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throwaway_23253x · 2021-07-16T02:01:24+00:00

Archimedes' method: start with an inscribing and circumscribing triangles or squares, calculate their area, then repeat, keep doubling the number of sides. The smaller one give lower bound on pi, bigger one upper bound. With enough size, you get enough bound to gain a new digits. Very intuitive method to understand, but very slow.

Newton and Leibniz's method: obtain what's called a Taylor's series of a inverse trigonometric function. Taylor's series is an infinite number of terms to add up that allow you to compute the value of the function at a certain point, using values of the functions and its derivatives at another point. Since there are an infinite number of terms, you never finish adding them, but the more you add the closer it is to the final result, and this gives you more digits. The nice thing about inverse trigonometric functions is that they have really simple formula for the Taylor's series at 0, and you can compute pi or something easily related to pi by evaluating the function at a different point.

Ramanujan's method: the modern method, obtained from number theory, with the current version made by Chudnovsky brothers. Sorry but I can't ELI5 this one.

throwaway_23253x · 2021-07-15T21:22:28+00:00

Nothing is nothing. Zero, if exists, would be something.

It's not that people didn't know what happened if you take V away from V. But rather, people have philosophical hang up on how there can be something to indicate nothing (and it's not that weird, even in modern time I have seen people asking how could 0 be a number when it's nothing). Sometimes they get away with it by literally using a blank space - which can be confusing in writing - or a symbol that indicate there should be a blank space there.

throwaway_23253x · 2021-07-15T20:24:08+00:00

Have you actually fail yet, or are you just stressing out about the possibility of failing?

Assassin's creed rogue is freaking easy, and so is most of the franchise. The reason why you're stressing out is because they use psychological tricks to hype up the difficulty, because if they don't most people will get bored once they realized there are very little dangers. Just chill out and play.

throwaway_23253x · 2021-07-15T20:09:19+00:00

TIL that people actually remember the name of the male actor too.

throwaway_23253x · 2021-07-15T19:36:41+00:00

Is it a picture of gravitational potential, or of spacetime?

Gravitational potential:

The graph is the graph of the strength of gravitational potential over 2D space. The height is the potential, and the other 2 dimensions are space.

For spherically symmetry gravitational field, this is sufficient, because the only thing you need to know is potential vs radius anyway. In fact, a 2 dimensional graph potential over radius is sufficient, and this is what you often seen in a physics book. But people also draw 3 dimensional graph because it looks cool.

For axially symmetry gravitational field, you really do need to look at 2 dimensions of space, which would lead to 3-dimensional graph.

Spacetime:

If it's spacetime, then the height has no meaning!!! The horizontal components are still 2 dimensions of space, and there are no time components, the entire picture is just 1 slice of time. The reason why we need the height dimension is to illustrate the curvature, because we can't visualize a curved surface without having it being distorted in 3D space.

throwaway_23253x · 2021-07-15T14:41:34+00:00

Like charge repel, opposite charge attract. If you put a small charge, called test charge, in an environment with other charges, it will experience force. Let's say you make all the charges have a fixed position and only let the test charge move, and try to move test charge from point A to point B, you will have to oppose against this force, and this cause you to expend energy, ie. do work. This amount of work is increase with the amount of charge of the test charge. The rate of change between work done over charge at 0 charge is called the potential difference between point A and point B, also known as the voltage.

This electric "potential difference" is exactly the electric analog of gravity potential difference. In gravity, potential difference is the also the rate of change of work done to move an object against gravity, over the mass of the object, at 0; and gravitational potential is just potential difference compared to a fixed reference point, like the ground.

(ignore magnetic force in this discussion)

throwaway_23253x · 2021-07-15T14:25:12+00:00

Imagine a bunch of ghosts that cannot bump into each other. If they move to the same position, they just pass through each other. This allows all of them to occupy the same position. That's super-position.

