A previous version got removed for spoilery titles, so hopefully this is a sufficiently non-spoilery title. Over the last couple of years various explanations have been advanced for the powers and motives of the Ring Builders and the Others, as I’ll call them here. I’m going to collate everything here, (mostly written by me) and unify it, and put my cards on the table in preparation for any revelations that remain in Season 5.
Tl,DR:

The Ring builders are made out of unobservable matter that exists in our universe. Brane theory and a version of quantum theory that links consciousness to quantum effects are both correct in-universe. This may or may not be the classic roger penrose objective reduction theory.
The ring builders can open wormholes through and into the bulk (higher dimensions) and construct an artificial brane-world (the ringspace). They can also influence consciousness by modifying the quantum vacuum.
The Others are beings native to the bulk (higher dimensions) and can force aside the ‘AdS (anti de sitter) layer shielding our braneworld at will to exert effects anywhere in space acausally.
They don’t understand that we are alive, let alone conscious, and attack us because ring builder technology such as the inertialess drive, USM projector, solar system destroying beam and ring gates emit their (vastly greater than observed) waste heat directly into the Others’ native dimension.
‘Fucking Duarte and his fucking tests’ did the same thing (but with a much smaller energy release), and now the Others are trying to stem the cause of the pollution.

Higher Dimensions

Original, and thanks to kabbooooom for extra info.
My guess is that the Goths are something like the 'bulk beings' that were humanity's distant descendants in Interstellar, that the gate space was another braneworld and that the ring gates are wormholes that lead into a small separate braneworld that is the ring space.
My physics degree didn't cover anything like this stuff, but there's some informed speculation about travel to other brane worlds in Kip Thorne's excellent The Science of Interstellar
The ring builders are much more like us than the Goths, but found a way to create an unusual kind of wormhole, one that leads into the bulk, rather than just connecting two parts of our spacetime.
That would make the PM builders capable of more than mere FTL travel, but I could well imagine that the Goths are the native inhabitants of the bulk and get angry at intruders. It would explain their ability to reach into our 4D space-time at any point and induce effects that look acausal - they're not actually acausal, just short-cutting the distance in our 3+1 dimensional spacetime. There is a problem - the 'bulk beings' explanation by itself doesn't make sense of the weird way the Goths affect human consciousness.
One thing that really stood out is that the light orb we see in PR that appeared on the Magnetar-class behaved like an optical illusion or hallucination - people who looked at it didn't have their vision of anything inside blocked, and saw both the orb and anything passing through it with equal clarity. That effect would make the most sense if it somehow hotwired the retinas or brains of whoever was watching it, like a Blindsight Scrambler. The effect is apparently inspired by chimerical colours. The fact that the light orb emits no radiation except visible light also implies its some kind of induced hallucination rather than a real object. It would fit with the abilities of the protomolecule to manipulate consciousness, which is vastly outclassed by the goths.
But, the really weird thing is that the light orb's effects transfer over to images of it - even though its just a photo of a blank white orb Captain Singh gets a headache when he looks at a picture of it. That suggests weird conceptual stuff going on, or else there's some kind of subtle memetic kill pattern in the apparently uniform orb, one that transfers even to photographs. Again, it reminds me of the Scramblers.
These effects all point to the Goths being relentlessly hostile and kind of evil, but that could easily just be a misinterpretation on our part. But they also point towards an ability to manipulate fundamental physics also working to manipulate consciousness, which is a very controversial idea right now.

The role of Consciousness

The missing piece that explains how the higher dimensional beings can directly affect consciousness has already been provided, and funnily enough it showed up in the show before it showed up in the books. This is an analysis I wrote of the 'technobabble' in S3E9, when proto-miller tries to explain how he works to Holden. At the time I hadn't read PR or TW, so had no idea the consciousness connection was going to be relevant:

I wanna know exactly what you know.
Oh, so you wanna talk about the non-local quantum hologram, the phase-conjugate adaptive waves resonating in micro-tubules in the brain, which of course requires some closed-timeline curves and Lorentzian manifold, and you catch up, I'll wait.
Those words aren't technobabble, they actually make sense as a vague description of how the protomolecule goop is affecting Holden's brain to produce hallucinations.
If Holden knew enough physics to understand what some of them meant it would scare the wits out of him. I can't even conceive of a technology this advanced; no human being can. Miller's off-the-cuff line was final proof that the Protomolecule builders are geological eras more advanced than humanity.
For starters, a closed timelike curve is a particle that loops backwards into its own past! If Time Travel, at least on tiny scales, is allowable (since FTL is equivalent to time travel and the ring is FTL, this shouldn't be surprising) all conservation laws are immediately broken.
A 'non-local quantum hologram' probably refers to using quantum non-locality to transmit information - faster than light, faster than anything, infinitely fast. This is theoretically impossible - despite what you might have heard, the 'nonlocal' way that some quantum systems seem to behave can't be used to transmit anything faster than light. Clearly whoever built the protomolecule doesn't care what humans think is impossible.
The remark about a Lorentzian Manifold refers to the general mathematical description of how space behaves in general relativity. 'A Lorentzian Manifold' could be any kind of space with any number of dimensions, as long as it follows the laws of general relativity. So bring it all together, a closed timeline curve is produced by some quantum gravitational process to acausally affect the microtubules in the neurons in Holden's brain.
Finally, the reference to 'adaptive waves resonating in micro-tubules' refers to an unproven hypothesis by Roger Penrose that links human consciousness to quantum effects - possibly something to do with how Miller's consciousness merged with the protomolecule goop, and with Holden? For all we know, the Protomolecule builders have advanced in this direction too.

What's the relevance of this to the Goths? Elvie's hypothesis from TW that human brains are a support mechanism for a quantum-based consciousness field is confirmed true by proto-miller (at least in the Show). I think it is very revealing that the extra 'technobabble' was included in the show (which has the authors as producers) but wasn't in the original Abbadon's gate. The protomolecule are more advanced in their ability to manipulate the quantum fields underlying consciousness, and the Goths are more advanced still than the protomolecule builders.
The Okoye-Proto-MillePenrose theory of consciousness as exotic quantum gravitational effects has another bonus point in its favour - it explains away some of the implausible aspects of the setting - the lack of general artificial intelligence built by humans.
If the Expanse is a universe where mental functionalism is false - where Roger Penrose is right and intelligent behaviour (not just consciousness, but human-level intelligence) requires more than computation, then we know why there is no AGI.
That would explain why, despite the fact that Mars' AI labs have been trying to advance the state of the art for 200 years, there has been no detectable progress since about 2050. In other settings, like the Xeeleeverse or Revelation space or even Star Trek/40k, there are similar explanations as to how binding laws prevented AGI arising.
If Penrose/Elvi/Proto-miller is right in the Expanse, it would explain a lot - the weird interface between the Goth Tech and consciousness, the fact that the protomolecule has acausal connections among itself, and the fact that the only AIs known in the setting are based on alien technology that is definitely doing more than computation and could be incorporating whatever weird physics Penrose thinks is necessary.
I've dug a little deeper into what Elvi actually said in the conference, and it does sound very similar to what the Investigator told Holden. I did a bit of Philosophy of Mind in uni as well - it's good to be able to put it to use for once.

“It’s about the nature of consciousness.” “That may be a wider context than I was looking for, Major.”
“Bear with me,” Elvi said. “Unless we’re reaching for religious explanations, which I’m not the person to comment on, consciousness is a property of matter. That’s trivial. We’re made out of matter, we’re conscious. Minds are a thing that brains do. And there’s an energetic component. We know that neurons firing is a sign that a particular kind of conscious experience is happening. So, for instance, if I’m looking at your brain while you imagine something, I can guess reliably whether you’re imagining a song or a picture by seeing if your visual or auditory cortex is lighting up.”
“All right,” Trejo said.
“There’s no reason to believe that a brain is the only structure capable of having that combination of structure and energy. And in fact, there’s a fair amount of evidence that the gate builders had a conscious structure—a brain-like thing—where the material component wasn’t at all the same kind of thing we use. Anecdotally, we’ve found at least one brain-like structure that was a diamond the size of Jupiter.”

What Elvi is getting at here is what philosophers of mind call Multiple-realizability. This is the claim (accepted by most philosophers and basically all neuroscientists) that there's nothing special about neurons per se, that experience and subjective, first-person consciousness can be supported by lots of different physical systems as long as they all are doing the same kind of thing. What that kind of thing is, we have no idea.
There is a stronger claim (popular among neuroscientists, and accepted by some philosophers like Dennett), which is mental Functionalism, that the kind of thing you need to be doing to support consciousness is just information processing of the right sort, and the particular kind of matter or energy you're doing it with doesn't matter. So an electronic computer, or a giant committee of people pushing papers around, could be conscious if it was running the right algorithms. Elvi is saying here that multiple-realizability is true but that Functionalism is false - consciousness doesn't exclusively require neurons but it isn't just computation either, it's something more. This is probably the majority view right now among philosophers, anyway. Generally neuroscientists are more willing to say the mind is pure computation.

“I don’t know what that means,” Trejo said.
“Like we don’t have a steel chamber in fusion reactors. We have magnetic bottles. Magnetic fields that perform the same basic function as matter. The older civilization appears to have developed its consciousness in a form that relied more on energetic fields and maybe structures in unobservable matter than the stuff we made a brain from. There’s also some implication that quantum effects have something to do with our being aware. If that’s true for us, it was probably true for them.

Here, Elvi is vague but apparently pointing towards Roger Penrose's idea that humanlike intelligence exploits a successor to quantum physics as we understand it today, to make decisions that are literally uncomputable by any normal information-processing system. Though she seems to be focussing more on the idea that it's having experiences that is supported by quantum processes, which isn't Penrose's idea per se.
The reference to energetic fields and unobservable structures is referring to the PM builders' own consciousness and how it is supported, again hinting that they might be constructed out of exotic forms of matter, but ones that still inhabit our home universe - perhaps there is a connection to Dark Matter or energy here?

“My thesis—the one I was working on before I came here—explored the idea that our brains are kind of a field combat version of consciousness. Not too complex. Not a lot of bells and whistles, but takes a lot of punishment and keeps functioning. Our brain may actually have a kick-starting effect, so when the quantum interactions that underlie having experiences break down, they’re easier to start up again. Does that make sense?”

It does make sense, sort of - the idea that quantum interactions underlie consciousness is a big pill to swallow, given what we know about quantum physics as it is now. The idea that these interactions could be detached from the underlying physical structure; it would need new and very strange physics, but this is science fiction, so who am I to judge?

ER=EPR and AdS Space

In proto-Miller’s speech, the “nonlocal quantum hologram” was more likely referring to the AdS/CFT correspondence, and indirectly to the ER = EPR hypothesis (which is much more interesting as far as the protomolecule is concerned.
Five second explanation: ER = EPR is the idea that the EPR paradox (which seems to show that quantum entanglement transmits information faster than light), is explained by there existing a tiny wormhole that connects entangled particles.
AdS space is ‘anti de sitter space’, a type of solution to Einstein’s equations that describes a ‘bigger on the inside’ space with constant negative curvature - as you travel along it distances seem to grow. This is contrasted with ordinary positively curved ‘de sitter space’.
That really is significant because that would imply that AdS space is a thing in the Expanse's cosmology, and as far as I understand it an AdS layer is a common explanation given for how the 3+1 dimensional space described in conventional general relativity can sit embedded in a higher dimensional space without obviously being affected by it (i.e. why force laws are still mostly inverse square). That again points towards the ringspace being a brane world and the connections being not ordinary wormholes but running through some higher dimension. AdS explains why you can’t easily notice the gravitational effects of higher dimensions.
What about ‘ER=EPR’? the Protomolecule’s “instant communication” was actually communication via microwormholes, and isn’t really instantaneous at all. This isn’t directly stated in the books but it is alluded to in The Vital Abyss when Cortazar is describing what the Protomolecule does on a nanoscale, and he deduces that on a macroscale it would create a wormhole because it would be a fractal like elaboration of what it was already doing at a smaller level. This would also explain how the protomolecule is able to use quantum entanglement for FTL communication in the first place, given that this is absolutely impossible on conventional QM.

Which theory of Consciousness?

Orch-OR (the theory directly linking consciousness to quantum gravity) is not taken seriously, it isn’t even a hypothesis, it isn’t even falsifiable, and we have far better theories of consciousness based on information theory and neurophysiology that have already produced confirmed and clinically significant results.
But, that’s the way the Expanse is heading unfortunately. It is worth noting several things here: that the AdS/CFT correspondence and ER=EPR would be directly related to consciousness if Penrose and Hammeroff were actually correct, because both are describing in a fundamental way how quantum information affects the structure of spacetime. And secondly, even if we assume that consciousness is not quantum in origin directly, both may still be somehow related because consciousness appears to truly be a phenomenon of information processing, and information (albeit quantum information) appears to underlie the structure of space time if the above models are correct.
And finally, it’s worth noting that one of the actual legitimate (but still very, very incomplete) theories of consciousness that we have - Tononi’s IIT - has an interesting informational relationship with quantum mechanics as well. For further info on that, read Tegmark’s Perceptronium paper.
My point is, even if the phenomenon of consciousness is not specifically quantum in origin, it may still be related to the universe on a fundamental level and a true “theory of everything” would likely incorporate that. Most of my colleagues do suspect that consciousness has a deep root in physics, and it is particularly telling that there seems to be a confluence starting between neuroscience, information theory, and physics. So even though I disagree with the premise of how the Expanse interprets that part of the story, I can still get behind the general idea that beings with complete power over space time would also have complete power over consciousness, because it may actually be information (it-from-bit) that defines both the structure of reality and consciousness, albeit at different spatiotemporal scales and orders of magnitudes.

