Alex's Adventures in Numberland - Alex Bellos [139]
A particularly useful visualization of randomness was invented by John Venn in 1888. Venn is perhaps the least spectacular mathematician ever to become a household name. A Cambridge professor and Anglican cleric, he spent much of his later life compiling a biographical register of 136,000 of the university’s pre-1900 alumni. While he did not push forward the boundaries of his subject, he did, nevertheless, develop a lovely way of explaining logical arguments with intersecting circles. Even though Leibniz and Euler had both done something very similar in previous centuries, the diagrams are named after Venn. Much less known is that Venn thought up an equally irresistible way to illustrate randomness.
Imagine a dot in the middle of a blank page. From the dot there are eight possible directions to go: north, northeast, east, southeast, south, southwest, west and northwest. Assign the numbers 0 to 7 to each of the directions. Choose a number from 0 to 7 randomly. If the number comes up, trace a line in that direction. Do this repeatedly to create a path. Venn carried this out with the most unpredictable sequence of numbers he knew: the decimal expansion of pi (excluding 8s and 9s, and starting with 1415). The result, he wrote, was ‘a very fair graphical indication of randomness’.
Venn’s sketch is thought to be the first-ever diagram of a ‘random walk’. It is often called the ‘drunkard’s walk’ because it is more colourful to imagine that the original dot is instead a lamp-post and the path traced is the random staggering of a drunk. One of the most obvious questions to ask is how far will the drunk wander from the point of origin before collapsing? On average, the longer he has been walking, the further away he will be. It turns out that his distance increases with the square root of the time spent walking. So, if in one hour he stumbles, on average, one block from the lamp-post, it will take him four hours, on average, to go two blocks, and nine hours to go three.
As the drunkard randomly walks, there will be times when he goes in circles and doubles back on himself. What is the chance of the drunk eventually walking back into the lamp-post? Surprisingly, the answer is 100 percent. He might stray for years in the most remote places but it is a sure thing that, given sufficient time, the drunk will eventually return to base.
Imagine a drunkard’s walk in three dimensions. Call it the buzz of the befuddled bee. The bee starts at a point suspended in space and then flies in a straight line in a random direction for a fixed distance. The bee stops, dozes, then buzzes off in another random direction for the same distance. And so on. What is the chance of the bee eventually buzzing back into the spot where it started? The answer is only 0.34, or about a third. It was weird to realize that in two dimensions the chance of a drunkard walking back into the lamp-post was an absolute certainty, but it seems even weirder to think that a bee buzzing for ever is very unlikely ever to return home.
The first-ever random walk appeared in the third edition of John Venn’s Logic of Chance (1888). The rules for the direction of the walk (my addition) follow the digits 0–7 that appear in pi after the decimal point.
In Luke Rhinehart’s bestselling novel The Dice Man, the eponymous hero makes life decisions based on the throwing of dice. Consider Coin Man, who makes decisions based on the flip of a coin. Let’s say that if he flips heads, he moves one step up the page, and if he flips tails, he moves a step down the page. Coin Man’s path is a drunkard’s walk in one dimension – he can move only up and down the same line. Plotting the walk described by the second list of 30 coin flips chapter 9, you get the following graph.
The walk is a jagged line of peaks and valleys. If you extended this for more and more flips, a trend emerges. The line swings up and down, with larger and larger swings. Coin Man roams further and further from his starting point in both directions. Below are