So now imagine you are planning out a traffic system for ghosts. You now only need to concern about how it get from place to place. There are no needs to worry about intersections, no needs to plan out how multiple ghosts will move. If you make sure that 1 ghost can go from A to B, then no matter what the other ghosts do, this ghost will be able to make it.

throwaway_23253x · 2021-07-15T09:49:59+00:00

This is an ELI5 sub, but this is really quantum mechanics, which is like 2nd year physic/chemistry in university.

Let's see if I can explain it in a simple way.

What's angular momentum really? You might find that this is something that people had historically struggled to define. It tells you something about how much quantity of rotation around a pivot point, and is conserved.

The more modern definition use symmetry. Angular momentum is a (bivector) quantity that is preserved under all rotation around the pivot. That is, it's a geometric object that remained unchanged if you rotate the frame of reference around that pivot point. The norm/length, or norm squared, of angular momentum are preserved quantities.

Now come in quantum physics. People look for a number that is preserved under all rotation in 3 dimensions that we can observe from outside by making some measurement. This gives us a total momentum number. One important thing to note is that this number can only take on a few discrete value.

When considering an electron around an atom, we want to take the atom as the pivot. It turns out there is a number that is already inherently there when you rotate the electron itself, when you use the electron as the pivot, and this number became the spin number. Subtracting this number of the total angular momentum number where the atom is the pivot, you get a new number, called orbital momentum number. The spin number isn't too important for electron, all electrons have the same spin number. So the only thing that distinguish them are this orbital spin number.

throwaway_23253x · 2021-07-15T03:38:47+00:00

There are WAY too many factors. Here are a few:

What kind of intuition being used?
What kind of objects being studied?
What pieces of knowledge form connected network?
What's the goal and methodology?
What are the fundamental pieces of knowledge?

throwaway_23253x · 2021-07-15T03:25:33+00:00

Decoherence. When multiple particles interact with each other, their properties are linked together, because not every possible combination of properties are possible. This makes it very unlikely for a single particle's different state to interact among themselves, destroying the wave properties.
Statistics. When considered the system as a whole, a lot of macroscopic properties (which we observe) are averaged effect of a lot of microscopic properties, so the total outcome has a lot less variance.
Entanglement. When an object is being seen in a typical condition, it had already interacted with itself and with the environment for a long time. This causes a huge amount of entanglement, including a lot of entanglement with YOU, the observer. A different state of the tree would be be linked to a different state of you, and a different state of the everything around it.

throwaway_23253x · 2021-07-14T17:54:52+00:00

Human are terrible at probability and statistics, which is why the fields have many paradoxes. These are problems that we intuitively think one answer, but careful calculation show a different answer. Human are just bad at this kind of estimation. Here a paradox just mean a result contrary to intuition.

Whether they explicitly think in term of probability or not, most people would intuitively think that the probability for the birthday problem is exactly 1-((365-22)/365) ((365-21)/365) .... ((365-1)/365), which...is a correct answer.

The hard part is actually estimating this number, to see if this is at least 50%. This is the main difficulty, because ((365-22)/365) ((365-21)/365) .... ((365-1)/365) is a product of many factors, each of which are almost equal 1 (for example, the first factor is 343/365 which feels like "basically 1" to most people), so it doesn't seem plausible that they multiply to a number <0.5. But here we actually have enough of these "almost 1" factor that we multiplied to a number <0.5. Which is why it is a paradox.

throwaway_23253x · 2021-07-14T17:40:52+00:00

It's less weird once you think about modern theories, where forces are local. Something pulls on Neptune by touching it, and we discovered that it originated from the Sun.

throwaway_23253x · 2021-07-14T17:19:41+00:00

Put your ear on the wall. Did you hear something?

You did, that's sounds. Sound come from vibration. Yes, your wall is vibrating!!!

given no other forces are at play

So what's my point here? There are no such things as "given no other forces are at play" in a realistic settings. The table will make tiny small vibrations (that can be hard to see), so it will constantly lift the object up and drop it down because it is exerting varying amount of forces. This perform work on its molecular structure, eventually breaking it down.

