Negative Spaces and Complexified Objects
Three years ago as I was reading “Topology through Inquiry”, I had a struck of inspiration. It lead me to rethink the fundamentals of Sets to account for the concept of a hole. I wrote everything down and until yesterday, these ideas were nothing but scribbles hidden between the pages.
The Aha-moment occurred as I was reflecting on complex numbers. The theme of my recent conversations revolved around the meaning and mechanisms of space complexification.
This essay will attempt to bridge these two temporally distant thoughts. The result, is to my estimation interesting enough to contemplate.
What happened three years ago is that while reviewing basic operations like Union and Intersection using Venn notation as substitute for symbols. I drew the Complementary operation based on a different intuition than what is common.
I recall that taking the complementary of a sub-Set is akin to removing the sub-Set from the whole. To me, it made perfect sense to draw a “hole” in its place.
For the sake of stylistic coherence, I will from hereon alternate in referring to holes as Absets. A name that combines ab- from absent and Set.
Vanilla Sets are made to represent things or elements which are grouped together. It defines the relations and rules of combination between these groups. So I asked the question, what are the rules for Absets?
Funnily, I “set” myself to explore the most immediate implications using the DeMorgan rules. The complementary of Union and Intersection operations seemed to yield some counter-intuitive results. The rules were inverted.
The Intersection of Absets acts like the Union of Sets, and vice versa. An unexpected result, yet it made total sense.
The dominant intuition when it comes complementarity, at least as popularized by Set Theorists, is that when we remove something from a Set, it shrinks to fit the size the whatever remains.
In contrast, I say that what remains is the rest plus a hole representative of what was removed. The underlying assumption of the reasoning forwarded in the previous paragraph is that Absets are mapped to Nothing — the hole simply closes.
The “absence of something” and “nothing” aren’t the same. The former implies “something” while, nothing implies well, nothing.
It is this distinction that vanilla Sets theory fails to model. A distinction, which from an informational perspective is significant.
To take these ideas a step further I had to do a reality-check.
I formed two Sets from the things that were on my desk. Then, I proceeded by removing certain objects from each Set. With that, I had Sets with sub-Absets in them.
Upon performing Union and Intersection operations while keeping in mind the holes I instantiated and by applying their rules of combination, I was surprised to find that everything adds up perfectly.
Imagination played a huge role in these experimentations. What is even more interesting is that the imaginary will keep showing in what will follow.
What I exposed so far is about three years old. Back then, I thought it was too early to say anything about whether Absets were interesting or not. Until, I made a certain realization about Complex numbers.
It started as I advanced a certain analogy during a conversation. My point was to forward the computational repercussions of using an extra imaginary dimension.
It goes as follow.
I have a room filled with objects and I need to move them around. In altering the configuration of the room, I simply do permutations. In fact, any other action is merely a combination of these.
Associate solving anything employing Real numbers with, having to rearrange stuff while it is not possible for you to leave the room. You realize that it will prove harder to solve certain configurations, take a lot of time and even look impossible.
Now if you had an extra room, you can use it to hold certain things temporarily to make the task easier. The very idea using this extra room is what Complex numbers are about.
This explains allegorically why most often, the usage of Complex numbers is the answer to a problem which resists Real techniques. Examples of this are countless.
Now beside the ease that comes with having an extra freedom into which we can push and pull data without interfering with our descriptive freedom, the “i” models something that links back to the first thought, that is, Negative Area.
Once I made the connection, it didn’t take me more than a minute to put things visually.
We represent each Complex number along a direction perpendicular to the other and then draw a square. Its length and height are is aggregation of its real and imaginary part. The full area of the rectangle will be divided into four parts.
If we look at it as a matrix, the anti-diagonal is imaginary while the diagonal is real. What is more interesting is that the second part of diagonal comes with a negative sign.
The product of two Complex numbers — or as I represented it above, the area of a complexified square — comes with the possibility of having negative areas. Their combination with the more intuitive positive areas is akin to the interaction of matter with anti-matter.
The observed result is a real square with a missing area. What is clear is that “i” which stands for the Square Root of -1 is by build made to annihilate areas and generate holes.
Complex number are indeed an interesting construction. They allow us to encode more of reality than we can actually see.
Connecting the dots superficially, I can invoke the fact that physics finds more and more reasons to say that Complex numbers are at the core of the most fundamental descriptions of nature formulated to this day.
Where Real numbers were meant to represent reality, they do fail in the most critical situations. People get perplexed by the need to resort to the “i”maginary dimension for something that is supposed to be about reality. To that, I have nothing to say but the following.
M y experience with the discovery of Absets was revealing. Sometimes there might be more than what is immediately seen, even if the description we derive from it seems to be enough. We can achieve more powerful descriptions if we close our eyes.
The more sophisticated structure of Complex number which takes advantage of an imaginary dimension offers more leverage in encoding and solving geometries.
The fine line that bridges the three years gapped thought which motivated this writing is that, Complex numbers do model Absets.
The prospect of using these ideas might be far, yet I believe that the outlook they provide is creative enough to breed further elaborations. Maybe that be an incentive for more interesting and diversified usage of complexified spaces in our descriptions of things and their reality.
