metacogitans

My personal take on Kaluza-Klein theory is to drop the "Cylindrical 5th Dimension" as an explanation for charge and instead view the intrinsic spinning/rotation of curved spacetime produced by a particle as being responsible for its charge. While trying to visualize the intrinsic spinning/rotation of spacetime curvature around the particle geometrically, it seems the spinning/rotating curvature would have to have a polarity; that is, the spinning/rotation is faster along the "equator" of the spacetime curvature (can't think of what else to call it) and slower towards its poles -- assuming its spherical. But if it has a polarity, then what's to stop it from flipping upside down and switching its charge? Then I thought that 'switching charge' is exactly what happens during beta decay with up & down quarks.. The 'polarity' of a quark flipping would change the orientation of the quarks, causing the spacetime curvature throughout the nucleus to change shape producing a 'ripple' in spacetime, and that 'ripple' might actually be the electron and anti-neutrino emitted by a neutron in the case of beta-minus decay. What do you think? Edit: Another thought - could W bosons actually just be anomalies in the Riemannian geometry of spacetime causing a quark's axis to wobble and possibly causing its poles to flip, causing beta decay? Here's all of what I typed up earlier: - 'Charge' can be conceptualized geometrically as the intrinsic curvature a particle produces in surrounding spacetime. Similar charges produce repulsion between each other similar to how two gears spinning in the same direction kick off each other. 'Rotation' or 'spin' might be what first pops in our head as a way to visualize charge - however, this is difficult to visualize in 3 geometrical dimensions. Assuming the curvature is spherical more or less, we might imagine that the spin/rotation of the curvature has a polarity, and the fluidic nature of spacetime helps to keep that polarity stable. Different hadrons could then be thought of as the various configurations between particles (quarks in this case) causing spacetime curvature with intrinsic spins/rotation and polarity. - The interactions of the Weak Force could be thought of as taking place when the polarity of curvature produced by a quark is challenged; in the case of beta decay, the polarity of a quark's curvature is challenged successfully, thereby changing the orientation of its curvature relative to the other quarks, switching its quark flavor. The loose spacetime curvature resulting from the switch in polarity is what becomes the electron and electron anti-neutrino emitted by the neutron during beta-minus decay. A scary possible implication of this is that in a distant part of the universe, due to manifold geometry, the polarity of up and down quarks in the atomic nucleus could be in a completely different direction - their down quarks could be spinning in the same direction as our up quarks; magically teleporting to this hypothetical distant part of the universe would cause an eruption of weak force interactions, destroying whatever was teleported by turning the matter making it up into something completely different. - For the geometrical explanation to return to gravity, the spacetime curvature produced by particles collectively should incidentally produce gravity at macroscopic scales. Now someone just has to draw what quarks curving spacetime in a hadron actually looks like, translate that drawing into tensor calculus with the curvature from electrons also, put 3.6 x 10^51 of those hadrons with about the same number of electrons in a simulated computer model which would be roughly the same number of hadrons that the Earth has, let it run, and see if stuff gravitates or not. However I don't think there even any supercomputers capable of running that; someone could try to estimate an average of curvature for a large number of particles, but that's what solutions for General Relativity already do and so that'd be back to square one. Edit: Looking up whether or not a supercomputer could run the model or not, apparently there's a supercomputer capable of 10,000,000,000 Million Instructions Per Second, if the calculations performed are done in appropriate fragments instead of trying to emulate real-time, then the simulation should be able to run the equivalent of a few minutes in real-time if the computer is left to run it for a day or two. That's all you'd need to show whether it produced gravity or not - and if there's no inconsistencies between the simulation and real life measurements at any of the scales, it could be called a working unified theory. I don't know if the orientation of spacetime curvature produced by quarks in a hadron can even be described mathematically though; I'm pretty sure it's dynamic, and the quarks carry a unique frequency between each other that would affect the outcome of the simulation but is too much information to be calculated for each hadron in the simulation (I don't even think a computer could accurately simulate the dynamic frequency of even one hadron. It might unfortunately be non-mathematical, or at least uncomputable.