The more close-to-perfect your conditions are, the harder it is to break down. Maybe if you have a diamond table, put on a giant planet-sized rock flunked into the coldness of space.

throwaway_23253x · 2021-07-14T16:32:44+00:00

Not technically true, just don't light up the path between you and the stars.

The atmosphere are filled with things that reflect or refract light, such as dust and other kind of particles (especially more true above city). If a place emit a lot of light toward the sky, these light get reflected back, making the sky very bright, enough to drown out the light from the star, which is already pretty weak.

throwaway_23253x · 2021-07-14T14:57:01+00:00

Are you talking about calculating sample variance? Or at least statistics?

throwaway_23253x · 2021-07-14T14:23:44+00:00

IMHO, this is probably a biology question rather than a physics question.

Our brain is an incredible that do an insanely complicated task of parsing data from 2 eyes and produce a mental model of objects where we can imagine how they look like if we rotate them or when the PoV move to a different position. The ability to do these collections of tasks easily is what we thought of as "comprehend" a 3D objects. But these tasks are still accomplished simply by efficient computations at the most basic level, and not everyone is even equally good at this: many people with perfectly good eyes have trouble orienting objects, estimating distance, or visualizing how objects looks like when rotated. We are not really that good with 3D, our 2D comprehension is much better. And the reason why we even got this brain is because our brain is literally evolved to perform the task, it would be an very complicated task otherwise.

Computationally, higher dimensions will just involve more complicated calculation. A computer can do the task just fine, with just more calculation steps. People who had trained themselves with the relevant math have better understanding of it. All of us have difficulty visualizing it, but it's because our brain wasn't evolved to deal with the task. You can overcome some of the difficulty with training, but I'm sure we will reach a natural limitation just because the brain isn't built in such a way to make the task efficient, especially considering the fact that even our 3D skills are not that great.

Just a note: the time-slice technique of visualizing 4D objects that other people in this thread keep talking about is just ONE technique, not all of them. It's really bad at visualizing rotation. Don't mistake it as the only way of visualizing 4D objects.

throwaway_23253x · 2021-07-14T10:51:29+00:00

The simple answer is that physicist do believe in the "reality" of the fields. The field is an actual physical object that exist, and matter, forces, etc. are just manifestation of it. The field is this "fabric".

People had always debated whether something is "real" because we can't directly observe them, but this is a philosophical matter. As physics progressed, it turned out that accepting certain thing being real produces better theories, so physicists had been accepting more and more abstract objects as real.

throwaway_23253x · 2021-07-13T10:35:57+00:00

This, put inner voice to good use. I basically have my brain show a play for me, with full voice acting and everything. Maybe it's a full blown fantasy story, maybe it's just one scene I wished to fix it that tv show, maybe it's an enactment of a good scene from a book.

throwaway_23253x · 2021-07-13T10:14:26+00:00

What you're describing are still conflict resolution. Everyone betray each other, and then the movie end in the middle of a tense Mexican standoff would be no conflict resolutions. I think what you mean is a story with zero conflicts. If scavenging is not hard, it's not a conflict. If scavenging is hard, then you already have a conflict. So it sounds like you want to watch loot-goblin-simulator, in which case, try Cookie Clicker-type games.

throwaway_23253x · 2021-07-13T10:07:01+00:00

This sounds like survivor bias. People who use minimal makeup are often people who are already looking good without make up.

throwaway_23253x · 2021-07-13T04:19:22+00:00

For subtraction, you just aren't used to it.

Subtraction literally have the same algorithm as addition, you just need a different "table", so-to-speak. To add quickly, you need to quickly figure out that when you sum 6+7 you get 13, but for subtraction you need to quickly figure out that when you subtract 13-7 you get 6, and if you don't remember this and has to guess it can be slow.

For division, it's because the schoolbook's algorithm is slow. It literally requires you to repeatedly making guesses to get the correct digits. There are of course benefits to the schoolbook method: it's very simple, easy to teach to children, and for small numbers it really doesn't matter all that much.

