Fibonacci Sequence

A Brief bio of Leonardo Fibonacci: A Markdown document - link

The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...

The next number is found by adding up the two numbers before it:

Fn = Fn−1 + Fn−2

Fn is term number 'n' (term/iteration = repetition)

Fn−1 is the previous term (n−1)

Fn−2 is the term before that (n−2)

First 10 iterations of the Fibonacci Sequence.

The Fibonacci sequence is an infinite sequence: - link to my project about Infinity on Kaggle.

Fibonacci Sequence generator: A Markdown Document with code chunks for a print and a plot in R - link

Fibonacci Sequence, from the print output of the above Markdown Document.

A spreadsheet with a chart showing the Fibonacci Sequence - link

Chart from the above spreadsheet.

*Golden Ratio and Fibonacci Sequence: *
The Golden Ratio, also known as the Golden Number, Golden Proportion, or the Divine Proportion, is a ratio between two numbers that equals approximately 1.61803. Usually written as the Greek letter Φ (Phi), it is strongly associated with the Fibonacci Sequence. As you progress in the Fibonacci Sequence, the ratio between consecutive Fibonacci numbers approaches the Golden Ratio. The Golden Ratio is derived by dividing each number of the Fibonacci Sequence by its immediate predecessor. F(n)/ F(n-1), where Fn is the term number 'n' and Fn−1 is the previous term (n−1). The larger the Fibonacci numbers, the closer F(n)/F(n-1) gets to Φ (Phi).

*Golden Rectangle and Fibonacci Sequence: *
If you draw squares with side lengths equal to Fibonacci numbers and arrange them adjacent to each other in order of increasing Fibonacci numbers, you will construct a series of rectangles. The rectangles formed by these squares will get closer and closer in proportion to the Golden Rectangle as the numbers get larger. A Golden Rectangle is a rectangle where the ratio of its longer side to its shorter side is the Golden Ratio.

*Golden Spiral and Fibonacci Sequence: *
When you have the series of squares based on Fibonacci numbers, you can draw quarter circles inside each square to get an approximation of the Golden Spiral. This spiral becomes smoother and closer to the true Golden Spiral as you use larger Fibonacci numbers.

The Golden Spiral(true): A Markdown Document with a code chunk for a plot in R - link

The Golden Spiral(true), plot output from the above Markdown document.

The Golden Spiral is a special case of the Fibonacci Spiral, where the growth factor is the Golden Ratio. The Fibonacci Spiral has a cyclic varying arm-radius angle and discontinuous curvature, while the Golden Spiral has a constant arm-radius angle and continuous curvature.

The Fibonacci Spiral can be visualised using a series of quarter-circles with radii that follow the Fibonacci Sequence.

Giphy of the Fibonacci Spiral - link

*Fibonacci Sequence in plants: *

In botany, phyllotaxis (from Ancient Greek φύλλον (phúllon) 'leaf', and τάξις (táxis) 'arrangement') or phyllotaxy is the arrangement of leaves on a plant stem. link

Plants that follow the Fibonacci Sequence in shape include the sunflower (Helianthus annuus) , the brown algae, and the sneezewort (Achillea ptarmica) plant. The sunflower (Helianthus annuus) has both Fibonacci and Lucas numbers of visible spirals, while the sneezewort (Achillea ptarmica) plant has a growth pattern that follows the Fibonacci Sequence in the number of branches. The brown algae also exhibits Fibonacci phyllotaxis.

The arrangement of sunflower seeds tends to follow the Fibonacci Sequence due to the way they pack efficiently into the growing head of the flower. The seeds spiral out from the centre, with each seed placed at an angle of approximately 137.5° (known as the Golden Angle) relative to the one before it. This angle is based on the Golden Ratio, which is closely related to the Fibonacci Sequence.

Sunflower seeds: A Markdown document with a code chunk for a plot in R - link

Sunflower seeds, plot output from the above Markdown document.

*Turing’s Sunflowers project: *

To mark the centenary of Alan Turing’s birth, the Museum of Science and Industry (MSI) Manchester, UK. Ran a project in 2012 which invited members of the public to grow their own sunflower (Helianthus annuus) and either submit their data online or to bring it into MSI for counting, which enabled the collection of a larger dataset; it also meant that wider aims, including participation and learning, became part of the project. link

*DNA molecules: *

The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. The numbers, 34 and 21, are numbers in the Fibonacci Sequence, and their ratio 1.61904 closely approximates Φ (Phi), 1.61803(the Golden Ratio).

Basic double helix plot incorporating the Golden Ratio: A Markdown document with a code chunk for a plot in R -
link (The plot is rotatable within the Markdown document).

Basic double helix, plot output from the above Markdown document (with cross links to simulate DNA rungs).

Addressing concerns about the relationship between DNA, the Golden Ratio, and the Fibonacci Sequence: A Markdown document - link

A list of natural and man made occurrences of the Fibonacci Spiral: A Markdown document - link

A nice spiral shaped weather system, off the southeast coast of Greenland: - link to Earth Nullschool, a global map of current wind, weather, ocean, and pollution conditions, as forecast by supercomputers, on an interactive animated map. Updated every three hours.

Summary:

The Fibonacci Sequence provides a discrete approximation to the continuous properties of the Golden Ratio, Golden Rectangle, and Golden Spiral. The concepts are deeply intertwined in nature, art, architecture, and mathematics.

---

My first exposure to the Fibonacci Sequence was as a child. I was caught doodling in my maths exercise book, which earned me some lines to write out, after the lesson was over. Eating into my lunch break I started my punishment, with a noisy, rumbling tummy. My maths teacher Mr Silverton, came and gazed at my doodles and smiled. He told me to stop writing and read a book he took off his bookcase, instead. The book was about the Fibonacci Sequence. My doodles were almost like the Fibonacci Spiral ! The rest, as they say, is history.
I dedicate this presentation to the memory of Mr Silverton, who is probably smiling as I type this.

Patrick Ford 🌀.