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Idea Seeds #09 – Fruit, Veg and ‘Phi"

Updated on December 24, 2016

The Story of Phi

Nature and the physical world are truly wonderful and made even more so when you see the amazing relationships between them and the world of ‘geometry’. In his book, ‘The Golden Ratio − the story about ‘Phi’ − the extraordinary number that is found in nature, art and beauty’, Mario Livio, Professor and Head of the science division at the Hubble Telescope Institute, explains why people from ancient to modern times and from all walks of life have been so fascinated by its beauty. The book is full of examples of where it is found and used and is truly a joy to read.

The Never Ending Number Called ‘Phi’

The size of the mundane sheet of foolscap paper that was used before A4 became the standard was chosen to be close to the shape of a ‘golden rectangle’. The length to width ratio of the sheet is 1,625, very close to 1,618033….., the never ending number called ‘phi’.

So what makes this shape special? If you remove a square with sides equal to the width then the shape of remaining bit will again have the shape of a ‘golden rectangle’. If you remove a square from this remaining bit the new remaining bit will again be a ‘golden rectangle’.

No matter how many times you do this the shape of the remaining bit is the same. All very boring stuff until you find that it is linked to how some of nature’s creatures grow.

The sectioned shells and a diagram of a spiral formed in a ‘golden rectangle’ are shown on the right. They should make this clear. Find a piece of A4 paper and follow the instructions in the diagram and draw your own spiral.

Pineapples and the Fibonacci Sequence

Where do fruit & Veg fit in? The next time you see a pineapple have a close look at it. You will see that each of the scales is hexagonal in shape like the shape of cells in the honeycomb made by bees. In the three diagrams of pineapples you will see that each scale is part of three different spirals. Some pineapples have five, eight, thirteen or twenty one spirals of increasing steepness on their surface. These numbers are all found in the ‘Fibonacci sequence’.


Leonardo Fibonacci and Rabbit Breeding

Leonardo Fibonacci (1170 – 1240) was born in Pisa, the site of the famous Leaning Tower of Pisa, in Italy. He introduced the Hindu-Arabic numbers (0,1,2,3,4,5,6,7,8,9) that we use today, to Europe. Numbers at that time were written in Roman notation (I,V,X,L,C,D,M). As there are no easy ways to add two Roman numbers together or multiply one by another it makes doing calculations impossibly difficult. The story of Fibonacci’s life and achievements is fascinating and it is well worth spending time to learn more about him and the people he influenced and who had influenced him. In his book ‘Liber abaci’ (Book of the abacus) he uses the following problem to explain the sequence that today bears his name: “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair from which in the second month that pair becomes productive?”

Closer and Closer to Phi

Each of the rabbits in the diagram represents a pair. The sequence for adult pairs is: 1, 1, 2, 3, 5, 8, and for baby pairs 0, 1, 1, 2, 3, 5. Adding these together gives the total pairs 1, 2, 3, 5, 8, 13….. that would then in the next six month be 21, 34, 55, 89, 144, 233, …. If you look more closely you will see that each number in the sequence is the sum of the two numbers that appear just before it. So the next number in the sequence given above is: 144 + 233 = 377. The sequence arises in a wide variety of situations and yields some fascinating results. If you choose one of the larger numbers in the sequence and divide it by the number that precedes it, the ratio is close to phi. If you had chosen 233 and then divided it by 144 you would get 1,61805… If you choose bigger and bigger numbers in the sequence and do the calculations you will see the ratio gets closer and closer to phi.


Leafy Spirals

If you have read my previous articles you will know about the ‘principle of least action’ and the role it plays in nature. Branches in plants and trees form the scaffolding that supports their leaves. Their placement has been optimised to give each leaf the best chance of getting its share of sun, rain, and air. In some trees and plants, branches grow from the trunk from points at spirally spaced intervals to avoid one leaf growing directly above another and shielding the lower one from getting its share of resources. In one of his thirty-seven volume collection titled ‘Natural History’, Pliny the Great, (23 – 79) a Roman scholar with wide ranging curiosity wrote about the “regular intervals branches were arranged circularly around their stems”. Leonardo de Vinci (1452 – 1519) noted that in some plants leaves were arranged in a spiral patterns that repeated in groups of five but it was the famous astronomer, Johannes Kepler, (1571 – 1630) who first saw that the phenomenon was linked to the ‘Fibonacci sequence’.

Clockwise and Counter-clockwise Spirals

Much has been written about sunflowers and the clockwise and counter-clockwise spirals that can easily be seen in them. If you count them you will find that the totals of clockwise spirals compared to counter-clockwise spirals are consecutive ‘Fibonacci numbers’.

Similar spirals can be seen in this beautiful picture of a ‘Euphorbia’ species of plant taken by Dr Graham Grieve at Ngele on the Transkei/Natal border In South Africa.


The Golden Angle

‘Phi’, ‘the golden ratio’ 1.61803 refers to linear measurements but it can also be expressed in circular measurements by dividing 360 degrees by phi. This gives angles of 222,5 and 137,5 degrees. The latter is referred to as the ‘golden angle’. Petals arranged in a rose bud are spaced at 137,5 degree intervals to ensure that petals do not line up along any radial direction which would be the case if they were spaced at say 120 degrees. Here again nature has chosen the best and most efficient way to pack the petals. I have great difficulty in restraining myself from pulling every rose bud I see apart to again see the beauty of it all.


The Pentagon

This brief introduction to a few of nature’s wonders will hopefully motivate you to look for other examples. Are the leaves on a cabbage arranged like a rose? How are the florets on a cauliflower packed together? How do onions grow? The questions are endless. Finding answers to some of them has brought me much pleasure.

‘The golden ratio’ and the ‘Fibonacci sequence’ crop up again and again in the strangest of places; Art, Architecture, Music, Fractals, and more. The closest geometrical shape to the ‘golden ratio’ is the five sided ‘regular pentagon’. Cut an apple in half at its ‘equator’ and look at the arrangement of the seeds. You will see that the tips can be enclosed by a pentagon as shown in the diagram on the right.


Fruits and boats that float or sink

If you don’t do so already, start helping in the kitchen. When you wash a fruit do some experiments on it. Drop it into a bowl of water and see whether it floats or sinks. You will find that some varieties of pears and peaches sink while apples float. See whether they float in a stable way. If, for example, you turn an apple on its side does it return to its original orientation? Does it do so sluggishly or rapidly? How do your observations link to boats and ships and submarines? How does a banana float and in how many positions is it stable? Can you relate the shape of a banana to the shape of the kayaks used by the Inuit’s in the Arctic? What effect does the shape have on the stability and how does this tie up with ‘Eskimo rolls’? Remember Richard Feynman’s wise advice: “Learn by trying to understand the innumerable little and simple things you see around you in terms of other ideas”.


Geometry and Visualization

When you cut spherically shaped fruit see how many shapes you can make and name. Hemisphere; spherical segments, spherical wedges, spherical caps; spherical zones etc. Use the fruit to make models of the earth, mark the poles and then slice through the equator and make four mores slices, two through the tropics of Cancer and Capricorn and two more through the Arctic and Antarctic circles. Make sure you can explain why these slices mark important limits. Think equinox, solstice, and midnight sun. The ancients designed the ‘armillary sphere’ to help them ‘visualise’ the ‘geometry’ of the universe. All the things named in this paragraph form a part of it so start making the links.

I wrote in a previous article that ‘Geometry’ and ‘Visualization’ go hand in hand and that they are high on my list of critically important ‘idea-seeds’. Be creative and experiment and your ‘visualization’ and model making skills will rapidly improve.

Cone Flower
Cone Flower


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