We have encountered many extraordinary phenomena during this course but fractals may top them all, self-similar geometric forms that repeat themselves on various scales, they can both contain infinite detail as we zoom in and the very counter intuitive phenomena of infinite length within a finite form, with all of this being the product of very simple iterative rules.
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</p><p>Transcription excerpt:
Fractals are sometime called the pictures of nature and if chaos theory, as the name implies, is the chaotic and unpredictable dimension to nonlinear systems then we might say fractals represent their orderly side. We see lots of order in our world, the earth goes round the sun today and it will do the same tomorrow and the next day for millions of years, if we take a butterfly we see that one side of it is almost exactly mirrored in the other, we see the same in the regular geometric forms of snowflakes. One way of understanding this order is through the concept of symmetry.
In mathematics, "symmetry" can be defined as an object or process that is invariant to a transformation, in more familiar terms, this means something that stays the same despite a change. So with our example of the butterfly, its external physiology had a reflection symmetry meaning we can simply flip one side over and we will get the other, the same was true of our snowflake and we could imagine some symmetry in the pattern of earth’s orbit.
Without these symmetries to the world the scientific endeavor would be very difficult, because science is about the creation of compact representations of the world, it is like we are creating maps of the world and it is only through finding these symmetries and encoding them in models that we can describe a wide variety of phenomena with simple equations, without these symmetries our scientific map of the world would have to be the same size as the world itself an thus useless. symmetry and asymmetry are then two of the most powerful concepts in mathematics and science for talking about order and chaos, they help us to understand these abstract concepts in a distinctly geometric and visual form.
So with the phenomena of chaos we saw this breaking of symmetry two things that started out similar became increasing dissimilar the symmetry between them became broken and the result was after a short period of time complete asymmetry. Fractals have what is called scale invariance, that is they have a symmetry with respect to scale, meaning the scale can change but the structure will repeat itself over various levels of magnitude. This scale invariance is also called Self-similarity and it is this type of symmetry under magnification, that gives fractals an amazing type of structure and order. Fractals are both mathematical constructs that derive out of iterative functions and real world phenomena, in this module we will present them as mathematical models and then in the next look at their real world counter parts.
As we have seen with most of nonlinear systems the core ideas behind
fractals is of feedback and iteration. The creation of most fractals involves
applying some simple rule to a set of geometric shapes or numbers and then
repeating the process on the result. For example the most famous fractal
called the Mandelbrot set, so named after the discover of the concept of
fractals, is a product of a simple iterative map on complex numbers. We will
not go into the details of complex numbers but the iterative map itself is
quite simple. There are many extraordinary things about fractals but the first
thing we will note is the infinite variety that these simple iterative functions
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can produce.
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