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Extras articles - harveyline

harveyline extras

Phi, the Golden Ratio, Fibonnaci and plant growth


Follow this link for an interactive app: Phlower


Phi is discussed in Martin Gardner's book: More Mathematical Puzzles and Diversions. (Penguin Books, 1961). Phi, denoted as ø, is an irrational number which can be defined as:

A+1 / A = A / 1

Which can be solved for A = (1+SQRT5) / 2 = 1.61803398.......

One of the curious properties of phi is the reciprocal is 0.61803398; it is the only positive number that becomes its own reciprocal by subtracting 1.

Phi was familiar to the ancient Greeks and was consciously used by architects in the construction of the Parthenon.

This site illustrates the where phi can be found in that building:



Phi was also recognised by medieval and Renaissance mathematicians, where it was termed the "divine proportion".

The nature of phi is often represented with the "whirling squares" diagram, where the ratio of successively smaller squares is phi. A spiral is typically drawn within the squares. In many instances of this diagram the spiral is approximated by quarter circles.


However, the true spiral is termed the "golden spiral" and is a special form of logarithmic spiral which has the general equation r = ae^b*theta. Where a and b are constants; for the golden spiral b = .306349 radians. As ever there is more at:


Jared Tarbull's website at:


includes a Flash animation which generates the "whirling squares" diagram and the spiral. He also has variations using “whirling triangles” and “whirling stars” to create the golden spiral. Jared can create apparent complexity through the application of few simple rules and recursive equations. The art-work is nicely understated too.

The logarithmic spiral is present in nature, notably in the shells of marine creatures, such as the nautilus. This type of spiral does not alter its shape as it grows, so as a nautilus grows, the shape of its home stays constant, although the volume will increase.


A creature will be energy efficient and create a shell which is optimal expenditure of energy and resources. But the logarithmic spiral is also seen in the spiral arms of galaxies. Why should this be? What balance of physical forces gives rise to this shape? The current thinking is that there is insufficient mass in galaxies to stop them flying apart and so Dark Matter is introduced to provide this apparently invisible mass. Another potential line of research......

An extensive list websites related to phi, Fibonacci and the golden mean can be found here: www.goldenratio.org/info/.

Which includes links to websites that deal with the arts, e.g.


Provides a review of the golden section in art, architecture and music.


This is an intriguing observation about how eyes are positioned in portraits. The author also makes the point that the derivation and/or application of phi in nature and man-made structures is sometimes taken to a point which cannot be verified scientifically.

Many studies have been made that unearth phi in unlikely places, from the Pyramid at Giza to the height of women's navels to the proportions of credit cards. It may be the shape that most people refer, but as Martin Gardener put it most people prefer a rectangle somewhere between a square and a rectangle that is twice as long as it is wide.

There is also an extensive article on phi in Wikipedia.


So where does Fibonacci come into this? An Italian mathematician of the 13th Century he is mostly remembered for introducing the Fibonacci sequence of numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89........

Where successive numbers are the sum of the previous two numbers.

The ratio between any two consecutive numbers approaches phi as the series progresses.

More at:


The material at


provides a good summary of the appearance of Fibonacci numbers in plants with a Flash app that illustrates how phi appears in the pattern of plant growth. The apparent spiral patterns in the seeds are illustrated. Two spirals are apparent, one clockwise the other counter-clockwise. The number of seed elements in each spiral is usually a value found in the Fibonacci sequence.


In the world of computer 3D modelling, the value of phi is used in the creation of parametric plants.

The site at:


has an extensive article on computer generated plants including L-system, phyllotaxis, and plant distribution. As a distraction, the L-system rules can give rise to the Koch snowflake.

The plant modelling application Xfrog uses phi as a parameter; as used in the creation of some of the artwork presented on harveyline.com. The following plant was made in XFrog; the image is from www.greenworks.de and was created by Jan Walter Schliep.


The site at:


provides extensive description, definition and discussion of phi, phyllotaxis and parastiches in plants.

Phyllotaxis is the distribution of plant structures (leaves, bracts or petals) during growth. The distribution can be, but is not always, in some sort of spiral. When the spirals are apparent they are termed "parastiches".


As the plant grows a new seed or petal or leaf is added by "turns". Each turn is either 225.49 or 137.51 deg,. The ratio between these two angles is phi.

The distribution of the seeds, and the enlargement of the nautilus's shell are the organisms way of using its energy budget efficiently. The seeds, or the leaves on a plant are distributed so that all receive maximum light or food in the the available space.

It is possible to imitate this plant growth and distribution of petals or leaves programmatically as presented in Phlower.

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