As far as I know the only line in either books or show that point to the objective wave-function collapse in microtubules a la Penrose explanation is that one line from Miller. Perhaps he was oversimplifying some vastly more complex theory of everything into English. Similarly, Elvi's own explanation doesn't go much further than "there seems to be some kind of deep connection between fundamental physics and consciousness, such that you need something more than information processing, and it might be related to quantum" - she doesn't specifically mention objective wave function collapse.
More here.

The USM Projector and Ring Builder Drive

Ring Builder technology seems to defy energy and momentum conservation, but those laws are absolutely involate. We must conclude that the energy and momentum goes somewhere.
Take the USM projector - although it's quite hard to calculate the energy needed to produce a field that strong, its is absurdly greater than anything we can conceive of coming from even an antimatter power source.
We know the field strength was on the order of 10^15 gauss (strength at a real magnetar) which is 10^11 T, and that the beam was extremely narrow and stretched out for at least a million km. A magnetic field which is like that, i.e. like a one way current, must have nonzero divergence so it breaks maxwell's equations which will require magnetic monopoles. The energy needed to create this has to come from somewhere.
It's hard to work out the exact area. Let's be generous and say the USM projector generates a cylindrical region with a cross-sectional area of 1mm and a length of 1 million km containing 10^11 Tesla, the field strength at a typical magnetar. Leaving aside the astounding power output needed to create this thing instantly, how much energy in total would it require? Just use the equation for stored magnetic field energy.
Energy = volume * 1/2 * B^2 / u_0 = volume * 3.979 * 10^27 J/m^3 = 3.979 *10^30 Joules(0.5(10%5E11Tesla)%5E2+%2F+permeability+of+free+space))
A USM shot is 1.6 percent of the gravitational binding energy of the entire Earth, or about 950 TRILLION MEGATONS of TNT. Every missile every fired by every faction that ever fought in all the wars in the Expanse's history wouldn't reach a thousandth the power output of one shot from the USM projector. The waste heat alone ought to explode the Tempest to less than atoms, the waste energy from the Pallas station shot ought to have given third degree burns to people watching the lightshow from Earth.
There is some kind of special exemption involved - my guess is that the Magnetar class is not actually directly inducing the field but rather generating some kind of self-perpetuating process, maybe a very specific kind of cosmic inflation, or some exothermic-type reaction like the conversion of matter to strange matter or vacuum collapse or something in that - I don't know, I'm not (really) a physicist. Aren't there theories that say cosmic strings or monopoles can produce more of themselves under the right circumstances?
It's the same deal as whatever moved Eros. That also exerted something on the order of 10^29 Joules to move the moon up to thousands of km/s, and the waste heat released was barely on the order of 10^19 Joules - like Naomi suggested all the way back in the first book, it was merely the faintest spillover of a vastly more energetic process.
If the tiniest fraction of the true power of the USM projector was released as waste heat it would have vaporised the Tempest and blinded people watching the lightshow on Earth. Again, maybe less than a billionth of the energy is released as waste heat, either on firing or on impact with the asteroid.
Where does that waste energy go? It goes to the same place that the ring gates open to, the same place that the Others come from: the higher dimensions of the Bulk, which is the reason that they are pissed at us.

That's why the goths are pissed, the waste heat probably gets piped to their dimension with fuck knows what consequences

Last night while binge drinking and reading through chains of articles on Wikipedia I had the best idea. I realized there are sections of the site I never explored. So I immediately closed List of animals with fraudulent diplomas and the n+1 articles on Permian fauna I had open, then went to the Wikipedia reference desk. Now the reference desk is pretty cool and it's one more reason why Wikipedia is an internet gem, anyone can ask a question about any topic, and anyone can answer.
Most of the questions in the mathematics section are what one would expect; people asking about things in homework assignments they don't understand, people asking for help deciphering arcane mathematics articles on the site, and people questioning their own understanding of things. Every now and then you'd find people asking why some proof of [insert famous conjecture here] published in [insert obscure journal here] wasn't cited on Wikipedia and why it wasn't accepted as a proof by the mathematical community. But I found something a little more spicy than that, I found a user that claims to have proven the Riemann hypothesis, the Collatz conjecture, the Goldbach conjecture, and created an elementary proof of Fermat's last theorem.
Is the following proof of Riemann Hypothesis correct?

Riemann Hypothesis states that the real part of all non-trivial zeros of the Riemann zeta function, or ζ(s) = Σ(k=1 to ∞) 1/k^s = 0, equals one-half. For the non-trivial zero, s, a complex number, we have s = a + bi where Re(s)= a = 1/2.

If I had $1 for every "proof" of the Riemann hypothesis I've seen where the writer starts by trying to find s such that 1+1/2^s+1/3^s+1/4^s+⋯=0 I'd probably be lounging on a beach in the Caribbean right now. The problem here is that the Dirichlet series for ζ(s) only converges when the real part of s is greater than 1, and this series is never 0 where it converges. So analyzing only this series will not be helpful.

Fact III: The sum of the complex conjugate pairs of non-trivial zeros, s = a + bi and s' = c + di where ζ(s) = Σ(k=1 to ∞) 1/k^s = 0 and ζ(s') = Σ(k=1 to ∞) 1/k^s' = 0, of the Riemann zeta function equals one according to the Fundamental Theorem of Arithmetic and the Harmonic Series (H):(Note: Euler and others have proven that there exists an infinite set of primes in H. And that the divergence of H is a key reason for that result.)

If s' is the complex conjugate of s=a+bi then why not just write s'=a-bi instead of s'=c+di? Or why not write s=σ+it instead, as this is a fairly standard way to write a non-trivial zero in literature on the topic? Sure, this isn't bad math per say, but it's pretty bad notation. Also, s+s'=1 always only if the Riemann hypothesis is true and this would have nothing to do with the fundamental theorem of arithmetic or the harmonic series! They have already assumed the Riemann hypothesis is true before they've done anything!
The bit where they talk about primes in the harmonic series is somewhat odd. It looks like they think the divergence of the harmonic series implies the divergence of the sum of reciprocal primes (which it doesn't, the implication is the other way around) and they seem to treat the harmonic series like a set.
After this our writer slaps his four facts together in some convoluted way that I can't decipher and declares victory.

Therefore, according to Facts I, II, III, and IV, we have:
k^(1/2) ≤ k^a ≤ k, k^(1/2) ≤ k^c ≤ k, and a + c = 1.
Hence, k^a = k^c = k^(1/2) which implies a = c = 1/2. Riemann Hypothesis is true! Riemann was right!

Then they make some final notes where they try to rewrite the harmonic series using some underexplained ideas about prime gaps and says

There are infinitely many more positive integers than there are prime numbers, or prime numbers have a zero density relative to the positive integers, and prime numbers generate the positive even integers efficiently so that gaps between two consecutive prime numbers increase without bound.

which is true in the sense of natural density for sure, so why not just say that? Using the phrase "infinitely many more" makes it sound like cardinality. Saying "so that gaps between two consecutive prime numbers increase without bound" makes it look like they're saying all prime gaps become larger as we increase through the sequence of primes, this isn't necessarily true although it's statistically something we should expect. The existence of arbitrarily large prime gaps is true though and isn't hard to prove, but they did not prove it in any of what was written and it's not the same as what they said.
Is the following proof of Goldbach Conjecture correct?

Keywords: π(*):= Odd Prime Counting Function and Fundamental Theorem of Arithmetic (FTA) Goldbach conjecture states every positive even integer is the sum of two prime numbers. (We count one as prime in the sense of additive number theory outside of the FTA.)

What? The parenthetical here is so strange. Additive number theorists don’t take 1 to be prime and they have no reason to do so.
The writer then tries to make a probabilistic argument from a system of linear equations defined over a set of odd primes less than an even number e>2,

Therefore, e ≠ p + q over S, (p,q є S) , implies the following system of equations over S, 1 = e - n1 * q1, 3 = e - n2 * q2, ..., pk = e - nk * qk, according to the Fundamental Theorem of Arithmetic where 1 < qj ≤ (nj * qj)^.5 ≤ nj for 1 ≤ j ≤ k where pj, qj є S and nj is a positive integer. Note: If qj = 1, then nj є S, or nj is an odd prime less than e.

and this last sentence is what they try to base their argument on. They attempt argue that for every even number e>2, the probability that an equation of the form p=e-1q doesn't show up goes to 0. Which would mean that it's likely that e=p+q.
Even if their probabilistic manipulations made sense this obviously still wouldn't prove the Goldbach conjecture. Showing that it's "probably true" isn't a proof that it's true. As if to attest to the writer's own doubt,

In addition, empirical evidence has confirmed the validity of the conjecture for all positive even integers up to at least an order of 10^18. Therefore, we conclude the conjecture is true.

If you proved it, why do you need to test it empirically?
Is the following elementary proof of Fermat's Last Theorem correct?

x^n+y^n=z^n for n > 2. I begin the proof by assuming there exists an integral (positive integer) solution to equation one for some n > 2. Equation one becomes with some algebraic manipulation, 2. x^n=z^n-y^n = (z^(n/2)+y^(n/2))*(z^(n/2)-y^(n/2)).

Okay.

Now that I have factored the right side of equation two, Fermat, the great French mathematician and respectable jurist, made I believe the next logical and crucial step.

Any evidence that Fermat did what you're about to do?

He factored the left side as well, x^n, with the help of an extra real variable, Ɛ, such that 0 < Ɛ < n . I have the following equation, x^n = x^(n/2+Ɛ/2)* x^(n/2-Ɛ/2) = (z^(n/2)+y^(n/2))*(z^(n/2)-y^(n/2) ). This equation implies x^(n/2+Ɛ/2)= z^(n/2)+y^(n/2) and x^(n/2-Ɛ/2) = z^(n/2)-y^(n/2).

Ah yes, if ab=cd then a=c and b=d. Everyone knows that! Eventually, after a few more lines, the author concludes

However, (1/4)^(1/n) is not a rational number, a ratio of two whole numbers, for n > 2. This implies the right side of equation five is not a positive integer. This contradicts my assumption that y is a positive integer. Thus, Fermat’s Last Theorem is true, and Fermat was right!

It's so easy now, Fermat's last theorem obviously just reduces to knowing 1/4^1/n is irrational for n>2. How did nobody see this before?
Is the following proof of the Collatz Conjecture correct?

Proof of the Collatz Conjecture: Suppose there exists a sequence, S’={n0, n1, n2, …} that does not converge to one, or nk ≠ 1 or nsub(k-r) ≠ 2^µ over S’ for all kϵ ℕ where r

It's obvious that hailstone sequences don't converge, so the “does not converge to one” bit is irrelevant. Here the fundamental error is same error as in their attempted proof of the Goldbach conjecture; they think making a probabilistic argument in favor of the conjecture being true is the same thing as proving it. Lots of other basic little details are also wrong, but I'll just look at one:

From a given positive integer, n, we obtain the maximum positive odd integer, n0 > 7, by repeated division of n by 2.

What is n here, the starting number? What if n is odd? We'd have to 3n+1 it first, not divide by 2. Even if n is even the first odd number we hit once we finish dividing by 2 is not the maximum odd number in its hailstone sequence, this is easy to see starting with n=22.

