Aristotle described Gnomon as adding an “L” shaped figure to an existing square, such that the resulting figure is still a square. Euclid extended the concept to a parallelogram, and finally Hero extended the concept to any figure that, after being added by any figure, retains its original form.
It is interesting to note that all Fibonacci numbers from 3 are Gnomons, that too alternatively Aristotelian and Euclidian Gnomons. That is 3 (=2^2-1^1) is Aristotelian Gnomon, 5 (=23-11) is Euclidian Gnomon, 8 (=3^2-1^1) is Aristotelian Gnomon, 13 (=35-21) is Euclidian Gnomon, and so on.
It is interesting to note that all odd numbers are Gnomons in the following form
What is even more interesting is that, in biology, when any form (such as a shell or a horn) grows in a logarithm spiral, each successive increment of growth is a gnomon to the entire pre-existing structure. See the picture below
The shell shows three major gnomons, at three growth phases, with different pigmentations. We have marked the boundaries by two arrows. This shell can be considered as a Hero’s Gnomon.
Clearly, this is a very interesting observation. If nature follows log spirals and gnomons in its structures, then, it would not be unreasonable to expect to find the same in data relating to some social or natural phenomena. If we look for the gnomons in our data, then it is possible that we may be able to unearth the basic simpler building blocks by which a complex social phenomenon is built. Once we chance upon the simpler building blocks, drawing conclusions and making predictions is the next step. Is it not the goal of every machine learning exercise to make useful predictions?
In the previous blog posting we sketched a logical reason why the biological world may be most comfortable in the log scales & how we can use this knowledge to our advantage. In this blog posting, we have sketched a path which biology may follow to build macro structures by marrying log scales with Gnomons. Popular media is full of stories and images of Fibonacci numbers in biological world, without offering any explanation why do they occur in the nature. Here, in two short blogs, we have shown that it is the property of Fibonacci numbers of being Gnomons, coupled with nature’s tendency to work on log scales, we find these patterns everywhere. We believe this deep understanding can be used to deconstruct any social and natural dataset to find the basic building blocks.