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eastathenaeum
4 days ago
We need to stop calling it real and imaginary and start calling it magnitude and phase, because that's exactly what they are.
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YES! Thank you so much for the link! I love how they related the rotations within the multiplication of imaginary numbers to their represenation as both matrices and quaternions...To ramble crazily for a moment: These mathematical connections are *very* important for an understanding of how Steinmetz approached the relationship between energy and time when it comes to oscillating electric waves.
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letslearntogether
2 weeks ago
In turn, one can visualize these rotating waves as repeatedly travelling along a type of Riemann surface. When approaching the situation topologically, it is equivalent to how a projection of the 4-D Hopf Fibration yields a 3-D Clifford Torus. Applying this model to Geometrodynamics gives us a way to unify all of the fundamental forces at all scales, providing a coherent spacetime for describing ER=EPR.
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letslearntogether
2 weeks ago
Simultaneously, one merges together a bunch of seemingly disparate fields of mathematics, Langland's Program-style...All of this is also connected to something that I've been wanting to share with you ever since you mentioned your work on Relativity: Einstein's version of Energy-Mass Equivalence (i.e.: E=mc^2) is actually a reduction of Heaviside's quadratic (i.e.: E^2 = p^2c^2+m^2c^4).
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letslearntogether
2 weeks ago
This has two solutions. The Klein-Fock-Gordon form of this equation shows how it relates to Schrödinger's work in QM. Similar to how the negative solution to Dirac's equation led to the discovery of "antimatter", the mathematician Luigi Fantappiè suggested that the negative solution to the Klein-Fock-Gordon equation implies the existence of a "reverse time", amongst other phenomena.
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letslearntogether
2 weeks ago
Modern research into Quantum Backflow seems to hint at it. Precious few have taken T-symmetry and these negative solutions to their fullest logical extent (e.g.: William James Sidis, Hannes Alfvén, etc.). However, its biological effects are measurable (e.g.: experiments demonstrating "pre-stimuli responses" in the body done by HeartMath, Vannini, et al.; Saxton Burr's work on "L-fields"; etc.).
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letslearntogether
2 weeks ago
...Phew, still barely scratching the surface there. It is challenging condensing this into something readable, so it probably looks like word salad. I hope it at least triggers some interesting ideas for you.
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eastathenaeum
2 weeks ago
You're most welcome. I'm grateful you found it valuable. That video in particular should have existed 12 years ago so that quaternions would have been intuitive to me before spending entirely too much time trying to understand it. It was a nice refresher and reinforcement that clicks alongside just being told that H is the union of two C's. If only many of the visuals available to us now existed then and before.
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eastathenaeum
2 weeks ago
To be clear, I never worked on relativity and never meant to insinuate it. I've only learned physics as an engineering student, and the rest is just bouncing around trying to understand every (physical/formal) scientific concept as well as I could via Wikipedia, YouTube, and downloaded pdf text books. There was a 2 year period when reading dominated my entire free time. I know of quite a lot, but I'm no physicist.
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letslearntogether
1 week ago
Right! A good visualization can clarify a lot. I understand too; I myself am not much of a "professional" anything, but it sure is fun to explore together! Part of me has always thought that anyone was [blank] whenever they did [blank], so everyone can be a scientist whenever they choose to practice the scientific method, for example...Anyhoo, thanks again! Going to go study more.
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ninjacoder58
1 week ago
That was deep, I'm just chillin in EONJ and your just going cosmic on us all 😂
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Finished creating pages. Even though I'm trying to solidify up to 5.2, I did it this way for convenience of adding wherever whenever or just when it clicks on how to present it. I have a real anxiety for coding this project and feeling stuck a lot because of minor details. So this way, stuff can fall together without piling up a bunch of screenshots and notes in the background and it just becomes easy in the end.
Also, I originally wanted to add a differential geometry section, but it is a very diverse field and I wanted only to cover the fundamentals, namely in only projecting one surface onto another. So eventually once I get to a certain point, some features of that can be weaved into sections here without having to get too far into it.