[https://pubchem.ncbi.nlm.nih.gov/periodic-table/png/Periodic_Table_of_Elements_w_Chemical_Group_Block_PubChem.png ] or [https://ptable.com/#Properties ]
If we are going off the Lewis definition of acids as electron pair acceptors and bases as electron pair donors, the problems of ion solubility (mostly H+ and OH- ions) can be appropriately distanced from the actual behavior of hydronium (H3O+) or hydroxide (OH-) complexes in water. In other words, we first ask what species exist in what concentrations in the solution of interest, then what will happen between the different species. However, we cannot completely separate the Brønsted-Lowry and Lewis definitions due to Le Chatelier’s principle, which would state that the presence of the products of dissociation tend to prevent additional dissociation events. However, if product ions start being consumed in other reactions, the effective result is to shift the equilibrium back towards the starting materials, and additional dissociation events will then become energetically favorable. The result of this is that the behavior of chemical reactions is best contemplated holistically and with a full set of executive functionality instead of being taught as a series of disconnected fragments that imply the existence of a much higher level of precision than is actually ever possible and must be stitched together by students working without the benefit of fully developed brains. As I go through the process of writing out this series of posts, I am getting the definite impression that the progress that has been made in our understanding of atoms and orbitals has mostly obsoleted the way that general chemistry is currently taught, and that the current state of teaching is centered around exams to the detriment of the students. My general chemistry education also had far too much emphasis on the Brønsted-Lowry definition of acids and bases instead of treating these as equilibrium problems.
So and before we go any farther, let’s get pH out of the way. A lowercase “p” denotes the mathematical operation of taking the negative log of a quantity for some reason, so pH is actually the negative (base 10) log of H where H is the ionic activity of “H+” in the solution of interest. As it turns out, this is actually the activity of hydronium complexes instead of lone protons, but unless you are trying to visualize what is actually happening in the solution the two can be treated as equivalent. Of course, if you’ve gotten so obsessed with applying equations to chemical processes that you are willing to ignore the three-dimensional picture, you’re probably also not doing anything of value, but anyway. In most cases, pH can be calculated with the concentration of hydronium in moles per liter instead of a more rigorous activity measurement, so in other words pH is mostly equal to -log([H3O+]). [I should also note that the difference between the concentration of hydronium and the concentration of protons is not particularly significant in acid-base problems because the protons in water will either react with other species or form hydronium. If you are calculating the concentration of protons in water at any given time, you are also calculating the concentration of hydronium.] If you’re willing to get pedantic there is a nearly infinite amount of additional complexity that can be brought in here, but I’m not emotionally invested in this and see no reason to care. Proceeding with pH=-([H3O+]), you may notice that we are only calculating the acidity of our solution and not the basicity.
However, due to the spontaneous dissociation/autoionization of water, acidity and basicity are closely related to each other. In a volume of water, the multiplication product of the concentrations in moles per liter of hydronium/H3O+ and hydroxide/OH- is a constant. At 25 degrees Celsius, this constant (Kw) is equal to 1.0x10^-14, and Kw=[H3O+]*[OH-]. In this notation scheme, the square brackets denote concentration in moles per liter, and square brackets are usually but not always moles per liter. In any case, the reason to care is that the assumptions here mostly hold true once we start adding additional chemical species to the volume of water we started with. As the number of ions in solution increase, other issues start to arise, but mostly what you need to remember is that this is a simplified model and not an absolute definition of what is happening on the molecular level. Where this model is valuable is in relating the concentration of hydronium to the concentration of hydroxide (both in moles per liter) in a mostly reliable manner, which means that if we know a value for one at a given time we can calculate the value of the other one. So, if you have a concentration of hydroxide and you want to know the pH, you can use Kw to calculate the concentration of hydronium, then take the negative base 10 log of the result to get to pH. The addition of the logarithm allows the comparison of numbers with vastly different orders of magnitude but also brings quite a bit of confusion. In any case, using these assumptions we can define interrelated pH and pOH scales to measure acidity and basicity as the density of hydronium and hydroxide in solution. You may notice that this aligns well with the Lewis definitions, although we are not considering any other possible Lewis acids or bases.
Once you get into organic chemistry and start trying to do reactions, having a trace amount of ions in your reaction mixture doesn’t get you anywhere, and all of the assumptions as previously defined get thrown out of the window. At high concentrations of ions/high ionic activities (which are mostly equivalent concepts), we get back to the idiosyncratic and non-intuitive behavior that we expect to see in chemistry. These conditions also favor the Lewis definitions, and if it seems like I am being a bit heavy-handed in mentioning the advantages of teaching the Lewis definitions to students as early as possible you would be quite correct. Fully embracing the Lewis definitions will require the more neurotic or tradition-bound individuals among the chemical community to let go of literally centuries of work that turns out not to be valid, but as before I have no particular emotional investment in Brønsted-Lowry and would much prefer to be taught the concepts in a way that actually makes sense.
In my list of topics I am supposed to cover acid-base equilibrium, which in the context of water (aqueous solutions) is how hydronium and hydroxide move into and out of solution. First looking at “HA” or a proton donor, we can either have the acidic proton attached to the conjugate base or not. The Lewis basic strength of “A-” determines how tightly the H+ is bonded and therefore how accessible it is to the surrounding water molecules. If the H+ is bonded too tightly, there is no chance of a water molecule ever removing it, and the compound is probably not going to be participating in any aqueous acid-base reactions. At this point I am really wanting to bring in some more organic chemistry concepts and talk about an example like ethanol (CH3CH2OH) as a compound with three distinct types of protons in three different chemical environments, with the hydrogen on the oxygen end (Eth-OH) as well as the two lone pairs on the oxygen being the most interesting electron pair acceptors and donors, but the current general chemistry syllabus as defined by the American Chemical Society (ACS) prevents this. Moving on to “BOH” in water, the strength of the bond between “B+” and hydroxide is also going to be important. As an example, the hydroxl group on ethanol has essentially no chance of being removed in an aqueous solution unless something quite energetic/violent happens, but the hydroxl proton can be stripped off or another proton can bond to one of the lone pairs on oxygen depending on the reaction conditions.
In the context of this post, I am basically trying to get into a decent position to talk about buffers. These are modeled by the Henderson Hasselbalch equation and are usually a combination of a weakly proton-donating “HA” with the “A-” part of that molecule paired with a positively charged counterion (counter-cation possibly). As an example cation, let’s choose sodium (Na+), which is a terrible electron pair acceptor because it is already in a noble gas valence electron configuration and adding electrons will be destabilizing. So, we can basically ignore the sodium ions unless we are interested in the total ionic activity for some reason, and at the same time the charges all balance out. If we select the correct “A-” and adjust the relative amounts of “HA” and “NaA”, we end up with a mixture that starts out at a pH that can be predicted via calculation. This is normal when adding proton or hydroxide donors to water, but where buffers are different is the ability to absorb proton or hydroxide inputs without the pH changing much. This is because of the presence of both protonated “HA” and deprotonated “A-” and is useful in situations were the molecules under study cannot tolerate large pH swings, which usually means proteins and other biological molecules. Selecting a buffer requires the concept of the constant of acidic dissociation (Ka) and the negative log of the same (pKa), but between this and Henderson Hasselbalch equation you should have plenty of keywords to play with. I am also supposed to be covering titrations here, but since these are as obsolete as Brønsted-Lowry and really shitty to have to carry out in the lab I’m not going to bother.

[https://pubchem.ncbi.nlm.nih.gov/periodic-table/png/Periodic_Table_of_Elements_w_Chemical_Group_Block_PubChem.png ] or [https://ptable.com/#Properties ]
In the last 14 posts, I have attempted to present the main points/useful information from a whole academic year of general chemistry. A significant fraction of the material taught in general chemistry is obsolete, but I am also skipping over any of the information that is actually beneficial to have somewhat memorized, all of the math, etc. Generally speaking, people don’t seem to have much trouble retaining information that is useful to them, so unless you’re having to pass a series of exams I would not worry about any of the details if you don’t want to. Maintaining a degree of rigor and intellectual honesty is important, but at the same time knowing a theory should enhance your understanding of the real world instead of detracting from it.
In any case, we have atomic nuclei with positively charged protons and non-charged neutrons surrounded by somewhat amorphous clouds of negatively charged electron density generated by a discrete number of negatively charged electrons moving around at high speed. How nuclei, orbitals, and electrons interact is chemistry, and given the complexity in chemical reactions that is evident (particularly in biology) it should come as no surprise that the behavior of electrons, elements, and molecules is also extremely complex. We as a species have spent many centuries of unified time and uncountable person-millennia of effort grappling with aspects of the complexity of chemical behavior, before discovering relatively recently that everything is derived from quantum mechanics and none of the simple mathematical models are particularly valid. The discovery of quantum mechanics started in the early 1900s to the 1920s or so in the physics community and has led to a progressive series of major improvements in the way we think about the world that is still underway. The information gained has led to our disastrous exploration of nuclear fission in heavy elements but also to the development of much more potent instrumentation, semiconductors, computers, and a better, if not necessarily more comforting, understanding of the universe that we live in.
Looking at chemistry specifically, our goal as a species needs to be to do as little chemistry as possible while still ensuring our survival. Where chemical reactions are unavoidable, we need to take care to ensure that the resulting waste is as non-toxic, biodegradable, and/or easily denaturable as possible. Simple molecules such as carbon dioxide can cause problems when emitted in bulk, and more complex molecules tend to be nastier in much lower quantities and concentrations (eg polychlorinated biphenyls/PCBs). As creatures with cellular machinery that is mostly made of organic molecules, we are going to be most interested in organic reactions despite our historical inability to make much sense of the complicated electronics and molecular orbitals of organic reactions. Unfortunately, this means that we will not be able to skip as many of the details, and if I want to try for complete coverage I would expect to see a few tens of posts. The main difference between general and organic chemistry is that a significant fraction (possibly even most) of the general chemistry material is obsolete and/or irrelevant, while the majority of organic chemistry material is both important and relevant. So this may take a while, and I’m going to wish that I still had access to the ChemDoodle software that is set up for organic structures. On ubuntu linux, the GChemPaint program seems similar and is free, and I guess that I’m about to find out how well that it works.
I will do my best to relate concepts back to the mental picture of how chemical compounds interact that you are hopefully building up as I introduce them, but as always things are usually going to be messy. The list of high level topics in organic chemistry as defined by my undergraduate study guide is as follows: structure, bonding, intermolecular forces of organic molecules, acids and bases in organic reactions, nomenclature, isomers, principles of kinetics and energy in organic reactions, preparation and reactions of (alkenes, alkynes, aldehydes, ketones, alcohols, sulfides, carboxylic acids, amines, aromatic compounds), organic reaction mechanisms, principles of conjugation and aromaticity, and spectroscopy. I have not yet decided if this is the order in which I would like to present these concepts, but hopefully you can see that this is a large amount of material. As a final note, organic chemistry is mostly the chemistry of hydrogen, carbon, nitrogen, and oxygen with trace quantities of several other elements participating at times. Organic molecules are interesting both because of the wide range of properties and behaviors that they exhibit and also because of our desire to understand our biology, and we are studying mainly the chemistry of the 1s, 2s, and 2p valence orbitals in small atoms.

After years of development and brainstorming, I am happy to announce that my SemanTeX package for LaTeX has just been added to CTAN and is now part of TeX Live and MikTeX. The manual is available here.
Its purpose is to allow for a more semantic, systematized way of writing mathematics, compared to the ordinary math syntax. The system is object-oriented and uses keyval syntax, and everything is highly customizable.
An example from elementary analysis: Suppose we want to take the complex conjugate of a function f and then derive it n times, i.e. f̅ ⁽ⁿ) . Using SemanTeX, we can write

$ \vf[conj,der=\vn] $

(The v in \vf stands for “variable”, so \vf is the variable f. It is usually best to create commands \va, \vA, \vb, \vB, ... for each variable you are using.)
Or suppose you want to invert a function g and restrict it to a subset U, and then apply it to x, i.e. we want to write g^-1|_U(x). This can be done by writing

$ \vg[inv,res=\vU]{\vx} $

An example from algebraic geometry: Suppose F is a sheaf and h a map, and that we want to typeset the equation (h ^-1F)_p = F_(h(p)), i.e. the stalk of h ^-1F at p is the stalk of F at h(p). This can be accomplished by writing

$ \vh[inverseimage]{\sheafF}[spar,stalk=\vp] = \sheafF[stalk=\vh{\vp}] $

(here, spar is the key that adds the parentheses around h ^-1F). All of these keys are defined by the user, and they can be modified and adjusted for all sorts of situations in all kinds of different branches of mathematics. Let us see how to set up SemanTeX to type the examples above:

\documentclass{article} \usepackage{amsmath,semantex} \NewVariableClass\MyVar % creates a new class of variables, called "\MyVar" % Now we create a couple of variables of the class \MyVar: \NewObject\MyVar\vf{f} \NewObject\MyVar\vg{g} \NewObject\MyVar\vh{h} \NewObject\MyVar\vn{n} \NewObject\MyVar\vp{p} \NewObject\MyVar\vU{U} \NewObject\MyVar\vx{x} \NewObject\MyVar\sheafF{\mathcal{F}} % Now we set up the class \MyVar: \SetupClass\MyVar{ output=\MyVar, % This means that the output of an object % of class \MyVar is also of class \MyVar % We add a few keys for use with the class \MyVar: definekeys={ % we define a few keys {inv}{upper={-1}}, {conj}{command=\overline}, {inverseimage}{upper={-1},nopar}, }, definekeys[1]={ % we define keys taking 1 value {der}{upper={(#1)}}, {stalk}{seplower={#1}}, % "seplower" means "separator + lower", i.e. lower index % separated from any previous lower index by a separator, % which by default is a comma {res}{ rightreturn ,symbolputright={|}, lower={#1} }, }, } \begin{document} $ \vf[conj,der=\vn] $ $ \vg[inv,res=\vU]{\vx} $ $ \vh[inverseimage]{\sheafF}[spar,stalk=\vp] = \sheafF[stalk=\vh{\vp}] $ \end{document}

https://preview.redd.it/pqftr2gz7g251.png?width=493&format=png&auto=webp&s=2273bb8139bbddfbd2cc51ff9135b711aade96ff
See the manual for more details. Comments, bug reports, and ideas are more than welcome! :-)

FOREX Strategies

What are FOREX Strategies?
https://preview.redd.it/ihmphstzguv51.jpg?width=960&format=pjpg&auto=webp&s=81f6b73c367d8695605514f8d32aaf3e2aeabc6e
You may have noticed that most of people confuse the terminology and refer to FOREX Strategies in the wrong way. There are methodologies, systems, strategies, and techniques. The most effective methodology is Price Language (Trend Tracking). Combined with a correct reading of mass psychology presented by the charts.
We know that in the Stock Markets there are thousands of strategies. FOREX, like the rest of the markets, presents you with the opportunity to apply similar strategies to win consistently. Taking advantage of repetitive psychological patterns.
First, the Price Language methodology has created great fortunes in FOREX, and the next fortune may be yours. But this methodology must be implemented within a framework of advanced concepts of Markets. Without forgetting the basics. And working hard day by day.
Second, a strategy is a set of parameters and techniques that together give you the advantage to act in any situation. Thus for example in war, generals have attack strategies and counterattack strategies.
FOREX strategies alike are entry strategies and exit strategies. All beginners should know these FOREX strategies for beginners. That way you will get a general idea of the game and understand that trading is a war against the Market and its Specialists. Only applying FOREX strategies revealed by the same Specialists and using their own techniques,
... you can survive in this war.
Do not fall into the trap of the many "systems" and "methods" that are offered on the internet about operating in the FOREX Market. They just don't work in the long run. They are strategies based on indicators for the most part. Using rigid parameters. That if they can work and give profitability during a certain period of time, they will always reach a breaking point when the market changes its dynamics.
Instead, take advantage of your precious time and learn the Language of Price or Price Action.
The Language methodology will allow you to adapt to each new phase of the Market. If you combine this knowledge with the appropriate psychological concepts, you can live comfortably from speculation in FOREX.

