Fractals for Christmas 🎄

Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same world differently.

–Benoît Mandelbrot, The Fractal Geometry of Nature (1982)

We often naively see dimension of a space X as the minimum number of information we need to provide to specify any point in X uniquely. For example one can think of specifying a point in a space that has only one point. So we need to provide no information and hence a point gets a dimension 0. Then, to specify a point on a line we need to provide just one coordinate. So a line gets a dimension 1. Now comes the interesting part — fractional dimensions. One of the first things (or rather a doubt) that can come across our mind when we hear the term “fractional dimensions” (especially after reading the naive introduction we presented here) is how can a fractional number of information be provided for specifying something. But what can happen is — there is some minimum number n_0 among integers of coordinates we need to specify, but less than n_0 coordinates may be good enough (in some sense) due to self similarity. Let us see an example to understand what we are saying here.

Consider the Cantor Set for instance. How can we specify a point in this space ? One way is to mention the coordinate of the point. Since this space is a subset of the straight line so mentioning one coordinate is good enough. Again since this space has more than a single point so specifying no information will not allow us to specify the points uniquely. So 1 is sort of a minimum threshold on the number of information (among integers) we need to provide. But again it is self similar because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated. So it seems (in some sense) that specifying less than 1 number of information can be good enough. This motivates us to think about some fractional dimension playing under the ground (and actually this was one of the examples and reasons that motivated the study of fractional dimensions).

In loose words, we call such spaces with fractional dimensions fractals.

And all these loose statements about dimensions can be made rigorous by bringing Hausdorff dimensions.

One can play with fractals and create a wide variety of patterns for creating beautiful shapes. For example the Koch snowflake can be created starting from an equilateral triangle and erecting equilateral triangles on the middle third part of each side of the shape we had at the previous stage.

Here is a X-mas tree created using fractals.

Merry Christmas to everyone !! Enjoy the day and the year ahead !!