The Golden Spiral and the use of Geogebra were central to this project. What follows is part of the material students were presented with and worked on.

Golden number φ - Fibonacci sequence- Golden spiral

 

In mathematics, two quantities are in the golden ratio if the ratio is the same as the ratio of their sum to the larger of the two quantities.

 

So, for two quantities a and b with a>b>0:

 

 

Where the Greek lettet phi (φ or Φ ) represents the golden ratio and its value is φ=1,6180339887........ . The golden ratio is also called the golden number. The letter phi (φ) is the first letter of the name Phidia (Φειδία). Phidias was an Ancient Greek scultor, painter and architect, who made the Parthenon statues that seem to embody the golden ratio.

 

Fibonacci sequence

 

The Fibonacci sequence is the integer sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,....... and is characterized by the fact that every number after the first two is the sum of the two preceding ones.

 

Fibonacci sequence and Golden ratio

 

The ratios of two succesive numbers of the Fibonacci sequence after the first one is:

 

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We see that the ratio of succesive Fibonacci numbers converges to the golden ratio φ.

 

An orthogonal rectangle whose side lengths are 13 and 8, is the golden orthogonal rectangle because the numbers 13 and 8 are succesive numbers from the Fibonacci sequence which converges to the golden nymber φ.

 

The spiral which is inscribed to the golden rectangle is a golden spiral and the Fibonacci spiral approximates the golden. The Fibonacci spiral is inscribed to the rectangle whose side lengths are 13 and 8. In the picture below we can see the golden spiral :

 

We create the golden spiral with using the geogebra

 

 

 

 

Golden spiral and nature

 

Approximate golden spirals can occur in the nature, for example: the Nautilus

 

 

which resembles the pattern of  the Ionian column