MATHEMATICAL THEORY AND THE GOJO SATORU HELLSCAPE
Read on! Here’s my last lil tidbit on physics. Disclaimer that I’m not a physicist or a mathematician! I’m just a massive nerd.
Just for interest! Interviews with Gege say that Gojo is able to manipulate relative distance (aka. space) between targets, but his own canon explanation of infinity states that he is slowing an approaching target.
Blue is achieved by reducing a given distance. 1000m can be collapsed to 10m and so forth in an infinite series (think of a dividend number series where the values are reduced as n goes up), which results in a rapid attraction towards a central point.
Red is achieved by amplifying a given distance. For example 1mm is blown up to 10m, to 100m, to 1000m- resulting in a rapidly expanding repulsive force.
Hence, his Infinity might be explained via a controlled adaptation of the above. The distance between himself and an approaching target grows smaller and smaller, never quite reaching zero and stopping at a given point (a quantifiable distance value or a self-determined limit) which I assume he can control as his barrier is wider at times and practically skin tight at others. He alters not time, but speed by proxy of manipulating distance.
All that’s relatively simple! What’s more difficult is the explanation of Purple. Gege states that Purple is a combination of Red (推) and Blue (拉).
[ 推 = 拉 ]
However, the laws of science dictate that a combination of equivalent repulsive and attractive force will result in a stable equilibrium. This state is low in entropy and minimises potential energy, able to resist external displacement by default. As a result, this isn’t exactly a destructive state.
[ 推 > 拉 ] - [ 推 < 拉 ]
The first would be an example of a badly executed Red. Both forces are exerted, but the repulsion of Red trumps over in the end. Vice versa for the second.
For Purple, I’ve read interviews where Gege specifically cites mathematical theory regarding imaginary numbers and Kullback-Leibler divergence.
Imaginary Numbers
The former illustrates the result of the quadratic equation x² + 1 = 0 , hence x = √-1. This solution however, is an impossibility considering that the square of a real number cannot be negative. √-1 is referred to as i, which is an imaginary number. In addition, consider also that there is an ambiguity with regards to whether the value of i is a positive or a negative value since the resultant value of a root is theoretically ambiguous by default.
(-2)² = +4 (+2)² = +4 Hence with x² = 4, x could either be - 2 or +2. Same goes for i.
Any number which become associated with the element i are known as complex numbers (e.g. 3i + 4, 2i). More on this later.
Kullback-Leibler (KL) Divergence
KL divergence refers to a likelihood ratio concerning the two probability values of P and Q. These two values can be distinguished via Y, which represents an observed value drawn from either P or Q. The KL divergence is thus, the value which is derived from obtaining Y from P.
This applies well to all the waffling about Gojo’s theoretical manipulation of space. Think of Infinity, where the perceivable distance from a target is observed to be the same (hand stopping three centimetres away from him) despite his ability to change it (endless extension with nearing vicinity - 1.0 → 0.5 → 0.25 → 0.125 → 0.0625 → ∞).
To draw information about a given point (e.g. X) from P or Q results in information gain and loss. Imagine two people standing at two different points gazing at the same target. Their perceived distances will be different (e.g. 5m as compared to 15m, an information gain of 10m from the latter point, and a loss of 10m from the former), hence these distances become asymmetric. This isn’t too relevant later on, but I thought it worth perusing since it was specifically mentioned in the interview.
So what does this mean?
IMO KL divergence doesn’t really explain Purple. Reading the interview relates all of this heavily back to the manipulation of space as it pertains to Red and Blue (and by extension, Infinity- if you ask me).
Personally I like to think of Purple as matter erasure. It’s a combination of attraction and repulsion which results in a destructive or consuming force (going off what is seen in the manga). What the manga shows us is already largely impossible considering the concept that combining Red and Blue would theoretically form a stable equilibrium that is inherently resistant to change. But this impossibility is what sticks out to me.
Complex Domain
My own justification relies on the idea of the complex domain. This refers to any subset of values which lie on the complex plane, which is a theoretical graphical depiction derived from mapping the coordinates of x (real numbers, -1 to 1 on a unit sphere), y (imaginary numbers, -i to i) and z (x+ iy, 0 to ∞). Mapping these coordinates will result in an Argand diagram. Stereographic projection of an Argand diagram gives us the Riemann sphere, which gives us a one to one correspondence of points of a sphere onto a flat plane. How this is done is by positioning a unit sphere over the origin point (0,0,0) and drawing a line from any given point on the complex plane to the north pole of the sphere. This results in the drawn line crossing one point (z) before reaching the north pole. This point (z) is that which is plotted onto the graph.
However, the north pole of the sphere itself does not correspond to any point on the complex plane as no line can be drawn from the plane to the north pole that would intersect it in a way that would derive a z value. Hence, a theoretical point at infinity is assigned to this north pole. The complex plane with this theoretical infinity point is referred to as the extended complex plane.
The point of infinity is a theoretical value used to symbolise a limit. An implied limit. Think of the equivalent of a vanishing point in perspective drawing. Do you see where I’m going? Gojo states that his technique allows him to bring infinity into existence. Similarly, Hollow Purple thus enforces (brings into existence) a vanishing point which defines a theoretical limit beyond an arbitrary distance where objects vanish. Hence allowing them to- well, vanish!
That’s all I got, folks. Let’s eat some grass together.