On the back cover of either Penguin Classic's "Schopenhauer: Essays and Aphorisms" or Dover Philosophical Classics "Schopenhauer: On the Basis of Morality" (I can't remember which), it states that Albert Einstein was inspired by Schopenhauer. I also remember seeing that stated on the internet a few times. Seeing as how Einstein is perhaps the most misquoted person of all time, I was wondering if anyone knew of any actual basis for this? Usually the best way is to just search the body of all letters, interviews, and writings of Albert Einstein for the specific term in question; does anyone know where to find a compilation of all things actually said by Einstein?

Alright. We know that gravity follows the inverse-square law; the intensity of the force falls off exponentially with distance, with the intensity approaching infinity as the distance between two massive objects decreases. In General Relativity however, this must be expressed in terms of warped spacetime. As the distance between two massive objects decreases, spacetime dilation approaches infinity - meaning there will always be 'more space' between two massive objects when traveling towards each other. Hence, dn-1 = dn • (1/w) Where d is the distance traveled by an object in an increment of time when traveling in the direction of another massive object. For bosons however, which are emitted from particles, the distance between the boson and the particle emitting the boson starts as infinitesimal, or what we could practically consider to be 0. The distance d which the boson travels away from its source particle during an increment of time could be expressed by: dn+1 = dn • w Since the speed which a photon travels is the same in all reference frames, the distance traveled would be the same whether traveling towards or away from a massive particle; so bosons are thereby able to 'close the distance' between particles and be absorbed, accelerating the particle. That was my 'correction' of the original post (please let me know what you think of it ); I'm still going to try showing how to derive it mathematically from other equations - it's going to take me a while though; I'm working on it.

I'll have to correct my post then. I was trying to give a simple geometrical description of spacetime dilation for individual particles is all. What are your thoughts on the math?

Well, whatever that grid would be called is what I meant. I guess I used the wrong terminology. What should I say then? 'Where the speed of light is measured as a constant locally'?

The velocity of the photon remains constant while the grid plotting its trajectory is warped due to gravity. (posted by user James S Saint on ilovephilosophy.com)

The basis for dn-1 = dn • (1/w) is Special Relativity and also the Pauli Exclusion Principle. With the Pauli Exclusion Principle, it could roughly be said that 'two fermions can not simultaneously occupy the same exact location'. With special relativity, an object can not travel faster than the 'light' (or in a broader sense, any radiation or force-carrying boson) emitted by the object - as if an object were to 'collide' with another object, the 'collision' would have to be mediated by a force-carrying boson; no matter how close together two particles are, radiation traveling between them will still be observed as traveling at the speed of light. If the speed of light is always locally constant, 'locally' would be a frame of reference wouldn't it? The speed of light is not objectively constant according to general relativity. A ray of light that passes through the center of a gravity-well would seem to have traveled a shorter distance than a ray of light passing alongside the gravity well for the same amount of time. If we clocked a ray of light traveling through the center of a massive body's gravity well to find the body's diameter, the circumference of the massive body would be shorter than what we would expect when taking the measured diameter multiplied by pi. Well, it was derived philosophically from the mathematical concepts of special relativity. Deriving it strictly mathematically will take me a while, but I can certainly try.

Going to draw pictures.. One minute please. Light's local reference frame. The dilation of spacetime described by general relativity It varies depending on the particle. Different particles have different warp constants, especially composite particles which could only be properly described with tensor calculus and not a simple constant. Bosons are the only particle capable of 'meeting' another particle, thereby accelerating them. As bosons mediate all interactions, fermions themselves must be intrinsically 'separated' from other fermions.

From light's frame of reference, let d be the distance light travels during an increment of time in the direction of a massive particle: dn+1 = dn • w Where w is the 'warp constant' of a particle, and w > 1 For massive non-boson particles: dn-1 = dn • (1/w) For massive non-boson particles, dn-1 > 0 What this means is that two massive particles can not 'collide', and there will always be spacetime between them.