Forex Trading Strategies Reddit - Basic FOREX Strategies

You have two basic FOREX strategies, one entry, and one exit. Both follow a general strategy that helps you capitalize on the collective behaviors of the Market. That is, of the total of participating speculators.
This behavior causes the formation of cycles that repeat over and over again. Driven by the basic emotions (uncertainty, greed, and panic) of the speculators involved that can be taken advantage of with the aforementioned FOREX strategies. Specialists identify these emotions in the order flow and capitalize on these events every hour, every day, and every month.
Basic FOREX Strategies - The Price Cycle
These repetitive cycles consist of 4 phases:

Accumulation
Upward trend
Distribution
Downward trend

https://preview.redd.it/6dvk2w0pduv51.png?width=300&format=png&auto=webp&s=a3ab65ca4eab6d20174b3327b862d8b59dcc13b7
The two trends can be easily identified by their notorious breakdown. And the two areas of uncertainty (accumulation and distribution), due to their notorious range trajectories.
This general behavior determines the core of our FOREX strategies.
You buy when the price of a pair has broken and has come out of one of its congestion formations (accumulation or distribution). You implement one of the Forex strategies, in this case, the entry one.
The multi-time technique will help you find the point of least risk when entering your initial buy or sell order. In the same way and using the same strategy but this time to close your position, the multiple timing technique will also show you how to close your operation obtaining the highest possible profit.
The most consistent way to extract profits in the market is by trading the start of trends within a cycle . Once confirmed by their respective breaks from the areas of uncertainty. This is the mother of all FOREX strategies . And in a market that operates 24 hours, we have more frequent cycles and therefore more opportunities.

Forex Trading Strategies Reddit - Advanced Forex Strategies

There are many advanced FOREX strategies that are generally used by professional speculators working for large financial firms.
Among these firms are banks, Investment Fund managers and Hedge Fund managers. The latter is an investment modality similar to Investment Funds, with the difference that Hedge Funds use more complex investment strategies. Its operations are more oriented to aggressive speculations in the short and medium-term.
Among the most common strategies is hedging (hedging), carry trade, automated systems based on quantum mathematics. And a large number of combinations between the different option strategies.

The Carry Trade

The central idea of Carry Trade is to buy a pair in which the base currency has a considerably higher interest rate than the quoted currency. To earn the difference in rates regardless of whether the price of the pair rises or falls.
Suppose we buy a $ 100,000 lot of AUDJPY, which according to the rates on the chart would turn out to be the ideal instrument in this example to use the Forex carry trade strategy.
As our capital is in US dollars we have to assume for our example, the following quotes necessary to perform the place calculations:
AUD / JPY = 80.00 USD / JPY = 85.00
What happens internally in your broker is this.

By placing as collateral $ 1,000 of your $ 50,000 of capital (assumed for this example), deposited in your account, you have access to $ 100,000 virtual (this is what is known as leverage); that is, you put in $ 1,000 and your broker lends you 99,000.
With those $ 100,000 virtual dollars, your broker borrows on your behalf ¥ 8,500,000 Japanese yen (85 × 100,000) at 0.1% annual interest from a Japanese bank.
With those ¥ 8,500,000 Japanese yen, your broker buys A $ 106,250 Australian dollars (8,500,000 / 80) and deposits it in an Australian bank where it receives 4.5% annual interest on your behalf.
One year later (and regardless of the profit or loss generated by the pair's movement), your profit will be the difference between the AUD rate and the JPY rate, that is:

Profit = (AUD rate) - (JPY rate) - (costs of the 2 currency exchanges) Profit = (4.5%) - (0.1%) - (0.1% to 1%)
The great advantage of carry trade FOREX strategies is that this percentage profit is applied to the $ 100,000 of the standard lot; the broker transfers all of the profit to you, even if you only contributed $ 1,000. On the other hand, if you carry out the inverse of this operation, this benefit of the Forex carry trade becomes a cost (swap), and you assume it completely.
Remember that FOREX carry trade strategies are recommended for pairs with considerable interest rate differences, such as the one we have just seen in our example.
These FOREX strategies should also not be used in isolation. The idea is that through technical analysis you identify when would be the ideal time to enter the market using your carry trade Forex strategy and multiply your profits considerably.

What FOREX Strategies Do Hedge Funds Use?

The FOREX strategies used by large fund managers do not constitute an advantage in terms of percentage results for them, nor do they constitute a competitive disadvantage for you.
The vast majority of them fail because of their big egos. In fact, there was a firm made up of great financial geniuses, including 2 winners of the Nobel Prize in Economics, who developed a strategy based on quantum mathematical calculations.
With an initial base capital of about 3 billion dollars, and after 3 successful years obtaining annual returns of over 40%, the firm Long-Term Capital Management, begins its fourth year with losses. To counteract these losses the geniuses decide to multiply the initial capital several times, while the losses continued.
The year closed with the bankruptcy of the fund, and with a total accumulated loss of 1 trillion dollars, due to the great leverage used. And all for not admitting that the FOREX Strategies of Long Term Capital Management were not in line with the dynamics of the Market.
There are an overwhelming number of opportunities in the stock markets to make money interpreting the Language of Price.
You don't need to use complex "advanced" strategies that have been created to handle hundreds or billions of dollars.
The reasons for using these FOREX strategies are very different from what a "retail trader" pursues with his small speculation business.
As you can see, you should not worry about wanting to integrate any of these advanced strategies into your arsenal. They are only beneficial for managing hundreds or billions of dollars, where the return parameters are very different when you handle small amounts of capital.
Do not worry about collecting hundreds of free FOREX strategies that circulate on the internet, that great accumulation of mediocre information will only serve to confuse you and waste your valuable time.
Spend that time learning Price Action,
… And you will always be one step behind the Specialists, identifying each new Market condition, and anticipating the vast majority of reversals of all prices.
Ironically, the most successful fund managers indicate that their most profitable trades are those based on the basic trend-following strategies of the Price Language. The same ones that you will learn in this Free Course.
Dedicate yourself to perfecting them and believe me you won't need anything else. As long as you have good risk management, taking into consideration the following points ...

Styles of Investments in FOREX

The Investment FOREX long term is not recommended for small investors like you and me. If we take into account the term investing literally as large investors do who buy a financial product today to sell it years later.
We both have a better niche in the short and medium-term.
You may have noticed that the big multi-year trends in the Forex Market do exist. But minor swings within a big trend are usually very wide.
These minor movements allow us to easily double and triple the annual return of the big general trend, motivating most traders to speculate in the short and medium-term.
These minor oscillations or trends that occur within the large multi-year trends owe their occurrence mainly to two reasons.
First, the FOREX Market presents 3 sessions a day each in different cities of the world with different time zones (Asia, Europe, and America). This causes more frequent trend changes than in the rest of the stock markets.
Second, the purpose for which it was created also plays a role. The modern Foreign Exchange Market, since its inception in 1972, was conceived by the global financial system as a tool for speculation. To obtain benefits in the short and medium-term (from several days to 1 year).
These two points are basically the reasons why we observe the immense speed with which the FOREX market changes trends.
For example, for those who live in America, in the early morning (Europe) the EURUSD pair may be on the rise, in the morning or afternoon (America) it may be down, and then finally at night (Asia) it may return to the rise.

Define your Own Style for your FOREX Investments

One of the first decisions you will have to make is to choose your style as a trader or investor.
There are 4 types of well-defined styles.
Most professional traders tend to have multiple styles, although they always identify with one primary style for their FOREX investments. Study the characteristics of the 4 main styles to make your investments in FOREX :
1. Long Term: recommended for anyone who is going to enter the market for the first time and who can dedicate a minimum of one hour per month to their investments in Forex. The period of an open position ranges from 1 year to 5 years.
2. Medium Term: recommended for anyone who is going to enter the market for the first time and who can dedicate a minimum of one hour per week to their investments in Forex. The period of an open position ranges from 1 month to 1 year.
3. Short Term: recommended for anyone who is going to enter the market for the first time, or who already has a certain time operating in the long and medium-term, showing constant profits, and who can dedicate a minimum of one hour per day to your investments in FOREX. The period of an open position ranges from 1 day to 1 month.
4. Intraday : recommended only for people with a fairly solid earnings record in the short term, and with a capital greater than $ 50,000. As we have noted, this option constitutes a full-time job.
People who start investing in FOREX , should start executing short-term (weeks) and medium-term (months) transactions only, and not pay attention to intraday oscillations (day trading).
If you are interested in being an intraday speculator, I recommend that you first exhaust at least a year doing operations in the short and medium-term to assimilate the correct strategies and to develop the necessary mentality to carry out this work.
The second option would be to participate in some kind of intensive training.
I remind you that self-educating is almost impossible in speculation. You are likely to accumulate a lot of knowledge by reading books and attending courses. But you will probably never learn to make money with all the incomplete "systems" circulating on the internet.

Mistakes to Avoid When Looking for Your Style

Many people who are new to FOREX investments make the mistake of combining these styles, which is a key to failure.
I recommend that if you are not getting the results you expected by adopting one of these styles, do not try to change it. The problem sure is not in the style, but in your strategies or in your psychology.
A successful investor is able to make a profit in any longer trading time than he is used to. I explain. If you are already a profitable operator in the short term, it is very likely that you will also be profitable in the medium and long term,
… As long as you can interpret the Language of Price or Price Action.
In the opposite case, the same would not happen. If you were a medium-term trader, you would need time to adjust to the intraday. The reality is that long, medium and short term traders have very similar personalities. The intraday trader is completely different.

The Myth of the Intraday in Investments in FOREX

If you are already successful in the short, medium and long term, you will notice that the sacrifice and the hours necessary in front of the computer to operate intraday is much greater. The intraday style will be useful to increase your account if it is less than USD $ 100,000 in a very short time in exchange for 8 to 12 hours a day of hard work but ...
You must first develop the necessary skills to operate the intraday.
The ideal is to combine all the styles to get more out of the Market and carry out more effective transactions and have a diversification in your investments in FOREX.
There are intraday traders that are very successful, but the reality is that there are very few in the world that make a profit year after year. If you want to become an intraday, you just have to prepare yourself properly through intensive training.
Otherwise, I recommend that you don't even think about educating yourself to adopt the intraday style. It is not necessary to go against a probability of failure greater than 99%. Unless
... your ego is greater than your common sense.
The main reason why this style of investments in FOREX is not recommended for the vast majority of us "retail investors" (the official term "retail traders"), is the high operational cost.
The real commissions in this market range between $ 2.0 and $ 2.50 for each lot of 100,000 virtual units. This means that a complete operation (opening and closing) is approximately $ 5.00, for each standard lot traded ($ 100,000 virtual).
Another fundamental reason is the advent of robotic traders (HFT = High-Frequency Trading), which tend to manipulate the market in the shorter intraday swings. Please do not confuse HFTs with automated systems that we find daily on the internet, and that can be purchased for a few hundred dollars and often for free on FOREX forums / groups.
These HFTs to which I refer, they are effective. They cost millions of dollars and have been developed by the large Wall Street financial firms to manage their investments in FOREX.
The reality of the intraday trader is that you execute orders for large lots at the same time, to profit from the smallest movements in the market. It is an activity based on reflexes. The slightest oversight or distraction can turn into a catastrophe for your FOREX investments.
I recommend that you start investing in FOREX using slow time periods such as H4 or Daily. For some reason, all Goldman Sachs intraday FOREX investments are made with algorithms.

Finally…

To choose your style as a trader and manage your investments in FOREX, first determine what your degree of experience is, analyze the points mentioned below and the rest you will discover when you execute your first operations.
The points that will affect your decision are:

Capital
Time available each day
Level of Experience
Personality

Discovering your style is a search process. For some it will be a long way to find the right time frame that matches their personality. Don't be put off by the falls. After all, those who continue the path despite the falls are the ones who reach the destination.
And I hope you are one of those who get up over and over again. The next lesson will boost your confidence when you discover the main reason that moves currencies ...

Fundamental Analysis in Forex Trading Reddit

The fundamental analysis in Forex is used mostly by long-term investors. Players as we saw in the styles of operators, start a negotiation today, to close it years later.
I always emphasize the importance that the mass media give to this type of analysis to distract the great mass of participants.
It is all part of a great mass psychological manipulation. For centuries the ignorance of the masses has been organized before the great movements begin.
The important news are the macroeconomic reports published by the Central Banks and other government agencies destined for this work. All reports are made up. 99% of them are corrected months later.
These events are tools to justify fundamental analysis and price cleaning movements. Any silly headline does the job. With this, it is possible to absorb most of the existing liquidity, before the new trend phase is projected.

