The Recurring Beauty

Recently, I stared reading one of the essential books for a computer scientists, namely Concrete Mathematics. After the first chapter, it really got me thinking "Oh My!! Recurrences could solve anything." This is of course not matched to the thrill of the challenge in funding recurring patterns in a structure.

Along my college years, colleges tend to step away from recurrences. In fact, many cases it is the easier approach to tackle. Furthermore, it could be in some cases the more efficient approach. This reminds me of many algorithm techniques that largely depend on finding substructures or small solutions that are used for generalizations. The most vivid example comes to me in the form of memoization in dynamic programming (DP). For me the corner stone of DP is finding the "magical" recurrence. Thus, from super slow, to super fast (lots of memory use though :D).

Other examples of how recursive thinking makes complex problems elegantly simple could be seen in graph theory. Just by examining some properties on nodes, arcs, degrees, huge complexities just boil down to some elegantly beautiful properties that are easily generalizable.

Such beauty could be easily seen when recurrences could end up in something that is visually breath taking. An example are Fractals, a stunning example of how simple math with recursion lead to amazing imagery.