Those are just the parameters of the thought experiment... The calculations for just one variant of the setup won't provide insight for all variants; we would have to make calculations for numerous variants of the setup to begin deducing the overall tendency, and we would realize it would have been easier just to look at the mathematical concepts themselves to figure out the overall tendency. With extreme conditions, the iron ball will either immediately get stuck against a coil or be too far away for the electromagnets to cause the ball to roll. But when the conditions are right for the iron ball to switch between rolling away from the direction of the electromagnets and towards the directions of the electromagnets, The majority of variants in the setup would favor attraction - that's my assumption anyways. I guess it only applies to magnetic repulsion, not all repulsion. No in the part of my post where you are quoting me here. I am also considering the scale of individual particles. But what does that have to do with my 'argument' anyways? If we were able to conduct an experiment where we measure the gravitational attraction between two clouds of electrons in space, there would be evidence if we measured there to be no gravitational attraction occurring between them. Since a value for mass is determined by a number of particles, then there is a relationship between charge and mass. If the overall charge of an object increases or decreases, then the number of particles in the object will have changed - so there has to be some relationship. In the text at the root of the chain of quotes, I said 'charged particles produce an electromagnetic field'. Since you quoted wikipedia, I get to as well: Static E and M fields and static EM fields "When an EM field (see electromagnetic tensor) is not varying in time, it may be seen as a purely electrical field or a purely magnetic field, or a mixture of both. However the general case of a static EM field with both electric and magnetic components present, is the case that appears to most observers. Observers who see only an electric or magnetic field component of a static EM field, have the other (electric or magnetic) component suppressed, due to the special case of the immobile state of the charges that produce the EM field in that case. In such cases the other component becomes manifest in other observer frames. A consequence of this, is that any case that seems to consist of a "pure" static electric or magnetic field, can be converted to an EM field, with both E and M components present, by simply moving the observer into a frame of reference which is moving with regard to the frame in which only the “pure” electric or magnetic field appears. That is, a pure static electric field will show the familiar magnetic field associated with a current, in any frame of reference where the charge moves. Likewise, any new motion of a charge in a region that seemed previously to contain only a magnetic field, will show that that the space now contains an electric field as well, which will be found to produces an additional Lorentz force upon the moving charge." https://en.wikipedia.org/wiki/Electromagnetic_field#Static_E_and_M_fields_and_static_EM_fields

Couldn't the refraction of light as it passed through the sun's corona explain the results of the 1919 solar eclipse experiment? Light bends as it passes through a medium; The sun's corona stretches far into space, with increasing density towards the sun's surface. Even past the corona, the sun's gravitational pull still attracts dust particles in orbit around it.

