The key insight for me on this topic also came from 3Blue1Brown, but in a different video: it’s that e^x is NOT best thought of as repeated multiplication, but instead as the function exp(x) = 1 + x + x^2/2 + X^3/6 + x^4/24 + …
After being relieved of the burden of that misconception, I was finally able to understand the role of complex numbers in the Fourier Transform.
People told me that over and over but it didn’t help — because it didn’t make sense why repeated multiplication would cause rotation!
Later in that video, we see a visualization of the rotation. I was able to grasp how the exp function could yield rotation where I’d never been able to understand why e*e*e*e… did.
The best explanation I've seen, which 3blue1brown probably avoided because it requires calculus, is that exp is the unique function whose value at 0 is 1 and whose derivative is itself. Then by the chain rule the derivative of exp(ix) is i*exp(ix), meaning that the direction of motion is perpendicular to the current position vector. Which naturally leads to circular motion because you're always staying the same distance from the origin.
With that definition it's easy to derive the Taylor series expansion (every derivative at 0 is 1), and you can think of Euler's formula not as telling you how to evaluate exp(ix) (it's already perfectly well defined), but as an introduction of cos and sin as shorthand for its real and imaginary parts.
I think you need to study the Euler equation to understand the relationship between goniometric functions and exponential functions when calculating with complex numbers. One easy to remember formula that connects those functions.
Indeed. The title of that 3Blue1Brown video is, “What is Euler’s Formula actually saying?”
That “easy to remember” formula had been presented to me countless times. It only made sense once I stopped thinking about it in terms of repeated multiplication.
The repeated "many folding", which would be better visualized as tendition, exponentiation ("tendaddition") pattern also breaks with fractions & at the most basic negative numbers: https://www.youtube.com/watch?v=mvmuCPvRoWQ&t=922s
Also: https://news.ycombinator.com/item?id=28524792 "log base" would be better named untendaddition similarly division - does not necessary "separate" - untendition & unaddition - does not "draw under". Etymologos to use "given" symbols over the datum names.
Further, the pattern matched "grouping" definitions (https://news.ycombinator.com/item?id=25278021) names have a better correspondence to the Greek scheme: diation, triation, tetration... while the "Group theory" definitions scheme lending to various geometries (hypercomplex: complex::hyperbolic, split-complex::elliptic, dual::parabolic) would match some other naming scheme.
That would not even kill all humans. It would possibly destroy all of civilization though. But even then, the coming generations would have artefacts to study and be inspired by.
That's an interesting thought. ACOUP did a blog post a while back [1] outlining the very specific circumstances that led to the Industrial Revolution and arguing that it's hard to imagine another way it could have happened. I suppose a rebuild would have to find a completely new route. Perhaps fields of Don Quixote esque wind mills connected to giant led-acid batteries rather than coal! With a couple hundred years gap between Armageddon and a substantial human population making progress, at least there would hopefully be decent timber reserves to work through the early phases.
> For more than 100 years, Machinery's Handbook has been the most popular reference work in metalworking, design, engineering and manufacturing facilities, and technical schools and colleges throughout the world. It is universally acknowledged as an extraordinarily authoritative, comprehensive, and practical tool, providing its users with the most fundamental and essential aspects of sophisticated manufacturing practice.
I think the level of scientific knowledge we have accumulated would be a tremendous short cut.
Even without books, the myths and obvious ruins of previous technological success would be a huge cultural guidepost for recovery.
Perhaps we would go through a 1000 year energy poor "dark age", and recovered populations wouldn't peak as high as ours, but I would expect that to be the worst case if an awareness of our history was not lost.
And maybe 10,000 years, after a complete cultural breakdown to hunter gathering with little functional memory of the past.
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Given rewrites of first versions are often much improved for having seen previous failure modes, it would be interesting to pop into the future of a recreated civilization and see how they might do things better!
Even having this written knowledge available to us it might still be impossible to recreate the technologies; so much knowledge exists in an active state only.
