In reality (in fact), the algebra of hypercomplex numbers is greatly undervalued and is only reduced to a mere “tool” that facilitates an action (simple action).
Well, from his beginnings with Hamilton, Tate, James C. Maxwell. It had a lot of resistance, which until then (even today, despite the postulate that best describes the universe [relativity]).
The world is thought to be Euclidean (straight) & 2D, It is enough to verify that each method, function, calculus, etc. is thought in terms of x and y, where the z-axis is an unreal or imaginary (magical) thing.
But this is not the case, B. Riemann taught (at the time) that in a sphere a right angle is not 90° in a sphere, and our universe is predominantly “spherical” (in terms of celestial bodies), Assuming that Euclid can only be accepted in 2Ds terms (Plane), but not in 3Ds (Cube) and 3Ds+1t (Sphere by Gravity efects).
Now… Why was quaternion (Hamilton) algebra not adopted as the main mathematical model?
Easy, because at that time, there was no way to put into practice that complex algebra of rotations, translations, forces, etc.
Until Maxwell introduced his treatise on electricity and magnetism (originally written in quaternions). that is where all your knowledge about vectors and their algebra comes from.
Heaviside (who was not a mathematician, scientist, physicist or teacher, just an engineer). He was the first to popularize Maxwell’s work, but in doing so he dismembered quaternions and separated the scalar from the vector, just to maintain the commutative property of multiplication (orthodox stuff)
From there came the great battle of the mathematicians, where vector notation won (losing itself in time until the appearance of quantum and its use by Pauli)
- Many marveled (like Riemann) at that type of algebra, but the fear of being discredited or treated as crazy by the scientific community (even Einstein), led them to create complex algebraic structures, only to keep the analysis and geometry in 2D, compacting events, effects of 3Ds+1Dt stuffs in 2Ds, making their interpretation impossible (“Knots” in 2Ds are just “Apparent Intersections” in 3Ds and Paths in 3Ds+1Dt). -
The problem of “dimensions” that falls so much mockery on those who use these algebras, is nothing more than the result of the ignorance and lack of understanding of those who do nothing more than repeat the copy-paste of Heaviside (in Maxwell’s time).
Mathematically, a quaternion has 4 “dimensions” because it has 4 components (nothing else).
And that misinterpretation is perpetuated, making others believe that you are talking about metaphysic and quantum healing waves.
But in a single quaternion, you can encode a lot of information, more in a dual quaternion.
All general relativity can be expressed in terms of dual quaternions, including much quantum physics.
But if there is something that a dual quaternion does not describe well (for reasons of symmetries), it is the chirality and irreversibility of time. (which according to all contemporary physics may be possible)
Which we know is not possible, a broken glass (even if you put together all the quanta of energy) will never be “unbreak” (Thermodynamics, arrow of time, etc.).
That brings us to the octotions, which due to their mathematical particularity (non-commutative and non-associative), makes them purely “evolutionary” (there is no going back, or at least to the same “exact” state).
A dual quaternion can be easily mapped into an octonion, so instead of operating dual quaternions (and making something reversible over time), they operate as octonions (becoming irreversible).
In this way, physics would correspond to what is observed in our 3Ds+1Dt plane.
- And no, we do not leave our universe to an 8-dimensional one. We only describe more complex things or movements in a simpler way.
In this way, 2 octonions can contain; Points, Planes, Volumes, Velocities, rotations, accelerations, trajectories, spins, the inner and outer face of an object (Normal) and be physically correct.
with your current algebra… Can you invert a sphere from the inside out and vice versa, without its BREP being “cut”, while rotating, translating, accelerating, increasing its volume and/or “energy”? (without violating the laws of thermodynamics)
So you don’t know much about “the probability cloud” called electron.
- And all this thanks to (or consequently) “losing” a mathematical property.
Then imagine a:
- New Voronois 3Ds+1Dt, physically correct.
- New “vector” databases for iA’s
- New transformers for iA’s
- New algorithms for inference and calculations in GPUs
- New ways of “seeing” physics
- New 3D physics simulations (which are impossible today)
I don’t want to say that I will do all of this (I don’t know), but when a human discovers that there are more tools (besides the hammer), and above all… understands their use… Amazing things happen.
“In my time”, the cell phone was a fictional thing, only super agent 86 could have that technology, today they are looking at how to limit it to 2-year-olds.