Fibonacci as a pattern of proportional relationships
Fibonacci as a pattern of shapes that we can learn to see. As part of this Fibonacci series of articles we have looked at the Fibonacci numbers and what the sequence represents. In this article we look at the visual presentation of that proportional relationship as patterns.
Patterns are what I’m good at, much better than numbers, so this bit about Fibonacci I really enjoy. this post is to help you and your young people visualise it as a pattern. It is a ‘self accumulating’ pattern. Like it is a self accumulating number sequence.
Fibonacci as a pattern in spirals
Much is always said about Fibonacci and nature so lets start by looking at the pattern that it is most associated with in nature – The Spiral.
The most accurate (nature is never perfect remember!) to fibonacci spiral in nature is the Nautilus shell. whose proportions appear to grow in reflection of the divine proportions also known as the golden ratio.
Other spirals occur in nature, these are sometimes referred to as a fibonacci too. These spirals will have growth proportions that relate to the number sequence or have spiralling swirls that total up to a Fibonacci number. These can all be called fibonacci spirals yet they differ proportionally to the golden proportion that governs the relationship between the numbers in the Fibonacci sequence.
The Fibonacci numbers as spirals present themselves in a pine cone and other spiralling sets of seeds. The total number of spirals in one direction and the other direction will often appear as consecutive numbers in the Fibonacci sequence.
Being in ‘right relationship’
When we first understand the presence of the numbers in nature, we could all be forgiven for thinking that nature can count!. But nature’s superpower is not counting. But the ability to be in right relationship to its neighbour and grow as a self accumulation process.(link to everything you need to know page/ video link)
As a pattern of golden relationships
Each of the numbers in the Fibonacci sequence can become a square shape, simply by cutting out of grid paper.
For example: the 6th number in the sequence is 8, the red square. This becomes a 8cmx8cm (use whatever measurement you would like here, the unit is not the important bit) or 8 x 8 square on grid paper.
Each square has a side length that equals a number in the sequence. (These images use just the first eight digits in the sequence). To make a fibonacci puzzle game, draw a diagonal line across each square and then work out how to make the spiral pattern.
The formula for the puzzle is as follows: the two grid paper squares for consecutive numbers put next to each other, make up the length of the side of the next grid paper square in the sequence. (it’s much easier to ‘see’ it in the photo, words just complicate it all really)
Orientate each square so that the diagonal line connects with the next square. You will have a created the Fibonacci spiral. Taking a curved line instead of the diagonal will make it feel more natural shaped and recognisable.
The spiral you have just drawn is a Golden Spiral, which is a built to the proportions of the golden ratio/divine proportions. As we know from the What really is the Fibonacci sequence? article that the number sequence is an expression of the divine proportion/golden ratio. Therefore a spiral to these proportions is referred to as a fibonacci spiral.
Fibonacci as a pattern of squares
Once this concept of Fibonacci as a pattern of squares has been grasped. It is easy to translate this pattern making into any other craft that is based on squares.
for example :
A Pattern of Golden Rectangles
Within the square pattern we also see it as a pattern of golden rectangles. The proportions of each rectangle match the Golden Ratio/Divine Proportion. This is the Phi relationship. (this article can explain more about PHI)
This Phi relationship governs the relationship between the measurements of the long and the shorter sides of each rectangle.
Fibonacci is a relational pattern
You can see from all these examples of Fibonacci as a pattern. Fibonacci is a relational pattern. It is an expression of a relational experience.
It is a relational pattern that can see in many forms in the natural world.
Patterns in maths
The fibonacci relationship is often expressed as fractions and decimals and digits. At its heart it is a relational pattern. It is the Phi relationship between each number as we discussed in the What are the Fibonacci numbers? article. The divine proportion. It is the ultimate social number sequence. Once you recognise that pattern and you will be amazed at the places you begin to recognise it. This understanding begins with seeing it creates many different versions of these patterns like in these activities shared here.
Fibonacci as a pattern of pentagram stars
Relating to the number Five, the pentagram star is one expression of the Five archetype. These stars have the divine proportional relationship within them. The Fibonacci numbers are an expression of the divine proportion. Because of this we can draw a 5 pointed star using the number sequence to help us.
This is an infinite pattern. The Five archetype within it holds the power of regeneration, so from one pentagram star we can keep drawing stars upon stars upon stars. The pentagonal pattern pack activity provides a series of printed sheets to be able to replicate this infinite star making ability.
Seeing the Five pointed Star in Nature FREE ebook
This is a new way of thinking about numbers. And this is a new way of looking at the natural world. Includes 60 colour photos and 20 diagrams to visualise the understanding.
One line of the pentagram holds the golden ratio in its points. No measurements needed. The small to the larger line is equal to the larger line to the whole. Known as the Golden Mean in geometric drawings, or the golden ratio/divine proportion as we have said earlier.
The link (below) is to an activity video – one of a series I made on Facebook – demonstrating the pentagram pattern making process. Download the pack at the Resource Library page.
Fibonacci as a pattern of triangles
Within the pattern of stars is Fibonacci as a pattern of triangles. The ebook discusses this. These are also triangles that hold the divine proportional relationship within them. Really, once you start seeing this relational pattern it is in available to see in many different ways.
Fibonacci as a pattern of straws
Use the Fibonacci numbers as straight lines. The best way to do this is to cut lengths of straws, each equal to a number in the sequence. Once the numbers get taller than the length of straw you will need to attach multiple straws together- using artstraws is the best way to do this.
For more crafts based on using the Fibonacci numbers, go to the Fibonacci crafts page.