Reaction!

Except in rare situations, the result of an economic report of the fundamental analysis is generally already assimilated in the graph. In most cases, there are financial institutions that already have access to this information and are organizing and carrying out their operations in advance.
The phrase buy the rumor and sell the news is a very old adage on Wall Street. And its meaning contains what we have just explained. For the investor who can interpret the Language of Price, fundamental analysis is of little importance. Well, in general, their disclosure does not indicate that you have to take any action in your open trades , as long as your entry strategy provides you with a good support cushion.
This reality of fundamental analysis causes a lot of confusion for investors who lack in-depth knowledge of the forex market.

Macroeconomic Data

The data published in these events is irrelevant. Both for speculators and for the people in general. They are false. They lack reliability.
The price can go up or down with the same result of the data. The main ones are:
- Interest Rates - GDP (gross domestic product) - CPI (inflation) - ISM (manufacturing index) - NFP (payroll) - Double Deficits (deficit = fiscal + balance of payments)
If you are initiated, I recommend you avoid operating near these events. It is only a matter of having the time pending. Use the economic calendar for Fundamental Analysis of Forex Factory.
There is a probabilistic advantage in operating these fundamental analysis events. But it takes preparation, experience, and practice. They represent a way of diversifying in the general operation of a speculator.

The Uncertainty of Fundamental Analysis

On many occasions after the disclosure of an economic report, the price movement of the currency pair that is going to be affected tends to move in the opposite direction to the logic of the report.
I show you an example of a fundamental analysis report. Imagine that the EUR / USD pair is trading at 1.2500, and the FED (US Federal Reserve) issues a statement announcing that it has just raised inter-bank interest rates from 0.25 points to 0.75 points. Very positive news for the US dollar that logically implies an appreciation of the currency and consequently an instantaneous collapse of the EUR / USD pair (up the dollar and down the euro)
However, minutes after the release of said fundamental analysis report, the pair after effectively collapsing to 1.2400, returns and returns to its levels prior to the report (1.2500). This situation is very common , but it is not so easy to identify it when it is occurring, but after the damage is done.
Traps like these devour the accounts of beginners who approach the market with little experience, with weak strategies, and especially with very little experience.
That is why I reiterate that you forget the fundamental analysis for now. Just keep in mind when operating, that there is no publication scheduled nearby. Just check the economic calendar for the day and forget about the numbers. Let the economists mess around with the data.

FOREX Market Correlation

The Forex market correlation exists between pairs with similar "base" currencies and not always under the same circumstances. The correlation in the Forex market that is most followed and that has the greatest impact on fundamental analysis is that of the US dollar (USD).
The USD is the most traded monetary unit with a volume greater than 80% with respect to the rest of the currencies. This fact determines why their correlation is the most important, the most followed, and perhaps the only one worth following in the fundamental macro analysis.
The 7 major pairs are usually in sync . These 7 pairs all include the USD and present a fundamental analysis correlation almost 75% of the time. Influencing the rest of the currency pairs.

Advantages of the FOREX Market Correlation

In the fundamental analysis the most basic FOREX correlation is the following. When the USD appreciates, the USD / CAD, USD / CHF, and USD / JPY pairs tend to go up in price. This indicates that the Canadian dollar (CAD), the Swiss franc (CHF), and the Japanese yen (JPY) are losing value against the USD.
We must bear in mind that this correlation does not occur 100% of the time. In fact, the JPY generally tends to move in the opposite direction , since in recent decades this currency has been used as a source of financing to invest in other financial instruments.
On the other side is the FOREX market correlation that generates a movement almost in unison in the other 4 major pairs EUR / USD, GBP / USD, AUD / USD, and NZD / USD. These tend to fall in price, homologous the appreciation of the USD. But not always.
In this case the fundamental analysis correlation works most of the time, between 65 and 85% of the time. Small differences are noted in the extent that each of these pairs experiences.
There is also a correlation in the secondary FOREX market, where the pairs of all currencies that do not include the USD participate, but I recommend you not to waste time on them for now. There are more important things about the Language of Price to know first.

FOREX Commodity Correlation

In this part I will explain to you in a basic way the Correlation Commodities - FOREX of the fundamental analysis.
There are three currencies that have a direct correlation with commodities. They are usually called: "COMDOLLS" which is short for "Commodities Dollars" (Commodities Dollars), since all three obey the dollar denomination. These are:
- The New Zealand Dollar (NZD) - The Australian Dollar (AUD) - The Canadian Dollar (CAD)
These three currencies make up the group of the 8 largest together with the euro, the pound, the yen, the franc and the US dollar. Together, they merge to produce the major pairs traded in the FOREX Foreign Exchange Market.
The FOREX Commodity Correlation has an affinity in most cases greater than 75%. And each of them has its different raw material of correlation. You will notice that the NZD and the AUD are two currencies that act practically in unison. Both present minimal discrepancies in their fluctuations in the short, medium and long term.
This is mainly because their economies are very similar and their economic and fiscal policies are too. Their main production items also show great similarities, despite the fact that the Australian economy is much larger than the New Zealand economy.
The raw materials that follow the movement of the AUD are mainly gold and copper. If you put the history of these three quotes during the last decade of the year 2,000 together on the same chart, you will notice a very similar upward movement between the three quotes. Pure correlation of fundamental analysis.
This strong correlation with commodities in the metals area for the AUD has provided Australia with an economic advantage enviable over the other major powers that have seen their currencies devalue sharply against the AUD. At the same time, they experience a constant decrease in the purchasing power of their citizens.
The NZD maintains a correlation with raw materials related to agriculture and livestock, mainly including milk and its derivatives. It is one of the countries that dominates the world export of these economic items, and also has important exports of metals , although in smaller quantities than Australia.
Finally, you have a correlation with raw materials in the energy area. For historical reasons the CAD, which is not the largest oil producer in the world, but an important supplier to the largest consumer that is the US, has seen its currency oscillate in line with oil prices.
To make long-term investments in the Foreign Exchange Market, it is necessary to take into consideration at least one Commodity Correlation - FOREX in your fundamental analysis.

Forex Technical Analysis Reddit

The technical analysis is the methodology that interprets the movements of the price. Specialists look for liquidity to fund their business. The repetition of the strategies used by the specialists in their work generate repetitive patterns.
If you were an analyst, you would develop the visual ability to identify such patterns on a graph. If you were a programmer you would quantify them mathematically using complex formulas.
And if you could learn to interpret the Language of Price, you would have the ability to anticipate 90% of all movements that occur on a chart. And in this business, anticipating is what will make you money.
Market prices are reflected and framed on a horizontal time axis and a vertical price axis. Prices go up or down according to the aggressiveness of the participating operators. In an efficient or balanced market these oscillations should be imperceptible.
But in reality this is not the case, since the Market works thanks to the digital printing of hundreds of billions of units of paper money systematically distributed by the Central Banks through the banking system. These resources serve as a tool to manipulate 100% of the movements that occur in the FOREX Market.
Are you looking for Technical Indicators? All technical indicators were created from the 70's. How do you think that for more than 200 years the speculators of the past accumulated great wealth?
With the Language of Price. The best timing is given by the price itself. Indicator-generated entry signals usually occur at the wrong time.
The basis of technical analysis is human psychology. Unfortunately, human beings are not perfect and are loaded with emotions that dominate their behavior in similar situations, creating repetitive and highly predictable behavior when it occurs in masses.
The study of technical analysis through indicators and subjective training, originates and shapes the collective thinking on which all the traps that specialists execute every day to maintain their business are designed. If the majority won, the Market would cease to exist.
Although you already know that the patterns are not generated by the masses , but the repetitive behavior of the Specialists in the face of the action response of the masses. It is very easy for speculaists, because they can see everyone's orders in their books.
And they also exert a great influence on the decisions of the masses through the mass media. It is what I call the war between the Egg and the Stone , if you hit me you win and if I hit you also you win.

The Deception of Modern Technical Analysis

Through the centuries thousands of people have been able to extract great benefits from the financial markets by applying the basic strategies of technical analysis and the psychology of the Price Language.
More than 200 years ago when the markets began to operate officially, fundamental analysis predominated, which was only used by large financial institutions. As this analysis tool began to become popular, these institutions began to apply the strategies of technical analysis.
In recent decades and with the massification of internet technology, technical analysis has begun to be handled by anyone who has a computer with internet access. The same financial institutions, which have been present for more than a century and as a result of this overcrowding , establish a strategy to confuse and misinform about the true strategies of technical analysis.
This has been accomplished in the following manner. Currently there are hundreds, if not thousands of technical indicators that have been developed by so-called "gurus" of technical analysis and that sell their magic indicators packed in a "system" or "method" that usually cost thousands of dollars, or simply with the publication of a book with which they generate large profits. Double benefit.
The aim is to confuse the initiates in speculation and create the collective mentality that will originate the same behaviors over and over again. About 95% of these new entrants completely lose all the capital they invest in their early stages as investors.
Leaving them with a negative experience and creating the idea and the image that financial markets are an exclusive area for geniuses with high academic levels and that only they can produce returns in the markets year after year.
The initiate, having lost all his original capital, turns to these “gurus” for help and teachings. You spend more capital on the products they offer you and the cycle repeats itself . Obviously, the vast majority do not relapse and completely forget to re-engage in the stock markets.
I hope you have not been a victim of this drama.
Now I will show you the simplicity of a FOREX technical analysis , without the need to resort to any indicator as a tool to determine an effective entry or exit strategy when planning your operations.

The Price Cycle

Previously you studied in the FOREX strategies lesson, that the typical price cycle when it is reflected in a graph, presents four very specific phases and very easy to identify if you perform a technical analysis with common sense . These are:

Accumulation
Bullish trend
Distribution
Bearish trend

Remember also that the most effective way to constantly extract profits in the markets is by taking advantage of phases 2 and 4 (the trends). Combined with a correct reading of the collective behavior of the masses of speculators interpreting the Language of Price.
You will be surprised by the simplicity with which thousands of people around the world and over the centuries have accumulated large sums of money by drawing a few simple lines and applying responsible risk management with their capital.

How to Identify Trends?

Being able to determine the trend phases within the price cycle is the essence of technical analysis since it is these two phases that provide you with the probabilistic advantage you need to operate in the markets and obtain constant returns.
In the most plain and simple language, in the world of technical analysis, there are only two types of formations: trends and ranges.
The trends, in turn, can be bullish if they go up, or bearish if they go down. The ranges, on the other hand, can be accumulation if they are at the beginning of the cycle, or distribution if they are in the high part of the cycle. As I had indicated in the topic of FOREX strategies when describing the price cycle.
This sounds more like a play on words, but I will show you the practical definition to simplify your life and then you will apply these definitions on the graph so that everything makes more sense to you.

Bullish trend: a succession of major highs and major lows
Bearish trend: a succession of minor highs and minor lows
Floor Range: equal highs and varied lows
Ceiling Range: equal minimums and varied maximums

https://preview.redd.it/vvmsshf0guv51.png?width=600&format=png&auto=webp&s=c321679a7dcc03f7184778be86379ef442fddf91
Some key points from the graph:

The start of this big uptrend was detected when the last high (thick green line) of the previous downtrend was broken to the upside, ending the succession of lower highs, while exiting the lateral floor formation.
The succession of major lows in the uptrend (thin blue lines)
The succession of major highs in the uptrend (thin green lines)
The end of the uptrend was detected when the last low (thick blue line) of the uptrend was broken to the downside, ending the succession of higher lows, while exiting the lateral ceiling formation.

A tool that will help you sharpen your technical eye and identify trends on the chart is the Currency Scanner. This application is very effective and will provide you with a much-needed boost in your operations to identify reliable trends. At first, we are not sure how reliable a trend is. You will receive great help to find opportunities with the Currency Scanner .

The Common Sense, The Less Common of Senses

The central idea of technical analysis consists in determining the price situation of a market, that is, in which phase of the pattern of its cycle it is currently conjugated with the collective thinking of the masses and the possible traps that the market would have prepared to remove. the capital at stake by the public.
To carry out a precise technical analysis, you will use the support and resistance lines, which can be static (horizontal) or dynamic (projecting an angle with respect to the horizontal axis).
Your common sense prevails here.
If you show a 10-year-old a chart, they will be able to tell you if the price is going up or down. You will most likely have no idea how to draw the lines, but you will be able to establish the general trend. Simply using your common sense.
By introducing indicators and other gadgets , the simplicity and effectiveness of the technical analysis created by your common sense evaporates.
The following graph conceptually shows you all the possible situations in which you could draw these lines to carry out your technical analysis of the place. You can clearly observe a downtrend delimited by its dynamic trend line and an uptrend on the right side with its respective dynamic delimitation.
https://preview.redd.it/5iehg0r6guv51.png?width=500&format=png&auto=webp&s=84c265a5d35da7ea970792c4bf40fe20b33bd8bd

Forex Charts Analysis

I want to remind you that the formations or patterns that develop on the charts (triangles, wedges, pennants, boxes, etc.) only work to execute trades that have initially been confirmed by the static support and resistance lines and to read the collective thinking of the masses.
Chart formations work, but you must know the Language of Price to determine when the Specialists will exploit a chartist figure, or when they will allow it to run. In fact, you will learn with the Language that you can operate a chart figure in any direction.
Much of the "mentalization" that the masses receive is to believe that the figures are made to be respected. Which is an inefficient way of working. Simply because you could wait days or months for a perfect chart figure to occur in order to perform a reliable trade. When in fact there are dozens every day.