Like the F1 engines on Saturn V. We couldn't just "recreate them" because all of the know-how was lost.
Written info is useless without the right sort of culture where the necessary experimentation and problem solving can occur to work out those undocumented details. That is all that matters.
However, that is also true today and the effects are visible through the variations in technological progress of different populations currently on earth.
If you can jump right to electric power, then hydro power can give you all the power you need to get civilization going again. I don't think that tech knowledge would be likely lost if people survive at all.
My guess is that all the required knowledge, including sufficient people, are squirreled away in the bottom of Cheyenne Mountain and probably a few similar locations around the world.
I guess it depends on how long before the collapse happens, but I think there's enough accessible fossil fuels left to sustain a less than 10-figure human population for under a century while it figures out what else to do. They'd probably have to live with far more local supply and distribution networks, but global communication should still be fine.
I think your reply is a little too strident. It’s not a “fake idea”. It’s just someone’s idea. I hadn’t heard of it but I googled it upon reading this thread and it seems to be the idea that people tend to collectively remember things inaccurately in the exact same way. That phenomenon alone seems pretty plausible and not that dumb or fake, right?
But what isn’t plausible is the part you left out that’s talked about in the video? That some people explain the Mandela effect through pseudoscientific bullshit like alternative timelines merging. You also seem to conflate the ridiculousness of what some people collectively remember with the Mandela effect concept as a whole. It’s a bit like saying hypnosis as an idea is dumb because people do dumb things when hypnotised? I also don’t understand what is deeply offensive about people misremembering that Mandela died in prison? They’re not saying (or even remembering) that he was a bad person, probably the opposite, they’re just ignorant of history.
But having said all that I am perplexed by the OP comment: “I can’t help but think the Mandela effect is an example”. An example of what? Cracks in spacetime? Because yeah that’s dumb, though probably a joke.
Chess GMs at Magnus and Hans’ level wouldn't even need to cheat on every move and certainly not on any opening where something like the first 20 best moves are all memorised. These guys would only need the engine to tell with them one or two moves per game to pretty much always win.
Why did Magnus start the game and then resign? Maybe he thought it would make a statement. Maybe he needed to technically show up to the game to avoid a penalty.
I’ve exercised probably 90% of days for the last ten years. Days when I don’t are just rest days or travel days. To be honest, I think it’s best just to make it a habit like brushing your teeth. Working out is just what you do, and it’s a weird day if you don’t. Being highly motivated is kind of fleeting and just boring routine is more important.
I worked out in the morning because "It's Monday, and Monday is a weekday, so that's why" is far more sustainable in the long term. Sometimes less thinking is better.
This said, for many people with jobs/kids/other commitments, regular workout is more of an exercise in scheduling than willpower. For that, the only advice I could offer is to use the mornings, and be done before anyone else in the house realises.
Perhaps I’m looking into the data too much, but it doesn’t surprise me that 0 is often the neutral white, black or grey, and that, because we use the base 10 number system, the pattern is the same for 10. These colours and numbers are both “clean”. It’s the same for 2: the most popular colours are blue, yellow, red then green, and that pattern is repeated for 20. Of course, this “pattern” is already broken since it should be 1 with the same pattern as 10 and 2 wirh the same pattern as 20. It would be interesting to see if patterns do indeed emerge with more data. I would certainly expect 100, 1000, 1,000,000 to often be white, black or grey.
For myself, I’ve always associated colours with days of the week. I don’t know if others have this, but it’s always felt more “obvious” what colours some days of the week are compared to others, like the feeling is stronger or the colour is more brilliantly defined. But I do wonder to what extent it may have just been me as a child tricking myself into thinking I’m special, and I’ve just forever embedded those colour memories, because it’s really not like if someone tells me “Wednesday” I automatically experience a powerful crimson red; I have to kind of ask myself: “what colour is Wednesday?” to experience it in my mind’s eye.
100%. I found the books “how to have impossible conversations” and “crucial conversations” helpful for me, but in a nutshell they’re basically “say things in a way that protect the other side’s ego”
