Monday, May 7, 2012

Interesting formula...and graph

Have you seen a mathematical formula, when graphed in two dimensions, can exactly reproduce the formula itself visually? Here is one:

               {1\over 2} < \left\lfloor \mathrm{mod}\left(\left\lfloor {y \over 17} \right\rfloor 2^{-17 \lfloor x \rfloor - \mathrm{mod}(\lfloor y\rfloor, 17)},2\right)\right\rfloor ,
where \lfloor \cdot \rfloor denotes the floor function and mod is the modulo operation.

When proper parameters are chosen for x and y, which at the same time satisfy this inequality, the resulting graph will look like this:

             Tupper's self referential formula plot.png,

which, as you can see, looks exactly the same as the original formula. Interesting, huh?

See more here. HT: Feng Dong.

Sunday, May 6, 2012

Two books on power & prosperity

One is Why Nations Fail by Daron Acemoglu and James Robinson. In this book, the authors explain why some nations are rich and prosper and others poor and remain so. In contrast to contemporary theories made popular by journalists or economists, their explanation is that, it's neither geography or cultural factors, nor the policy makers' ignorance about what's best for the society as a whole that determine the wealth of nations. Instead, it's the nation's economic and political institutions that matter (in the authors' terms, "inclusive" vs. "extractive" institutions). The authors make the book extremely readable and engaging by using lots of compelling examples and putting together hundreds of years of historical accounts in a coherent and logical manner. Read a couple of book reviews herehere and here, or listen to one of the authors discussing it here

The other is Private Empire: ExxonMobil and American Power by Steve Coll. Unlike Acemoglu and Robinson's take on power and prosperity from the why perspective and at the national level, Coll's book reveals how power and prosperity can emerge and co-evolve via the private empire building process. Watch the author discussing the book here.