Japanese Candles

Of all the tools you have to carry out technical analysis, perhaps the best known and most popular is the Japanese technique of candles (candlesticks).
Candles are mainly used to identify reversal points on the chart without resorting to confirmation of horizontal trend lines and only using a previous bar or candle breaks.
Its correct use is subject to a multi-time analysis (multiple temporalities) and a general evaluation of the context proposed by the market in general at the time of each scenario.
Later I will show you all the important details to take into account so that you use Japanese candles in a simple and very effective way.
Do not forget ... Trading in your beginnings based on formations (chartism) and candlestick patterns conjugated with hundreds of tools and technical indicators, constitutes the perfect path to your failure. Before using any strategy or technique I recommend you focus on learning the Price Language, which includes 3 basic things:

The Price: structure and dynamics
Market sentiment: relative strength, external shocks, etc.
Psychology: flexible mindset and risk acceptance

After you acquire this solid foundation, I guarantee that you will be able to trade any trading system that exists, any strategy, technique or chart figure in a profitable and consistent manner.
Specialists make money every day at the expense of the collective behavior caused by the use of these strategies and techniques. With which you will only manage to lose your capital and your time by putting the cart in front of the horse.
People who do the opposite, at best become,
... Philosophers of Speculation, or indocile Robot Assistants or Expert Advisors.
To make money in any market condition, range or trend, you must use the technical analysis based on the Price Language and combine it with a correct psychological reading of the price. This knowledge can only be acquired through proper education and lots of supervised practice. Like any other career in life.
I hope you've found this guide helpful!

Hello all,
I've never learned about quaternions before, and it finally came up that I need to understand them to implement rotations in my game (being made in Unity). I decided to take the time to learn and understand them well, and one thing that helps me retain what I learned is to share/explain it to others in a way that is simple to understand.
So that is why I'm making these posts. This first part will be explaining what quaternions are - i.e. their mathematical definition. The next part will be a simple tutorial on how to use quaternions to rotate game objects in Unity.
DISCLAIMER: I myself have not gone in that much depth on the subject, I learned just enough to understand the basics, and this post will be a summary of that. Also, it's best to have a basic understanding of Linear Algebra and complex numbers to fully understand everything this post covers.
While I will be linking the sources that I used to learn from, I want to give a special shout out to this site right here: https://eater.net/quaternions/. They explain quaternions beautifully and simply using interactive videos that greatly visualize the concepts. I highly recommend checking that out.
That said, I hope you will find this and benefit from it. Please let me know in the comments if I have made any mistakes in my explanations or if I could improve on them in some way.

Basic terms

When discussing the rotation of an object, there are two types of coordinate frames that are important to know: the inertial frame, and the body frame.
The inertial frame can be thought of as the fixed frame of reference of an object - the rotation will always be applied from the inertial frame.
The body frame is the frame that shows the object's position and rotation relative to the inertial frame.
So when applying a rotation, what it essentially means is to perform the mathematical transformation of a vector from the inertial frame to the body frame.
See Figure 1 below for a simple visualization:
Fig. 1: The inertial frame (blue) transformation to the body frame (yellow) [1]

The need for quaternions

3D rotation is much easier to understand when using Euler angles: you have the three separate axes that you can rotate on in 3D space: x, y, and z. For each axis, you would pretty much handle the rotation on the plane perpendicular to that axis - which essentially is like performing 2D rotation for each dimension.
To rotate all 3 axes at once, you would basically put the formulas for rotating each individual axis in a matrix and multiply that matrix with the vector (from the right) to end up with the resulting rotated vector. See the figures below for a better understanding:
Fig. 2: The inertial frame - the frame of reference of an object [2]
Fig. 3: Rotation matrix from the inertial frame to body frame [2]
However, Euler angles cannot solve every case of 3D rotation, as it can suffer from the "Gimbal Lock." This occurs when the angle on one axis approaches 90 degrees, which results in the object only being rotatable in 2 dimensions on the other axes, as they are positioned in a way such that rotating on one of the two remaining axes has the same range of angular motion as the other.
To eliminate this problem and have the ability to rotate in all 3 dimensions at all times, we turn to ~~black magic~~ quaternions.

What are quaternions?

Mathematically speaking, a quaternion is a 4-element vector that can be used to encode any rotation in a 3D coordinate system. Generally speaking, it is actually a special kind of complex number, where the imaginary part is a 3-dimensional vector.
Example: q = a + v => a is the real part, which is a scalar, and v is the imaginary vector defined as: v = (xi, yj, zk), where i, j, and k can be thought of as unit complex vectors in the x, y, and z axes, respectively.
As a 4D vector, it can be represented like this: q = [a x y z]
When used in the context of 3D rotations, you can think of the imaginary/vector part as being the axis that the body will rotate on, and the real/scalar part as being the angle of rotation. More specifically, if theta is the angle of rotation, and v is the axis vector, then the rotation quaternion q = [a b c d] - called the attitude quaternion - is defined as follows:
Fig. 4: attitude quaternion [3]
Before discussing the formula for quaternion rotation, we must discuss quaternion multiplication. Quaternion multiplication, although long and tedious to do by hand, is not very different from the multiplication of two complex numbers in rectangular form. NOTE: the following is only one way to define quaternion multiplication, but it is by far the simplest to understand.
The Hamilton Product of two quaternions is essentially done by expanding each quaternion and multiplying them using the distribution law, similar to multiplying polynomials in algebra:
Let q1 = a1 + b1i + c1j + d1k and q2 = a2 + b2i + c2j + d2k be two quaternions. Their resulting product is:
q1q2 = (a1 + b1i + c1j + d1k)(a2 + b2i + c2j + d2k) = ... = (see result below)
Fig. 5: resulting product of two quaternions [3]
NOTE: order matters! Like matrix multiplication, quaternion multiplication is not commutative, meaning q1q2 != q2q1.
All that said, the formula for rotating a vector from the inertial frame to the body frame is:
Fig. 6: formula for quaternion rotation [3]
Where q^bi is the attitude quaternion as defined earlier, (0 vI) is the inertial frame vector to rotate, represented as a quaternion with a 0 real part, and (q^bi)^-1 is the inverse of the attitude quaternion.
The inverse of a quaternion is similar to the conjugate of a complex number, where you simply invert the sign of the imaginary part. For a quaternion, you invert the sign of all the components of the vector part.
Example: q = a + v => q^-1 = a - v

This all seems counter-intuitive

...is what you might be thinking - because that's what I thought as well when I first read all this.
My main questions were:

Why does the attitude quaternion multiply the input angle by 0.5?
Why do you bother multiplying the input vector by the attitude quaternion if you're going to immediately multiply that result by the inverse of the attitude quaternion? Wouldn't that negate the initial multiplication?

To answer the first question: the attitude quaternion halves the input angle because of the nature of the rotation formula, in which you sandwich the input vector between the attitude quaternion and its inverse. If you don't halve the angle, you essentially end up rotating the vector by double the input angle.
The second question is much more difficult to explain simply, so I will do my best to put it in my own words according to what I've learned:
Before the actual explanation, I have to define what a stereographic projection is. In simple terms, a stereographic projection is a mapping of an entity of a higher dimension onto an entity of a lower dimension. A simple, practical example of stereographic projection is mapping the Earth's globe onto a 2D map of the world.
When you perform a quaternion multiplication, geometrically speaking you're performing a stereographic projection of a 4-dimensional hypersphere (which is beyond our perception) onto 3-dimensional space.
So when you perform the first multiplication from the right - i.e. the attitude quaternion by the input vector (0, v) - the resulting projection causes vector v is rotated by the angle of the attitude quaternion, which is 0.5 * theta, on the attitude quaternion's rotation axis - i.e. its "vector" part. However, the object represented by the input vector would be distorted due to that projection, if you were to apply that transformation on all the points that make up that object.
Therefore, to undo the distortion effects of the hypersphere projection, we need to multiply the result mentioned above by the inverse of the attitude quaternion. This will rotate the object by the theta * 0.5 again while applying the "reverse" projection, undoing the distortions caused by the previous transformation.
This results in the complete rotation of the object by angle theta. The overall multiplication first rotates it halfway while distorting it, and then it rotates it the rest of the way undoing the "damage" done by the first multiplication.
You can get a better explanation and visualization of the above section at https://eater.net/quaternions/.
Thank you for reading this whole thing if you got to this point, and I hope you benefit from this as much as I have benefitted from writing it!

Sources

I will be splitting the sources in two sections: one for all the sources that I got information from to write this, and the other one for the figures that I directly took screenshots from. The one I can't stress enough to check out if you want to understand quaternions is: https://eater.net/quaternions/.
Sources of Info

Euler Angles (Wikipedia): https://en.wikipedia.org/wiki/Euler_angles
Euler Angles (chrobotics): http://www.chrobotics.com/library/understanding-euler-angles
Gimbal Lock visualization #1: https://youtu.be/zc8b2Jo7mno?t=67
Gimbal Lock visualization #2: https://youtu.be/Mm8tzzfy1Uw?t=285
Quaternions (chrobotics): http://www.chrobotics.com/library/understanding-quaternions
Quaternions (Wikipedia): https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
Quaternions explanation with interactive video: https://eater.net/quaternions/

Figures
[1]: https://en.wikipedia.org/wiki/Inertial_frame_of_reference#Newton's_inertial_frame_of_reference
[2]: http://www.chrobotics.com/library/understanding-euler-angles
[3]: http://www.chrobotics.com/library/understanding-quaternions

https://preview.redd.it/70mc32hunrd51.png?width=700&format=png&auto=webp&s=cdaf9778b650c0edac10a0cf38519b9a7e919f17
Since ancient times, the world of art and the world of science have been interconnected, and often artists have also been scientists. The universal human desire to understand and to describe the reality around us makes this connection persist through times, and science discoveries are still providing a rich source of inspiration to contemporary artists.
The intersection between art and science can be seen in the works of the famous Italian artist Cesare Catania, also known as a sculptor and a civil engineer. The majority of his work involves creating large paintings that represent a mix of tradition, like oil painting on canvas, and innovative materials and technologies. His work draws inspiration from such scientific areas, as mathematics, anatomy and music.
One of Cesare Catania’s most famous artworks, a contemporary tapestry “The Three Men” (2016–17), demonstrates how science achievements, even the historical ones that we used to study at school, are integrated in our everyday life.
Cesare Catania’s “The Three Men” is an aesthetically engaging painting that wants to embrace all the knowledge from the ancient Greeks to today. This artwork, full of fundamental mathematical references, should be considered as a tribute to three greatest scientists whose discoveries changed for ever our perception of the reality.
The left side of the painting is dedicated to the 18th century scientist Amedeo Avogadro and his revolutionary discoveries in the macroscopic world of «grams» (or atoms). The central part represents Pythagoras who was able to overcome the barriers between the straight and curved world, defining the PI quantity. The man on the right side of the painting is Heisenberg, the cornerstone of quantum mechanics that defines the boundaries of the knowledge of conjugate physical quantities. Heisenberg becomes for Cesare Catania the man that marks the break regarding the previous scholars of classical mechanics.
Finally, the tribute to the ‘900 scientist is given by Cesare Catania with representation of a stylized microscope and the Greek letter “delta”, commonly used in physics and mathematics to express the changes in quantity. Using numbers, geometrical shapes, graphs and letters he creates a fascinating image that clearly expresses his passion for the pictorial and engineering arts.
This large contemporary art painting, as well as other 40 masterpieces by Cesare Catania, are currently being exhibited on his official website as part of the virtual exhibition “Retrospective 2020”: an updated selection that brings into a new focus the topics elaborated by Cesare Catania in the last few years.

