Do you know how many times you use “Probably” in a day? The word is a part of our colloquial expression because society embraces uncertainty, disorder, and randomness as natural. Whether it’s a rolling die, a tossing coin, or an event, uncertainty is everywhere. This is why the society believes that a butterfly’s wings in Brazil can set off a tornado in Texas.
But mathematicians have illustrated a very contrary certainty always, a certain pattern of randomness. Imagine a robotic beetle placed in a twisting tube. The creature executes an infinite random walk by walking forever, as it moves randomly one step forward or one step back in the tube. Assume that the tube is infinitely long. What is the probability (chance) that the random walk will eventually take the beetle back to its starting point?
In 1921, Hungarian Mathematician George Pólya proved that the answer is one – the infinite likelihood of return for a one-dimensional random walk. If the beetle were placed at the origin of a two – space universe (a plane), and then the beetle executed an infinite random walk by taking a step north, south, east, or west, the probability that the random walk would eventually take the beetle back to the origin is also one.
Let me take you through some more random patterns, which have intrigued mathematicians for ages. Parrondo’s Paradox.
In the late 1990’s, Spanish physicist Juan Parrondo showed how two games guaranteed to make a player lose all his money can be played in alternating sequence to make the player rich. This appeared like a new law of nature that may help explain, among other things, how life arose out of a primordial soup, why investing in losing stocks can sometimes lead to greater capital gains.
Coastline Paradox. British mathematician Lewis Richardson considered this phenomenon during his attempt to correlate the consequence of wars with the nature of the boundary separating two or more nations. He found that the number of country’s wars was proportional to the number of countries it bordered.
Champernowne’s numbers. David Champernowne found a strange sequence of numbers that does not trigger some of the simplest, traditional statistical indicators of non-randomness. In other words, simple computer programs, which attempt to find regularity in sequences, may not “see” the regularity in Champernowne’s numbers. This deficit reinforces the notion that statisticians must be very cautious when declaring a sequence to be random or patternless.
Golden Ratio. In 1509, Italian mathematician Luca Pacioli, a close friend of Leonardo da Vinci, published Divina Proportione, a treatise of a number that is now widely known as the “Golden Ratio”. This ratio appears with amazing frequency in mathematics and nature.
Monster Groups. In 1981, American mathematician Robert Griess constructed the Monster – the largest and one of the most mysterious of the so-called sporadic groups, a particular set of groups in the field of group theory. The quest to comprehend the Monster has helped mathematicians understand some of the basic building blocks of symmetry and how such building blocks, along with some of their subfamilies, can be used to solve deep problems involving symmetry in mathematics and mathematical physics. We can think of the Monster group as a mind-boggling snowflake that exists in the space of 196,884 dimensions!
On one side, patterns in randomness should validate that market pattern seekers might be doing it right. On the other side, it should also tell them that patterns are sure and mathematical, if you can’t prove it mathematically, it is a mirage and you are indeed one of those lost beetles that will take a lot of time reaching home.