In the previous post I used an oak tree to describe self-similarity. I’d like to give a few more examples here, just so you can be more familiar with it.
As I live in a city where there is a big river (the Shannon) I often think of self-similarity as manifest in watercourses.
There are firstly, raindrops settling onto blades of grass which seep downwards into the topsoil and form tiny rivulets a few mm wide and deep, which then join together to form little gullies. These eventually unite to form bigger flows which become small streams – then bigger streams – then smaller rivers, tributaries of larger rivers which then flow into big rivers which flow into a big estuary and then to the sea. They are all unique and different but they are self-similar.
Waves are another example of self-similarity, whether they are in the ocean or cascading over a waterfall. So are the flames in an open fire. Both form patterns that we seem to be drawn to and as such have an almost mesmerising effect on us. How many of us have looked at waves or flames for a long time – transfixed by the constant motion and pattern of the ever-changing-but-still-staying-the-same movement.
Fractals also abound in the arts – probably more in songs than in any other art form. It’s the constant repetition of the melody in the different verses and, of course, the chorus that make songs attractive. Imagine a song with different meter and rhyme and/or melody for each verse – it would not be that attractive.
Getting away from nature and songs, a man-made example of the manifestation of fractal geometry would be a big highway or motorway from which main roads exit, then many secondary roads joining the main roads, still smaller roads extending from the secondary roads and still smaller lanes and boreens and then driveways to people’s houses and even paths to their doors. Once again it is easy to see the self-similarity as the pattern of a small road with boreens extending from it is similar to but not the same as a big motorway with main roads exiting.
In the corporate/business world, it is easy to see self-similarity (though probably not as spectacularly as a motorway) in a big central headquarters with loads of staff, and smaller offices/premises with lesser numbers of staff with less responsibility, and then still smaller premises to deal with issues that arise at ground level. Most armies are organisationally even more self-similar than civilian businesses, with corps, divisions, brigades, battalions, companies, platoons, and sections.
In the next post I will discuss fractal geometry and self-similarity as they apply in the family – our main concern.