I have made a post somewhat similar to this before but I've since collected a lot more information about these topics and so I believe warrants yet another post.
Myth #1: Women weren't allowed to work.
Most documentation from the past heavily implies that people attempted to avoid working as much as possible. Even so, many women did work (mostly because they had no choice but to) and there has never been a law restricting women from working.
"Though creative research has produced evidence of women working at male-dominated occupations in the eighteenth century and before, there is undoubtedly more documentation out there, in newspapers, diaries, legal proceedings, and prints. What is more compelling is the lack of documentation that women were not allowed to work. Although religious practices and social norms might have restricted certain activities in some parts of the world, there were no laws prohibiting women from working a trade."
At the time, only members of a trade guild (associations of producers that trained craft people, maintained control over production, regulated competition and prices) were allowed to sell their goods or practice their skill within the city. Non-guild members were prohibited from selling competitive products. An aspiring master would have to pass through the career chain from apprentice to journeyman before they could be elected to become a master tradesman. They would then have to produce a sum of money and a masterpiece before they could actually join the guild. If the masterpiece was not accepted by the masters, they were not allowed to join the guild, possibly remaining a journeyman for the rest of their life.
Women were not barred from any of the trade guilds in Britain at least as far back as the 1400s. Apprentice forms for guildwork had spaces left blank to insert 'he' or 'she', though girls who chose this traditional route to mastery of a trade (which was extremely long and arduous) were usually orphaned and had low prospects for marriage, thus they were going to need to work.
Married women, though, had the privilege of circumventing the long, onerous, arduous process of trade certification as they could receive the master tradesman status from their husbands and thereby continue his work, even take on apprentices to train (which suggests that the social assumption was that wives would learn enough about their husband's business and trade to be able to at least continue to oversee it after he died). Lots of widows continued to run their husbands' businesses long after their death.
The Worshipful Company of Blacksmiths in London lists sixty-five "brethren" and two "sistren" in its 1434 charter.
A 1770 publication called The Tradesman's True Guide or a Universal Directory for the Towns of Birmingham, Wolverhampton, Walsal, Dudley and the manufacturing village in the neighborhood of Birmingham carried exhaustive lists of tradesmen and tradeswomen alphabetically by name and by trade, and there are women listed in every trade from butcher to wire drawer.
https://www.history.org/foundation/journal/spring04/women.cfm
Really, very little I've seen shows that women were maliciously "kept out" from trades or professions that they wanted to enter into. It may have been a social norm that most women did not seek out careers as they do now, but looking at the histories of the first women to enter into medicine, law, etc, show that they hardly faced any resistance in doing so.
In 1847, Elizabeth Blackwell became the first woman to attend medical school in the US and was the first woman on the Medical Register of the General Medical Council. She sent applications for admission to twelve of the most promising institutions in the northern states, and her application was accepted by the medical department of Geneva University. Feminist revisionist history "explains" that she was only admitted to the college by the evil men as a joke, but as will be amply demonstrated this is the opposite of what actually happened.
In fact, the student body sent her a letter to welcome her:

At a meeting of the entire medical class of Geneva Medical College, held this day, October 20, 1847, the following resolutions were unanimously adopted:–
Resolved – That one of the radical principles of a Republican Government is the universal education of both sexes; that to every branch of scientific education the door should be open equally to all; that the application of Elizabeth Blackwell to become a member of our class meets our entire approbation; and in extending our unanimous invitation we pledge ourselves that no conduct of ours shall cause her to regret her attendance at this institution.

Resolved – That a copy of these proceedings be signed by the chairman and transmitted to Elizabeth Blackwell.

She describes this letter as one of her most valued possessions, and in her autobiography she states that the men of the college supported her 100% and beyond.
"The behaviour of the medical class during the two years that I was with them was admirable. It was that of true Christian gentlemen. I learned later that some of them had been inclined to think my application for admission a hoax, perpetrated at their expense by a rival college. But when the bona-fide student actually appeared they gave her a manly welcome, and fulfilled to the letter the promise contained in their invitation."
She also notes that the general public also had a favourable view of a woman entering into medicine.
"The admission of a woman for the first time to a complete medical education and full equality in the privileges and the responsibilities of the profession produced a widespread effect in America. The public press very generally recorded the event, and expressed a favourable opinion of it. Even in Europe some notice of it was taken, and 'Punch' showed his cordial appreciation by his amusing but friendly verses."
http://digital.library.upenn.edu/women/blackwell/pioneepioneer.html?fbclid=IwAR09FZZlnVMVha0PNLYnQQDSnptVDEoGxokEf10QHPCGgxZMly5HyfDlgQI#III
There are also other examples of women entering into "male" fields and encountering little resistance. There's the example of Arabella Mansfield, the first female lawyer in the United States. Although by Iowa law the bar exam was restricted to "males over 21," Arabella Mansfield took the exam in 1869, passing it with high scores. Mansfield challenged the state law excluding her, and the Court ruled that women may not be denied the right to practice law in Iowa, admitting Mansfield to the bar.
Or Ada Lovelace, who was encouraged in her mathematical ability from a young age by her parents and received the best of tutors to increase her ability. She worked with Charles Babbage on his Analytical Machine, though they both "fell into historical obscurity, and the Analytical Engine was unknown to builders of electro-mechanical and electronic computing machines in the 1930s and 1940s when they began their work, resulting in the need to re-invent many of the architectural innovations Babbage had proposed."
Or Amelia Earhart, who was flown "like a sack of potatoes" by pilot Wilmer Stultz across the Atlantic in her 1928 transatlantic flight, and immediately became a celebrity, wrote a book about the experience, was used to endorse products, and dubbed 'Lady Lindy'. There is a plaque that marks where the plane landed in Wales. All the fervour over it was solely because she was a woman who crossed the Atlantic regardless of the fact that she didn't do any flying at all in her 1928 flight (She did, however, go on to complete a transatlantic solo flight later on in 1932).
https://en.m.wikipedia.org/wiki/Amelia_Earhart
Or, for a very long-ago example of this, Murasaki Shikibu from Japan around the year 1000, who listened at the door while her brother received lessons in Chinese writing and through that learned it herself. She was far from reviled for being a woman who knew Chinese and was commissioned by the Empress to write the Tale of Genji, a classic work of Japanese literature often called the world's first novel.
Oh, the o p p r e s s i o n .
Myth #2: Marriage was an institution created by men to oppress women. Men were disproportionately given privilege and power and women were treated like his property in marriage.
Again this is a bastardisation of the past. Feminism has to stop treating men as the enemy and as the source of all women's (and also men's) problems. They need to realise that things weren't always as they are now, and realise that it was not Patriarchy or men that restricted the choices of both genders and forced them into strictly defined roles, but that these roles came about because it was the best way of arranging societies in the past - a past that was much harsher and more unforgiving than it is now. A past in which there was more work to do than there were hours in a day. A past where men spent most of their short lives working like mules to support themselves and their families and women spent most of their short lives incapacitated by childbirth and child-rearing. A past in which we sent children into coal mines that ruined their lungs and into factories which lost them limbs. Things sucked, and the reason it was bearable at all back then was because of the gender roles.
Men were not the ones who relegated women to the domestic sphere or kept them dependent on men. Biology did. A woman at that time could not simply strike out on her own and do whatever she wanted. For most of history we had no reliable birth control, no safe abortion, no maternity leave, or social security, and most public sphere work back then involved hard labour. And if you were a woman back then who wanted to be able to have sex, then you would have had to plan your life around the idea that you would be getting pregnant, giving birth, and breastfeeding for most of your adult life - and that would have meant you could not be a reliable enough worker to support even yourself, let alone a child, and it would inherently have made you dependent on other people to help you. Until the last 100 years or so, most women could not have lived a life of children and public sphere work, independent of a man (and not all can today, realistically).
Even with support, most women in poor and middle class families had to work because that was the only way to make enough money to live.
So in a time when single motherhood was a one-way trip to the gutter, marriage was a practical arrangement. The function of marriage was not to oppress women, but to positively assign paternity inasmuch as possible, and then obligate the men so assigned to their children and the mothers of their children. For life.
The taboo on sex outside of marriage (which was applied to men no less than women) made it difficult for an unscrupulous man to cheat the system. These rules surrounding marriage also mitigated the problem of paternity uncertainty, which maximised men's willingness to invest in their kids. "I know these are my kids, so I will work hard to feed and shelter them." And the reason why we kept lifelong monogamous marriage for so long is because it increased the resources which would have been available to offspring, providing a healthier environment for children to grow up in.
So when a man married a woman, he was legally obligated to feed, clothe and shelter her and any children produced in the marriage to the best of his ability. Unlike what feminists seem to think, he could not sell her. He could not return her for his money back. He could not drop her at the local midden heap. He could not destroy her the way he could his actual chattels. He could not trade her for a better one. He could not legally neglect her.
Just to make things crystal clear, here is a summary list of entitlements that women enjoyed in marriage (and after marriage) during the old "patriarchal" system of the past.

When a woman married, she had to hand over her property to her husband's care (again, NOT because she herself was property, but because he was administrator of the family), but he OWED her not just a living, but the best living he was financially capable of providing for her. A wife was entitled to be maintained by her husband.
Though women did not have the right to enter into contracts in their own name in marriage, women had the privilege of the Law of Agency, giving them the legal right to purchase goods on their husbands' credit as their agent. If the wife racked up debts that the husband couldn't pay, she was immune from liability for the debts she racked up - that liability fell to the husband instead.
Women had the entitlement of being protected from prosecution for any number of crimes if they could prove their husbands were aware of said crimes. In which case, he would be prosecuted in her stead--not just held AS responsible as she was, but held solely responsible for her actions. This allowed married women to displace accountability for a large number of offences onto their husbands. "She's out of control and does what she wants, regardless of my wishes," was not a valid legal defence for those men.
Women were exempt from paying taxes, and there was an entire female underground economy of barter and trade that was not subject to taxation or government interference. Even when married women gained the right to hold property within marriage, it was their husbands who were responsible for the taxes owing on it, and it was their husbands who were held 100% financially liable for providing all necessaries to the family, including the wife, even if she made more money than he did.
After divorce, men were held fully financially responsible under the law for the ex-wife and kids. Divorced women were entitled to be supported by their ex-husbands to a level commensurate with the husband's for potentially the rest of their lives unless she remarried.