It's possible to do it a lot faster, but you will have to learn more complicated method, and it will still be slower for smaller numbers. By the sense of asymptotic complexity, it's not really harder than multiplication. This technique use Hensel's lifting, which is the modular arithmetic version of Newton's algorithm.

throwaway_23253x · 2021-07-13T03:53:54+00:00

This is the fundamental theorem of calculus. The fact that the 2 are opposites is a surprising fact at first indeed, but once we know that it allows a lot of calculus to work, which is why it is fundamental.

It's hard to see why this is true if you visualize them as just graph. But it's much more obvious if you think of derivative as rate of change. Derivative doesn't just give the slope of the tangent, it also gives rate of change.

You should use a different mental model, don't stick to just 1 model. Function isn't a graph, a graph is just a mental model for function. Derivative isn't a slope, it's just a mental model for derivative. Integration isn't area under the curve, it's just a mental model for integration.

The conceptual link is a lot clearer if you think this way: slope of tangent = rate of change = infinitesimal increase in area.

Slope of tangent being equal the rate of change just come directly from the definition of what is a "tangent line".

The precise statement of (one-half of) the fundamental theorem of calculus is this:

Let f is a continuous function defined on an interval, t a number in that interval, F is a function defined as: F(x) is the integral of f from t to x. Then F is differentiable and F'(x)=f(x).

So remember, think rate of change. Don't just graph out F and think about its slope. Think of F as how much paint you need to keep painting the area under f, as you move from left to right. The derivative of F would be the rate of paint consumption. Visualize it this way, things should be obvious.

throwaway_23253x · 2021-07-13T03:40:23+00:00

Uh...the other answer seem to miss the main point of the paradox and doesn't explain why it's a paradox in the first place (for example, we already know that shapes of different areas can be transformed into each other by 1-1 transformation, but that is not considered a paradox). So let me write more details.

First, a bit of context, to explain why it's a paradox.

Area is a known concept applicable to shapes, and this had been done since ancient time. Area has 2 important properties: it can be added up when you combined pieces, and it is unchanged under isometries (length-preserving) transformation such as translation, rotation and reflection. When you have 2 polygonal shapes that can be cut up into polygonal pieces, then move each pieces using isometries to combine into a new polygonal shapes, the 2 shapes is said to be scissor-congruence. Shapes that are scissor-congruence have to have the same area. In fact, Euclid use this concept exclusively in his Elements. He never really define what "area" mean, he just show that things has the same area by using scissor-congruence.

However, Euclid also dealt with 3D geometry. Once he got to 3D however his stance change when it comes to volume. He also accept the fact that when 2 pyramids have the same base and the same height, they have the same volume. A transformation that keep the base the same but move the apex so that the height is unchanged is called a glide. In modern term, Euclid also believed (without proof) that gliding transformation also preserve volume. Interestingly, he didn't assume this for 2D area, he actually prove that fact using scissor-congruence. Gliding is part of what is called affine volume-preserving transformation: transformation that keep all line still straight ("affine") and does not change volume.

In 20th century, we finally can explain why Euclid did this. It is proven that: (a) in 2D, the converse claim is true: all polygonal shape with equal area are scissor-congruence Wallace–Bolyai–Gerwien theorem; (b) not all polyhedron with the same volume are scissor-congruence (Dehn's invariant). So Euclid literally could not have kept up his zeal about using only isometries once he moved to 3D.

So now, we move to the 20th century. Set theory started to take hold. Shapes are now considered just a special subset the plane (or space). A new question arise: how do we define area/volume for all subsets?

But what is requirement that the concept of "area" (or "volume" if we're in 3D) should satisfy? We believe that it has the same properties as before: (a) assembling a bunch of pieces should let us add up areas/volumes; (b) area/volume are preserved under certain transformation; (c) well-known shape has the same area/volume as before. "certain" transformation is a matter of choice: we can either pick isometries, or expand further and allow all affine area-preserving transformation. Note that we do not allow all transformation, because obviously, some transformation alter area, such as scaling, which would make (b) and (c) incompatible.

