### The Golden Ratio (by Erik Willison)

#exploremaths

The golden ratio is found throughout architecture and nature. Examples of this are the Notre-Dame Cathedral and the Great pyramids. In nature we can see it in sunflowers, plants with spiral leaves and even in spiral galaxies.

Firstly architecture, The Notre-Dame Cathedral was built way back in 1163 and was designed by some random guy that was of great importance to somebody; anyway, this spectacular work of art has golden rectangles scattered throughout it. the three primary examples are:

1. The ratio of the height of the first floor to the height of the second floor (starting from the top of the first floor) is the golden ratio, or pretty close to it (outlined in red)

2. The height of the second floor to the height of the third floor is a golden ratio (outlined in blue)

3. The ratio of the length of the top left box including the space between it and the top right one to the length of the right one by itself is another golden ratio.

My second example of the golden spiral in architecture is the Great Pyramids of Giza. Each of these pyramids use the golden section. The diagram below is so good that it is basically self-explanatory, but for those who are lost I shall explain. Inside this pyramid you can see a triangle reaching from the centre of one edge to the centre of the base and up to the very top of the pyramid. The Egyptians built this pyramid with dimensions that just so happen to be able to be simplified into the diagram below where Phi is the hypotenuse. The dimensions that the Egyptians used were:

– A base of 230.4m

– An estimated original height of 146.5m

The ratio between these two numbers is 0.636. Now I know that 0.636 is not the golden ratio and may seem quite far off 0.618etc.; but the height only needed to be 0.0376m taller and the ratio between these dimensions would fit the golden ratio to those three significant figures.

Onto nature. Nature proves to be mathematical in every sense and likes to use the golden spiral.

Starting off with the sunflower; the seeds of a sunflower grow in a particular pattern although they may seem to appear scattered randomly. A sunflower needs to decide how far it should turn around before it grows a new seed from its centre, the angle that it turns will determine how much of the flower’s circular centre will be filled with seeds, and obviously the aim is to optimise this number. Through micro-evolution, or something like that, the sunflower came to a decision: it will have each new seed that grows from its centre 137.5 degrees (or close enough to that) appart. This value seems random but it is infact the corresponding angle to the golden ratio, also know as the golden angle. and that is where the sunflower uses golden mathematics, optimizing its seed growth.

Secondly, in nature, we have plants with spiral leaves. These types of plants simply use the golden angle to grow their leaves so that no new leaves block out the sun of exiusting leaves on the plant, also this leaf formation helps dew and rain that fall on the plant flow directly towards its roots.

Finally, on the biggest scale in the universe, we have galaxies. A spiral galaxy, like our own, has arms that are meant to wrap around it as the galaxy turns from the centre. For some unknow reason, the arms of a spiral galaxy seem to defy this, and with that newtonian physics, as the stars on the extremities of the arms move at a higher velocity; thus maintaining the golden spiral in our galaxy.

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