### Visualization of the Unified Theory of Randomness

Intro

So a few weeks ago, I was discussing with a family member about fractal visualization. We both got really pumped about it, and then they told me about an article they read in Quantum Magazine. It was about how these scientists had created a unified theory of randomness, and could visualize it with fractal shapes, noise, and 3D software. I found this really interesting, even more so after making the mathematics article(HERE). So I decided to go ahead and research it. Here we go.

What is Randomness?

Randomness is lack of patterns or predictability in events. It also contains the idea  that symbols, steps, numbers, etc involved in it do not contain order. So how do we use randomness? In math a lot of equations involve and include the use of a random variable. What this variable will do in an equation, is include the calculation of something random (we don't know or can't predict) with the variables we can, and therefore we've suddenly calculated all of our possible outcomes. You can also randomly select groups of things, numbers or people to create an unbiased census in surveys. Which if you need accurate results in  your data collection.

Monte Carlo method also uses random variables. This is a computational algorithm that relies on random number sampling. This algorithm is used in creating probability generators,  numerical integration equations, and solving equations or problems that would take longer with other forms of math. As well as much more. In some ways, being random can help us achieve a better result in anything we do.

So how can we predict randomness, and what is the Unified Theory of Randomness?

There is a theory that random shapes can be categorized into different classes. These classes can have distinct properties of their own, and also have similarities with other types of random objects. This theory helps form what is called geometric randomness. This study was first formulated by two professors at the University of Cambridge named Scott Sheffield and Jason Miller. They've proven that objects such as trees, paths, and rocks have relationships in their shapes. Using the formulas of these shapes you can develop even more equations.

Professors Scott Sheffield and Jason Miller have gone on to take this information even further, and now have successfully mapped out and visualized the relationships of random two-dimensional planes and geometry. (I've linked their paper below.)  Now using this form of geometric randomness, they've shown how 3D objects can be visualized on random 2D planes.

The Visualization and Mapping of Objects

When visualizing random objects we also have to take into considerations the shape and appearance of normal ones. A normal piece of geometry could be classified as a cubes, spheres, lines, or oval shapes. These shapes have order. As for most of them, if you have the information and location of the points of the object, you can calculate their shape. But how does this relate to a 2D plane, and what would it look like?

Let's use an airplane as an example. Let's say on a piece of paper you wanted the plane to fly to one corner to another. If you drew a line between these two corners or points it would be a straight line. But if the plane had to fly from Toronto to another place on the globe,  the plane would have to follow the curvature of the Earth to get there. So that line would now become a curve. A straight line would be in this case be the visualization of a basic curve in a 2D environment.

But what would happen if the 2D plane was to change it's shape? Would this also change the appearance of the line? Yes. If the 2D plane was warped or moving surface, then the line would become more swirled and erratic. However, you can still predict where these lines would form. You would just need to know the starting and end points of your chosen line, and some of the point locations of the random plane.

How Would This Apply to Physics?

String Theory

String Theory is an accepted model of how gravity, electromagnetism, and nuclear forces interact with each other. Einstein first proposed this theory to help explain how classical physics and quantum physics interacted with each other. As well as how the universe operated as a whole with time, light, and theoretical particles. Michio Kaku a Theoretical Physicist, is credited with completing Einstein's incomplete study into the theory of everything, which then gave the name string theory.

String Theory describes particles as points, and the path that they travel across the universe as lines. These lines are then called strings, which are described as being one dimensional. The theory also helps define how these strings interact with each other as they travel across the universe.

Now as mentioned above, in order to map the shape of these lines and random behaviors we need to know the shape or points of the plane the shape is traveling on, and the start and end points of the shape. However, no one knows the true shape of the universe as it is always expanding changing shape. We also are currently unable to track and observe the start and end path of particles as they move around us. These strings would be considered under our randomized theory as we can only predict them through theoretical quantum equations. However, for particles that we do understand such as electrons or protons, we have more luck. Because we have measured the frequencies and motions of these particles we can actually predict how they would move on electromagnetic level.

How Can We Use This in Visual Effects?

In our chosen 3D software we can by default create basic shapes. We can generate objects that have recursive fractal shapes such as trees or leaves with l-systems. We can also generate basic curves such as Sin, Cos, and Tan, and other mathematical equations. But how can we generate randomized lines and objects?

We are fortunate that with a software such as Houdini we know the size, measurements, and starting position of our 3D space. We are also are fortunate that we can customize the measurements of the space we are operating in, and the points where we would like something to start and end. The only difficultly would be mapping the random growth from point A to point B, and introducing random factors.  You could use something like The Super Formula or a Fractal formula to create the line traveling from point A to point B, but then this shape won't be considered random.

However, there is something called random fractals which you can visualize. These random fractals have a direct correlation to Stochastic geometry. Which is the study of random spacial patterns.

The simplest random spacial pattern you could generate would be a Boolean Model. Which operates on the probability theory that each random point on your random plane will overlap and create recursive shapes and lines to appear. This theory operates for both 3D and 2D planes, and you can find the equations for it HERE.

Something that has already been used in computer graphics is something called Mathematical morphology. This is the study of image processing and it creates images out of set theories and random functions. It was first used to create binary black and white images, and to track that image's pixel data. This morphology technique is also used for function visualization and color image processing.

Finally, you can also visualize something called Random Graphs. Random graphs map the random probability placement of points over any random process that generates them. This feeds into something called network science, but I'll let you look into that on your own.  These random processes are called random trees, which are formed with random variables that multiply over time. The Erdős–Rényi model would be your best bet for generating a random graph and you can find it's equation HERE.

Overall, I think this is a cool route to explore for physics sake. We could probably generate some really cool shapes and sequences from these equations. There are way more than what I've mentioned here, but a few of the concepts were a bit to complicated to condense.  I'll leave the creativity up to you, and I hope you enjoy reading up on these theories more.

References

Applications of randomness: https://en.wikipedia.org/wiki/Applications_of_randomness

Liouville quantum gravity and the Brownian map:

Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding: https://arxiv.org/abs/1605.03563

String theory: https://simple.wikipedia.org/wiki/String_theory

String theory:

RANDOM FRACTALS:

Stochastic geometry:

Boolean model (probability theory):