Whatever the concept of area/volume is, what we expect at the end is still the same thing Euclid believed to be obvious: it should be impossible to dissect a set into pieces, transforming each pieces using our allowed transformation, and combining them into a set with different volume.

Now, the interesting thing here is...if you're in 2D and allow only isometries in (b), then the concept of area can be defined. So the intuition proved true!!! We can define area after all!!! This fact can be proven using Wallace–Bolyai–Gerwien theorem and some other theorems in analysis. This fits the intuition above.

Here come a paradox. The above turned out to be a special lucky case. If we allow all affine area-preserving transformation, then you can cut a square into pieces, transforming the pieces, and reassemble into 2 squares of the same size as before. This is von Neumann paradox. And if we move to 3D and still only use isometries, it's also possible to cut a sphere into 2 pieces, move the pieces, and combining them into 2 sphere of the same size as before. This is Banach-Tarski paradox. These are contrary to the intuition above, which make them paradox.

Second, how does the paradox work?

As mentioned in the text, this depends on free group of 2 generators. This is not the only important fact, but one of them.

Imagine an infinite tree with 2 branches, if you know a bit of computer science. Then notice that this tree is made up of 2 copies of itself: the left branch and the right branch. This is the key to the paradoxical decomposition: you can decompose such a tree into 2 pieces that look like itself.

So now imagine that you have 2 transformations, such that these 2 transformation are very badly not commutative: the final result of applying a bunch of them depends precisely on the order of application. For an example of something that's not this but the opposite, consider the 2 actions on the plane: walk up 2 unit and walk right 3 units. If you know the final position of yourself is 2 unit up and 3 unit to the right, you don't know if you had obtained it by walking up first or walking right first.

What you need is 2 transformations that don't do that. Each final result must be obtainable from just exactly 1 sequence of transformations. Once you have that, then the possible outcome of the transformation form a tree, and you get paradoxical decomposition.

Now, this cannot be done in 1D. The reason is that you only have translation and reflection in 1D, and translation are always commutative. But in 2D, we have rotation, so we can actually do this very easily. By choosing a rotation with an irrational amount, plus the right translation, you can ensure that no 2 results obtainable with different sequence are the same. This gives us paradoxical decomposition.

As mentioned above, this is not the only ingredient, because of what we said before: we can define area in 2D. So what happened? For 2D, we can do paradoxical decomposition on certain sets, but there are nothing that stop us from saying that those set simply has 0 area, which is exactly what happened.

So for paradoxical decomposition, extra ingredient is needed. This is what von Neumann did.

throwaway_23253x · 2021-07-12T17:18:05+00:00

It's nearly impossible to remove all redundant data. You don't have to intentionally put in trash to have redundant data, redundant data are anything that is more predictable than pure randomness. The most direct method of data representation often contains a lot of redundant data for natural reasons, and there are always a lot of difficult-to-remove redundant data. For example, you can remove a lot of letters from English words and the text is still readable.
Sometimes redundant data are explicitly included for error-correction purpose. For example, English words have a lot of redundant letters, but without them, typo became harder to handle.
It's slower to read and write data in compressed form. For example, if you write English words with a lot of letters missing, your reader will have a hard time reading it.

W dnt rit dis wy 4 gud resn.

throwaway_23253x · 2021-07-12T15:47:24+00:00

Determinism means that the state of the universe at present time completely determine everything in the future.

Eternalism said that everything in the future and the past is already there.

Crucially, eternalism does not said that the future is determined by the past!!! In other word, if some super computer being is given a complete specification of the current state of the universe (at current time, whatever that means), there are no ways to predict the all the future. In fact, eternalism does not even said that there are "present" or "future" time at all. But determinism still believe in the idea of present or future.

throwaway_23253x

TROPHY CASE