Here is the setup; the end of the coil used as input switches at periodic intervals, switching north and south. In the next image, the black dot represents the imaginary 'point of contact' with the center of the iron ball, and the black arrow is the general direction the ball would travel during that instant of the wave affecting it. This direction is always 'to the side', and hence into another electromagnet's 'lane' in the chamber where that electromagnet is most influential. Other things worth noting: - The iron ball realistically can't have an exact north and south, and the distribution of polarity throughout the ball is uneven - which means it would likely 'wobble' when moving through the chamber. - If the intervals when the electromagnet's north/south switches were timed right, the north/south end of the ball would move into a different position right as the electromagnets would switch leading to repulsion, and this may cause repulsion to be 'favored' for several intervals, but I would still suspect it to ultimately favor attraction. - As the iron ball gets closer to the electromagnets, the stronger their effect on the momentum of the iron ball, while the further away the iron ball gets, the weaker the effect. Attraction might also be favored for this reason, as it would gain more momentum from the magnets the closer it is. Although, this could actually lead to repulsion ultimately being dominant, as if the ball rolls too far away, the magnets wouldn't be strong enough to get it to roll back towards them. - Variables like the size of the coils and the ball, the weight of the ball, the strength of each electromagnet, the number of coils, how frequently the electromagnets switch between north and south, how far the ball is placed etc., would likely lead to varied results, with some favoring repulsion. What I think would be most interesting though would be variations where the iron ball is moving around in the chamber for some time before either attraction or repulsion becomes apparently dominant. The paradox is if attraction is favored more often than repulsion, but I suppose that isn't really a paradox, as magnetism fundamentally arises from the geometric orientation of charged particles, and attraction/repulsion are dependent variables. So attraction is not being 'favored', it is just north/south re-positioning themselves. But if the combined electromagnetic activity of the entire planet is composed of discrete particles with either a positive/negative charge, then considering a particle in space, wouldn't the particle's tendency be to move out of line with similar charged particles and in line with oppositely charged particles, resulting in attraction? Since repulsion always moves something "to the side" slightly, wouldn't repulsion always move a particle into line with an oppositely charged particle where attraction is occurring? In fact, wouldn't particles do this anyways following a path of least resistance? Also, what do you mean by cube law and not square law? Isn't it square-cube law and inverse-square law? And we only know gravity is proportional to mass from observations made at a macroscopic scale. But is there actually any evidence that charge doesn't effect gravity? Also, we could equivocate units of mass to include charge by looking at an object's molar mass and getting a number of particles. I don't know. I started wondering about it after thinking that the results of the 1919 solar eclipse experiment could be explained as the diffraction of light as it passes through the sun's corona. The more I think about it, there are other problems with 'warped space-time' as an explanation for gravity, like it would lead us to assume that objects would fall faster if they're at a higher temperature, since the 'movement' from the thermal energy of the particles would be faster-farther through warped space-time in the direction of a massive body. Since it is all a product of the same fundamental force 'electromagnetism', aren't all electric fields and all magnetic fields technically 'electromagnetic' fields as well? I understand the addition of the word 'magnetic' when the magnetic effect is something different is confusing, but what else am I supposed to call it when talking about the electromagnetic force in general? It's just a technical inconsistency of language, like If someone were talking about a new model for Kaluza-Klein theory with some variation to the equations involving quantum mechanics, you could still say he is talking about "Kaluza-Klein theory" even though Kaluza had nothing to do with the addition of quantum mechanics to the theory.

Is this secretly a challenge to see if I can be consistent in giving an explanation for all of electromagnetism without saying something that I need to be corrected on? - Charged particles produce an electromagnetic field. Similar charges repel; opposite charges attract. - Composite particles usually have a 'polarity' resulting from the geometric orientation of their constituents. - Polarity results in magnetic attraction/repulsion between composite particles. - In ferromagnetic compounds, molecules share polarity to a significant extent that repulsion/attraction can be demonstrated at our macroscopic scale. - By running current through a coiled conductive wire, a magnetic field is created with a north/south - The north/south can be reversed by switching the end used as input for the current. Now you answer my question ( ) ... I just know I'm going to have to make drawings in microsoft paint to describe my 'experiment' in the original post. Here, I'm going to try doing it in just text first: - + - + - > repulsive electromagnetic effect here < (-|+) <-- iron ball + - + - ^electromagnets Because the iron ball can't be precisely centered with the negative electromagnet, the repulsion repositions the iron ball for attraction. The catch is, the electromagnets switch back and forth between positive and negative. My point however, or 'hypothesis' if you will, is that no matter how quickly the electromagnets are switching between positive and negative, or the strength of the electromagnets, the iron ball will favor attraction towards the row of electromagnets -- or, if the electromagnets are too weak, it will not move at all. Repulsion will never be favored - hence, the paradox: attraction is dominant when immediate intuition would tell us they balance out. Or am I wrong? I am posting this thread as a question; I'd really like to know if/why that's not true. There is a reason for this thought experiment; I'm wondering if electromagnetism can be used to explain gravity. Take the Earth for example, every proton and neutron is by itself already polar, having multiple constituents with different charges (up and down quarks). The sum of all particles making up the Earth then could be thought of as a large number of electromagnets -- as they move around, they could be compared to the shifting electromagnets in my example. A particle near Earth, influenced by all the particles making up the Earth which act as little electromagnets, would have a net result of electromagnetic attraction - as attraction is favored.

