Golden Section

Description

The Golden Section (aka Golden Ratio) was first described by Euclid, the ancient Greek mathematician, and since then its relationship to the world around us has been extensively explored - and debated! It is a number often found when looking at the ratios of distances in simple geometric figures, such as the Pentagon, and appears to relate to proportions found in nature.

It was discovered that the ratio of 1.618 can be obtained by continuing the Fibonacci Series (ideally to infinity!). It was given the term Golden Section in the mid-19th century and assigned the Greek letter φ (phi) at the beginning of the 20th century.

The Rectangles constructed in this proportion are known as Golden Rectangles and we seem to find these visually pleasing.

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Activity: Draw assorted rectangles scattered randomly on a sheet of paper: some long and thin, some more squat and some approximating the Golden Ratio or adjacent numbers on the Fibonacci Series.

Ask a group of friends to state which 1, 2 or 3 rectangles they feel are the most well proportioned; the most pleasing to the eye, and record the results.

It is probable that the Rectangle(s) most closely approximating Golden Rectangles will come out on top.

This activity is described with others in Kurt Rowland's book 'The Shapes We Need' (from the excellent 'Looking and Seeing' series).

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A particular property of Golden Rectangles is that when a Square is added to the longer side another Golden Rectangle is created - and this process can be repeated to infinity.

When a group of Golden Rectangles is constructed in this way, a Spiral can be drawn through opposite corners of the Squares. This closely approximates a Logarithmic Spiral, which is also known as the Growth Spiral or the Golden Spiral. It can be approximated by drawing quadrants in each square as shown.

As plants grow, each new seed has to find space for itself and yet stay closely packed to its neighbours to maintain support. This is achieved by growing outwards in a spiral. The angle each new seed or leaf grows in relation to the one before it is critical: any angle that is a simple fraction will after a while, make a pattern of lines stacking up, which will leave gaps (e.g. if 1/3 of a revolution each time the result would be 3 floppy seed arms which would be destroyed after the first gust of wind or leaves stacked on top of each other and putting each other in the shade).

The Golden Ratio is not only an Irrational Number but is about as far from being a fraction as it is possible to go. The result is that after each revolution the new seed or leaf is positioned at a different point on the circumference than its predecessors. This ensures that new leaves for example, get the best possible access to sunlight without obscuring other leaves.

As a consequence there are many examples in nature of growth patterns similar to a Golden Spiral. Furthermore, counting leaves and petals can reveal many instances of numbers which are part of the Fibonacci Series.

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