https://ora.ox.ac.uk/objects/uuid:2e88e3f6-b270-4228-b930-9237c00e739f/download_file?file_format=application/pdf&safe_filename=Item.pdf&type_of_work=Journal%20article
That doesn't sound at all like women were treated like property to me.
I'm sorry, but if I have property, I have no legal duty of care toward the actual property I own, nor any obligation to commit to own it until death. I can smash my property on the ground and treat it however I want and it would be my right to do so. Men could not do any of that to the women they were married to. They had obligations towards them to provide for them - even after divorce, and to protect them - even from the consequences of their own actions. Wives had no such obligation towards their husbands - they were entitled to be the ones who were supported. Because the purpose of marriage in the past was not to oppress women and make them the property of men, but to ensure that women and their children could be provided for and protected, and to place the responsibility for that provision and protection on the man even at cost to his own wellbeing, health, and even life. This has been the case since time immemorial.
"Husbands, love your wives, just as Christ loved the church and gave himself up for her."
Ephesians 5.
https://www.biblegateway.com/passage/?search=Ephesians+5%3A22-33&version=NIV
In fact, many of the supposed "privileges" men had in marriage in the past were to help them carry out their obligations and to prevent them from becoming wholly onerous.
If a man had an obligation to be accountable for provision for his wife and children and to maintain family finances, it stood to reason that he also should have the entitlement of having control over any assets of his marriage, including those his wife brought into the marriage, because he was the one who had a responsibility to keep the entire family afloat and to increase their holdings.
If a man was responsible for the protection of his wife and children to the point where he could be legally required to stand between his wife and the law within numerous contexts, and be punished in her stead (whether she contracted debts he couldn't pay or committed a crime he would have to answer for), it makes sense why he would be considered head of household and why his family would need to obey him as well as abide by the restrictions he placed on them.
If, upon divorce, a man would be held solely responsible for the maintenance of his wife and child, it makes sense why he would get custody of the child that he was supporting, as well as ownership of assets so he could satisfy that obligation.
We gave men these rights and privileges because they were the tools that he needed to satisfy his responsibilities. And in that way there was a balance. "Women "got less" because their obligations and accountability were less, their responsibility was less. Men "got more" because the buck stopped with them, whether that buck consisted of a sack of coins or his blood."
Myth #3: Men were allowed to beat their wives with impunity.
In the past men were allowed to use "mild correction" on their wives under the law (as he was socially and legally responsible for the actions of his wife to the extent that he could even be held accountable for the crimes and offences she committed), but he could not beat her black and blue. Men had restrictions on what corporal punishment they could use on their wives.
In Colonial America, when men went beyond what was legally acceptable, they received punishment through the pillory. “The pillory was employed for treason, sedition, arson, blasphemy, witchcraft, perjury, wife beating, cheating, forgery, coin clipping, dice cogging, slandering, conjuring, fortune-telling, and drunkenness, among other offences. On several occasions, onlookers pelted the pilloried prisoner so enthusiastically with heavy missiles that death resulted.”
https://www.history.org/foundation/journal/spring03/branks.cfm
The whipping post also was a favourite form of punishment for wife-beating. Many examples of vigilante justice, sometimes lethal, enacted against wifebeaters can be found in newspaper clippings from the 1800s and onward, with the full endorsement of society.
https://unapologeticallyantifeminist.video.blog/2019/05/21/correcting-notions-about-domestic-violence-in-history/
We can assume that this was practiced in Europe as well. Likewise, Americans practiced shivaree and Europeans practiced charivari as a way to enable social censure of wife-beaters. Charivari was a historical folk custom expressing public disapproval of personal behaviour, where a mock parade was staged through a community accompanied by a discordant mock serenade. Sometimes the crowd would carry an effigy of the targeted man to a substitute punishment, e.g. burning. Sometimes the man who physically abused his wife would be abused by the community.
They would chant:
Old Abram Higback has been paying his good woman; But he neither paid her for what or for why, But he up with his fist and blacked her eye.
Now all ye old women, and old women kind, Get together, and be in a mind; Collar him, and take him to the shit-house, And shove him over head.
Now if that does not mend his manners, The skin of his arse must go to the tanners; And if that does not mend his manners, Take him and hang him on a nail in Hell.
And if the nail happens to crack, Down with your flaps, and at him piss.
However, if a husband was beaten by his wife - the husband, in contrast, was also the subject of the charivari for essentially allowing it to happen. In France about 1400, husbands beaten by their wives were “paraded on an ass, face to tail.” In England, a mural in Montacute House (constructed about 1598) shows a wife beating her husband with a shoe and then a crowd parading the husband on a cowlstaff.
Samuel Pepys recorded in his diary, 10 June 1667: “in the afternoon took boat and down to Greenwich, where I find the stairs full of people, there being a great riding there to-day for a man, the constable of the town, whose wife beat him.”
A Frenchman who traveled in England reported in 1698: "I have sometimes met in the streets of London a woman carrying a figure of straw representing a man, crown’d with very ample horns, preceded by a drum, and followed by a mob, making a most grating noise with tongs, grid-irons, frying-pans, and sauce-pans. I asked what was the meaning of all this; they told me that a woman had given her husband a sound beating, for accusing her of making him a cuckold, and that upon such occasions some kind neighbour of the poor innocent injur’d creature generally performed this ceremony."
https://www.purplemotes.net/2013/01/27/charivari-sex-inequality/
Hmmm. So while a man could give his wife mild correction under the law, if he went too far with wife-beating the community would frown down upon him and punish him with the pillory, whippings and charivari. THERE WAS NO SUCH EXTREME SOCIAL OR LEGAL CENSURE FOR A WIFE WHO BEAT HER HUSBAND. IF SHE BEAT HIM, HE WAS THE ONE WHO WAS PUNISHED.
Myth #4: Husbands were owed sex with their wives. Marital rape was not illegal, and marriage as an institution was based around rape of women.
The idea at the time was that there could not be such a thing as marital rape because as a married person you are owed sex from your spouse, i.e. conjugal rights. This applied to both men and women. People who believe that conjugal rights "allowed husbands to rape wives" and not the other way around haven't actually done any research.
It was based on the idea that both parties had a right to sex in the relationship, and this was exercisable in law by each partner in a marriage.
The Bible said:
"It is good for a man not to touch a woman. 2 Nevertheless, because of sexual immorality, let each man have his own wife, and let each woman have her own husband. 3 Let the husband render to his wife the affection due her, and likewise also the wife to her husband. 4 The wife does not have authority over her own body, but the husband does. And likewise the husband does not have authority over his own body, but the wife does. 5 Do not deprive one another except with consent for a time, that you may give yourselves to fasting and prayer; and come together again so that Satan does not tempt you because of your lack of self-control."
https://www.biblegateway.com/passage/?search=1+Corinthians+7&version=NKJV
Do note the specific passage where it says "The wife does not have authority over her own body, but the husband does. And likewise the husband does not have authority over his own body, but the wife does."
And here's a quote from an article detailing the history of sex and marriage.
"A hundred years ago or so, if you took the statute books literally, the only people entitled to have sex with each other were people who were legally married. All other sex was against the law. Fornication (sex with unmarried people) was illegal; so was adultery (sex with a married person—but not the person you yourself were married to). Cohabitation—living with someone in a sexual relationship—was against the law, except for people who were legally married. Enforcement was, shall we say, something less than perfect. At times, it was almost nonexistent. And the rules were riddled with exceptions. Sodomy, on the other hand, was quite a serious crime; and the full weight of the law sometimes came down heavily on people who had gay or lesbian sex."
"Married people were not only entitled to have sex; they were, in a way, required to have sex. Of course, nobody checked on whether John and Mary were having sex every night, or once a week, or once a month, or never. But if either John or Mary did not perform at all, for whatever reason, the frustrated partner had grounds for divorce; or even perhaps an annulment.
https://verdict.justia.com/2017/01/24/sex-necessary-legally-speaking
Husbands faced divorce if they were impotent and unable to consummate the marriage, though charges were usually made years after the wedding day. There was a similar charge of frigidity for wives, but it seems that wives charging their husbands for impotency was far more common. Husbands had to show an erection to a court audience and sometimes attempt to perform sex with their wives as well.
Source 1 Source 2
Being incapable of performing sex for your wife could merit corporal punishment. One medieval husband wrote about his unhappy marriage and his impotence in a book called The Lamentations of Little Matheus.
"My wife wants it, but I can’t. She petitions for her right. I say no. I just can’t pay."
"Even given his sexual incapacity, Matheolus was subject to corporal punishment:
"Acting as her own advocate, Petra {Matheolus’s wife} puts forward the law that if a shriveled purse {scrotum} can’t pay because it’s empty, under statute recompense for that injury is corporal punishment."
https://www.purplemotes.net/2015/05/03/matheolus-church-wife/
So what does this show? It shows that both parties had an obligation to provide the other partner with sex during the marriage, and could not deprive their spouse of it. If it was an issue, it was not an issue that solely affected women.
Myth #5: "Men had the vote and women did not. Women's suffrage was a matter of women as a united group fighting for their rights against the evil men that kept it from them. Suffragettes were champions of equality who got women the vote. Etc, etc."
Really, I'm very tired of hearing this one.
In Britain, the right to vote was not only restricted by gender, but also by class. Property owners had the vote, the vast majority of men did not. In 1830, just 3% of the population had the right to vote.
During this time, there were a few groups who were fighting for the vote to be extended to them, and one of these groups were the Chartists, who were active from 1830 to 1868. They were fighting for broad electoral reforms including:

a vote for all men (over 21)
the secret ballot
no property qualification to become an MP
payment for MPs
electoral districts of equal size
annual elections for Parliament.

They presented three huge demonstrations to Parliament, and their requests were denied every time. The restrictions were relaxed but not abolished. Benjamin Disraeli said when introducing the Second Reform Act of 1867: "We do not live – and I trust it will never be the fate of this country to live – under a democracy. The propositions which I am going to make tonight certainly have no tendency in that direction."
There were also the suffragists and the suffragettes, who were fighting for the vote to be extended to women. Suffragists emerged in the mid-19th century and believed in peaceful, constitutional campaign methods. In the early 20th century, after the suffragists failed to make significant progress, a new generation of activists emerged. These women became known as the suffragettes, and they were willing to take direct, militant action for the cause.
However, a huge amount of women disagreed with women's suffrage. So many women objected to the franchise being imposed on them that in the mere 18 months leading up to a 1910 parliamentary debate on women's suffrage, anti-suffragettes had managed to collect 300,000 signatures from women who objected to the franchise being imposed on them. Meanwhile, in the 16 years leading up to that debate, suffragettes only managed to collect 193,000 signatures of women who wanted the vote.
A segment from the transcript of the debate:
"Still, for what it is worth, one finds that from 1890–1906 only 193,000 women signed Petitions to this House in favour of female suffrage, and that during the last eighteen months, a period in which a strong anti-suffrage association has been in existence, 300,000 women, some of them eminent women, have signed petitions against it, and to the effect that these proposals shall not be forced upon them."
And...
"A rather interesting illustration was furnished by an inquiry which was instituted by "The Sheffield Independent," a newspaper which I think itself is sympathetic with these proposals. This inquiry was addressed to the women householders of Sheffield. Twenty-three thousand papers were sent out, and of the replies 9,000 were in favour of woman suffrage and 14,000 were against it. The representatives of this paper reported that in many cases their emissaries were chased with violence from the houses by the female inhabitants under the impression that they were collecting statistical matter as the emissaries of the suffragist party."
It may be difficult to understand why so many women would reject the vote so vehemently, but this was an era where full citizenship rights like the right to vote were granted only to those with citizenship responsibilities like conscription (and sometimes not even then). In that historical context, the vote was actually viewed not just as a right, but also an obligation. So it was never really about men in power shutting women out of the franchise. It was about government having to decide whether to impose a responsibility on women that women as a whole clearly did not want.
https://api.parliament.uk/historic-hansard/commons/1910/jul/11/parliamentary-franchise-women-bill
Regardless of the lack of support for women's suffrage among women, suffragettes became extremely militant in their activism. In 1903 Emmeline Pankhurst founded the Women's Social and Political Union (WSPU) with her daughters Christabel and Sylvia, and the WSPU carried out a nationwide bombing and arson campaign. Between 1912 and 1915, hundreds of bombs were left on trains, in theatres, post offices, churches, even outside the Bank of England; while arson attacks on timber yards, railway stations and private houses inflicted an untold amount of damage.
https://www.historytoday.com/history-matters/sanitising-suffragettes
Instead of being beneficial in supporting the vote for women, it was disastrous. Lloyd George said: "The action of the militants is ruinous. The feeling amongst the sympathisers of the cause in the House is one of panic. I am frankly not very hopeful of success if these tactics are persisted in."
Terrorism aside, Emmeline Pankhurst was also resistant to acquiring the vote for working class men, realising that this would be much tougher than winning the vote for "respectable" AKA upper-class women, and underlying the resistance appears to have been her contempt for working-class men. According to Sean Lang in his book on Parliamentary reform: "The Pankhursts became stridently anti-male, ruthlessly dropping even the most loyal of their male supporters from the WSPU, and claiming, as Christabel did in her 1913 book The Great Scourge, that men were “little more than carriers of venereal disease”."
https://conservativewoman.co.uk/men-won-votes-women-not-suffragettes/
When WW1 came around in 1914, suffragettes also threw their support behind the White Feather campaign with the aim of shaming men into enlisting. Those young women went in for the kill, striking men at the very heart of their masculine identities, the bestowing of a feather telling them, "If you don't go off to be maimed or die, you are no longer a man in the eyes of some brassy chit you've never even met before and will probably never see again." And many men went, because a woman's censure - ANY woman's censure - had the power to drive them straight into the teeth of death.
By mid 1915, it was clear that not enough men were signing up to replace the fatalities on the front line, and the Military Service Act was introduced in January 1916. By March 1916 compulsory conscription for all single men aged 18-41 was in place. In May 1916, the Act was extended to include conscription for married men.
http://www.eastsussexww1.org.uk/attitudes-towards-conscription/index.html
It was only at that point that the British government considered extending the vote to all men, as most of the propertyless, working class men that were conscripted into war during WW1 did not have the right to vote, and this was finally starting to be seen as the injustice that it was. In the trenches class divisions were eroded to the extent that class no longer had any real meaning and every man was the equal of each other in terms of sacrifice and service to one's country.
Men got universal suffrage in 1918 with the Representation of the People Act, which enfranchised more than five million men over the age of 21 without regard to property or class. Women of all classes were piggybacked onto the Act, with an age restriction temporarily in place to prevent women from becoming a supermajority voting bloc, given that about 1 million British men had died in the war. Regardless, this brought more than eight million women over 30 into the electorate. And just ten years later, the 1928 Act (the act which gave women electoral equality) gave the vote to women at age 21, which added another five million women to the electorate.
https://www.telegraph.co.uk/men/thinking-man/has-everyone-forgotten-male-suffrage/
Introducing the 1918 Bill, the Home Secretary George Cave said: "War by all classes of our countrymen has brought us nearer together, has opened men's eyes, and removed misunderstandings on all sides. It has made it, I think, impossible that ever again, at all events in the lifetime of the present generation, there should be a revival of the old class feeling which was responsible for so much, and, among other things, for the exclusion for a period, of so many of our population from the class of electors. I think I need say no more to justify this extension of the franchise."
The Act was achieved not by the suffragettes’ terrorist tactics, but by fighting of men in WW1, when thousands of men from their many different backgrounds came together to fight the common enemy. This finally achieved what the Chartists, the Levellers and the peasants before them had struggled for in terms of eroding barriers of class. Without the sacrifice of men who endured the horror of trench warfare in WW1, class barriers would have persisted and at most the vote would have been extended only to elite, wealthy, propertied women.
In a later study of attitudes to conscription in 1939, one woman responded: "In the last war [(WW1)] I never told one man that he ought to go. I tried to put myself in their place. If I’d been them I wouldn’t have gone. When I saw them going through the streets – all the young men – I used to think that they were being driven like cattle to the slaughter house. And they were, weren’t they? Fodder for the guns."
The common idea that all men had the vote, and that women did not, and that women's suffrage was a fight of women who were united against men working as an oppressive force to keep them down is an affront to these men who died in a literal war that secured universal suffrage for everyone (many of which, in fact, did not have the vote). It also ignores and erases the will and sentiment of many, many, many anti-suffragette women who actively rejected the vote and protested vehemently against it being imposed on them.
Furthermore, the view that suffragettes were a valiant, fair-minded, equality-oriented group who got women the vote and whose activism merely amounted to firecrackers in barrels is an affront to any right-thinking person. The real violence and extremism of the suffragettes is barely, if at all, recognised by the general public, and this is partly due to attempts by suffragettes to sanitise their history.
"Laura E. Nym Mayhall, in her work on the militant suffrage movement, identified the importance of "a small group of former suffragettes" in the 1920s and 1930s, who created a highly stylised story of the WSPU and the history of suffrage in England, which emphasised "women’s martyrdom and passivity". The group set about compiling the documents, memoirs and memorabilia that now form the basis of the Suffragette Fellowship Collection held by the Museum of London."
https://www.historytoday.com/history-matters/sanitising-suffragettes
And here we are now, in crazy world. According to the dominant narrative, all men always had the vote since the dawn of recorded human history, and they refused to give it to women because penis. The demonstrations of the Chartists, and the dead bodies of these men that were sacrificed to the unpopular idea of male suffrage (and who got votes for women in the process), have gone down the memory hole. The terroristic actions of the suffragettes, engaged in for the benefit of wealthy and privileged women only and for which they were barely punished, are the noblest of acts for the noblest of causes. And of course, the only reason anyone could EVER have opposed them was because of misogyny.
edit: wording
edit 2: Added some extracts from w1g2's comment.

Conjugate - MATH

conjugate mathematics define

conjugate mathematics define - win

The conjugate of -9 - √5 is -9 + √5, but the conjugate of -√5 - 9 is -√5 + 9

Explaining the Weird Physics