|Response to article on Golden Ratio by M.V.N. Murthy in the Sep-Oct 2013 issue of JM
Meghna K.K., The Institute of Mathematical Sciences, Chennai
You have mentioned in your article that the golden ratio is seen often in Nature. In particular, the golden spiral is most beautiful. It can be constructed as follows. Look at the figure (from http://www.yorku.ca/lbianchi/nats1800/lecture10a.html).
Here ACIF and BJKF are golden rectangles. Using a compass, with centre at B, draw the arc AD; then, with centre at J, draw the arc DK, etc. The resulting curve is a golden or logarithmic spiral.
If you have difficulty constructing the golden rectangles, another possible construction using only squares of different sizes (marked in the figure) is shown below.
Golden spirals are seen in many places. Arms of spiral galaxies spread out in this shape.
Also, many mollusc shells exhibit this shape. See the picture of the nautilus mollusc's shell. It has been cut in half to show the chambers clearly arranged in a golden (or logarithmic) spiral. Such a pattern is not just by chance: it allows the animal to grow without changing shape.
Many leaves of plants are also arranged in spiral shapes. The picture shows the criss-crossing spirals of the plant Aloe polyphylla.
Golden spirals are so elegant that they are found in paintings and in architecture. One of the most famous examples is the double spiral staircase in the Vatican Museums in the Vatican City near Rome. It was designed by Giuseppe Momo in 1932.
Note from Editors: Thank you for your information, which is most useful to our readers. Now that they know how to make a golden spiral, maybe they would like to make their own art-work!