Creating Randomness

Human beings have always shared a profound connection with unpredictability. In ancient times, people sought out random events to forecast their destinies. Over time, these early techniques evolved into games of chance and were even utilized in governance. The Enlightenment period introduced new methods for incorporating randomness into scientific explanations. Clearly, the generation of random numbers serves various human needs.

But what exactly does "random number" signify? Here’s a somewhat technical definition from Wolfram MathWorld. Don’t be concerned if parts of it seem complex; I'll clarify it afterward.

> A random number is one selected as if by chance from a particular distribution so that a large collection of these numbers reflects the underlying distribution. Typically, these numbers must also be independent, meaning there are no correlations between consecutive selections.

In essence, this means that a random number is inherently unpredictable. However, we can generally assert which numbers are more likely than others. For instance, I can confidently state that a 7 will not appear when rolling a standard six-sided die. Though rolling a die is a random act, it adheres to specific rules. Many random numbers follow a distribution that indicates the likelihood of different results. Yet, possessing knowledge of the distribution does not enable us to predict a single outcome.

The latter aspect of the formal definition is crucial. Consider the scenario where you've rolled a die three times, landing on a 1 each time. You might think that rolling a 1 again is unlikely. However, a key principle of randomness is that the chance of rolling a 1 remains constant, regardless of previous outcomes. This notion can be counterintuitive, as it defies common sense. Even after rolling three 1s, the probability of rolling another 1 does not change. For a process to be genuinely random, this rule must hold.

Distributions need not be evenly balanced. If you've played the game Settlers of Catan, you're likely aware of this fact. Instead of rolling one die, let’s consider the scenario of rolling two dice and summing their results.

My examples have revolved around dice for a reason. Historical records indicate that humans have utilized dice for over 5,000 years! They have been integral to divination, casual gaming, and gambling throughout civilization. Let's delve into the history of dice.

The First Random Numbers

Numerous historical sources indicate the use of dice in ancient times. The Bible references dice multiple times, typically termed as "casting lots." Early versions of dice were often crafted from bones or clay and frequently appear in archaeological findings. Dice achieve randomness due to the many minor variations involved in rolling them. Given the unpredictability of hand movements and the arrangement of air molecules at the time, accurately predicting the outcome is virtually impossible.

While the standard cube die is the most prevalent, many other forms exist. If you've played Dungeons and Dragons or other board games, you are likely familiar with various types of dice. Most of these are Platonic Solids, which possess properties conducive to randomness. The ten-sided die, often formed by kites, is an exception, typically paired to generate random numbers between 1 and 100.

However, dice can take on even more unusual forms! My personal favorite is the Möbius strip die, humorously designed to yield only the number 1. Throughout history, people have devised various methods to create different types of dice. Early dice were fashioned from bones and exhibited unconventional shapes, yet effectively generated random numbers.

> “As an instrument for selecting at random, I have found nothing superior to dice.” — Francis Galton

Randomness manifests in far more profound ways. While humans have devised useful tools to simulate it, nature employs its own mechanisms. Large-scale phenomena like evolution depend heavily on randomness to produce beneficial mutations, and uncertainty also pervades at the atomic level. We have merely begun to explore these intricate phenomena; let’s discuss the basics.

Quantum Randomness

Approximately a century ago, physicists made a groundbreaking discovery: upon delving deeper, they found that certain physical processes are inherently random. Numerous experiments have validated this concept of true randomness. Initially, scientists resisted this revelation as it contradicted classical physics that had prevailed for centuries. Most notably, Einstein was troubled by this notion, famously stating:

> “God does not play dice.”

Interestingly, this quote is often misattributed. What Einstein actually said was:

> “[t]he theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.”

Einstein dismissed the random interpretation of quantum physics, yet he has repeatedly been proven incorrect. The randomness inherent in fundamental matter carries intriguing implications that scientists continue to investigate. The realization that we do not inhabit a clockwork universe is both unsettling and reassuring. Let’s consider an example to elucidate this concept.

If you’ve studied physics, you might be familiar with a beam splitter. It’s a device commonly employed to divide a laser into multiple beams, each of reduced intensity. Now, imagine lowering the input to the point where only one photon strikes the beam at a time. Even with perfect knowledge, predicting the photon’s path upon hitting the beam splitter remains impossible.

At this scale, we witness one of the rare instances of true, pure randomness. While we could theoretically predict how a die would roll with sufficient information regarding hand movements and air conditions, this does not apply to a beam splitter. No amount of data can determine the outcome.

Physics presents other illustrations of this phenomenon, such as in radioactive decay. Although we can estimate the average rate at which an atom decays, predicting the emission of each individual beta particle is impossible. Similar to the dice, prior events bear no influence on the timing of the next particle's decay; only a predicted distribution of results exists.

Researchers have sought to simulate processes like those described above on computers for various studies. While a coin toss or dice may suffice for isolated examples, often we require the simulation of multiple random events, as illustrated earlier. However, computers operate deterministically, so how do we address this?

Getting More Sophisticated

Computers primarily employ two methods to generate random numbers. The most common is known as pseudo-randomness. These algorithms start with a number called a seed (a term familiar to Minecraft enthusiasts). This initial number is used to generate subsequent numbers and typically derives from the millisecond value recorded by the computer during execution. The algorithm repeatedly utilizes this seed number in an equation, yielding a lengthy sequence of random numbers.

While pseudo-random numbers are practical and relatively easy to generate, they have limitations. The sequence will almost inevitably repeat after a certain point (usually in the millions). Additionally, if someone knows both the seed and the equation, they can replicate every single random number generated. This makes pseudo-random numbers generally unsuitable for cryptographic applications; however, with additional effort, they can be made more secure.

More intriguing—yet complex—are methods that harness natural sources of randomness. Although these sources are not perfectly random (as they could be predicted with sufficient information), our current computational capabilities fall short of making accurate predictions. Once data is gathered, an algorithm can utilize it to produce a genuinely random sequence of numbers. This method is more labor-intensive but significantly more reliable than the previously mentioned pseudo-random approach.

I often refer to this process as “harvesting entropy.” Common sources include atmospheric noise, Brownian motion, electronic circuit noise, and the two physical processes I previously mentioned. Numerous innovative techniques have emerged for this method, and I’ve included additional examples below.

The study of randomness and its generation remains a vibrant area of research. As computers play an increasingly dominant role in our lives, randomness becomes an ever-more essential tool.

Going Further

I hope this exploration has been enlightening! Randomness has played a crucial role in nature, and humans have leveraged it throughout our history. If you're interested in delving deeper, I’ve provided a few links below for further exploration.

• RANDOM.org is a fantastic resource dedicated to randomness, featuring numerous useful random number generators and insights into their data sources. A similar site, LavaRnd.org, creatively uses lava lamps to generate random numbers!
• For a formal introduction to probability, I highly recommend Probability: for the Enthusiastic Beginner.
• To browse a fascinating collection of unique dice, check out this shop.
• The Stanford Encyclopedia of Philosophy offers an insightful comparison between Chance and Randomness, making for a compelling read.

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