Happy $pi$ day everyone! But it is not just an ordinary $pi$ day, no, it is the ultimate $pi$ day!
Random Walk with $pi$
The Journey of Pi
On the first of January in the year 0, a heavy burden was placed on little boy Pi. He got the task to spread the word about the beauty of $pi$ in as many countries as possible until 3/14/2015 (exactly 736036 days). Each day he could walk/swim exactly 50km in one of the cardinal directions, where the direction was given by the days digit of pi in base 4. That is if the digit is a 0 or 2 he travels 50km east or west. If it is a 1 or 3 he travels 50 km north or south. He starts his journey on (0,0). Which countries will he visit?
Pi actually missed quite a few countries. From a total of 244 countries/territories/colonies, Pi did not visit 111. Even my home country Germany! It is interesting to see, how he “avoided” a big part of Europe and also some African Countries which are not that far from his point of departure.
Due to his weird traveling route, Pi did not only miss some countries but also visit several points more than once. The following map shows, how often he visited certain locations. The darker the color, the more often he was there.
The most frequently visited point by Pi was -100 Longitude and 43.5 Latitude: The Middle of Nowhere, South Dakota, USA. And he went there 57 times… It also becomes apparent, why he missed so many countries. He was sailing around quite a lot on the seven seas. So at the end, was Pi just a Pirate?
Yes, i admit this scenario is a bit whacky. But mathematically still quite interesting. For example the world consisted of 259200 reachable locations in my example. A random walk with 736036 steps could have visited each location almost 3 times! There is a lot more interesting stuff behind this scenario, which I will talk about in later posts.