three or five turns of the spiral there is always a leaf aligned with the first and, depending on the species, this will be the second, the third, the fifth, the eighth or the thirteenth leaf.
Practical use
The Fibonacci sequence can be used to estimate tasks in technology projects, and in ‘Agile’ software development.
A critical part of managing an Agile team is estimating the time that tasks will take to complete. Functionality and features are grouped into a logical process sequence called a “User Story”. For example, “I want to invoice a customer, in an accurate and paperless way, and send the invoice digitally on the day the work was done,” We call this the “What, How, Wow.” The business outcome being to collect cash more quickly.
A points system is often used for forecasts. While you could use a different scale for estimating the functional components of this user story, such as shirt sizes (XS, S, M, L, XL), using Fibonacci’s sequence up to 21 (1, 2, 3, 5, 8, 13, 21) is a great choice, because it’s exponential: when estimating time, the shorter the time span the more certainty there is. Longer tasks are more complex, and the estimates are less precise. An exponential scale (growing at an increasing rate) provides detail for small tasks (“Is it a 1 or a 2?”) and forces uncertainty for large tasks (“Is it a 13 or a 21?”).
Perhaps “1” could equate to half a day, or two hours. You then have a basis for estimating effort (in hours/days). You can constantly refine this – nothing is set in stone, and no-one has a crystal ball.
The Golden Ratio and Fibonacci often get attributed to architecture: the Pyramids and the Parthenon, and to art: Da Vinci and Picasso. Some of this is conjecture. No matter. Truth is much more compelling than facts.
James Neophytou