Say you have an iron ball set in a chamber, with a row of electromagnets on one side of the chamber wall; Each electromagnet switches back and forth between emitting a positive or negative electromagnetic field into the chamber, and the row is setup so that each electromagnet emits an oppositely charged electromagnetic field into the chamber from the electromagnet(s) next to it. Now, here is the paradox: Will the iron ball ultimately roll towards or away from the row of electromagnets? My intuition would suggest that the electromagnets would cancel each other out -- or that imperfections in the construction of the setup would lead to a favored outcome. But, my hypothesis is that the majority of the time, attraction would be favored, as a repulsive effect can only result in momentum to the side of where the repulsive field is strongest in the chamber and into the path of an attractive field instead; as the waves from the source of a repulsive field spread as they propagate, with small imperfections in the setup causing the iron ball to be repulsed to one side or the other instead of staying centered with the source of the field.

I hope I can maybe shed some light on Special Relativity; there are two seemingly-contradicting principles that are both true: 1. No matter how fast you are travelling in one direction, if you turn on a flashlight, light will still seem to be travelling away from you at the speed of light c 2. Light travels at the same speed locally. Or, the speed of light is constant in a vacuum. Since both of those principles have to be true, somethings gotta give in the example in #1: time dilates to compensate for you still observing light travelling at the same speed as usual when you're travelling at a relativistic speed - everything else in the universe would be happening faster than usual. If you had a clock with you, it would be behind everyone else's clock when you got back. Because you'll always observe light travelling away from you at the speed of light, you can never travel as fast as the speed of light; and if you try to accelerate up to 300,000km/s, time dilation prevents you from ever reaching it. The Lorentz Factor describes this mathematically:

Here we go, I found one of the good ones, heh check it out: I got this for an answer now (I cut off some of the decimals, didn't need 5 dozen of them): 198,193,345,664,504,055,673.88288950657 Here's the link to that calculator if anyone else wants to check out: http://web2.0calc.com/ I really like how it professionally organizes your equations automatically up above the calculator; and your answer can be copied to your clipboard out of the entry bar (copy/paste is a HUGE bonus imo). Thank you very much, that clarified everything I needed to know I was looking for acceleration; I forgot to say meters per second "squared". I'll check out orbital speed next, thanks for the link

Where G is the Gravitational Constant 6.674×10−11 N⋅m2/kg2 m1 and m2 are the masses of the two objects in kilograms and r^2 is the distance between them squared I tried a test problem using the Earth and the Moon, and had some questions. My values were: moon:7.34767309 × 10^22 kilograms Earth:5.972 × 10^24 kg distance between their centers of mass: 384,400,000 meters I ended up getting this for an answer using the calculator built into the Google search engine: 2.9285715e+37 which didn't look right, so I did it again but to do it faster I only used a few of the decimal places in the numbers, and got 7.4778663e+28 I've had problems with this calculator before when trying to get it to use scientific notation; I did use parantheses correctly where appropriate and correct order of operations; it's just a poorly coded calculator. Can anyone recommend me an online calculator or calculator software that works well, and will display many digits of decimals? Being able to use the symbols / * ^ ( ) and able to have many sets of parantheses inside each other. All of that is absolutely Essential for a calculator to avoid having to enter the equation one piece at a time. I have a Texas Instruments Graphing Calculator but it's out of batteries I've wanted calculator software on my computers for a long time though, might as well find a decent one now. Anyways, a few other questions: If the number 7.4778663e+28 is close to correct for the gravitational force between the Earth and the Moon, I'm guessing that I have to divide it by the kilograms then of whichever celestial object to get a value in meters per second? What do I do exactly with "N * m^2 / kg^2"? is r^2 in the equation really the distance between centers of mass squared? Why is the symbol r being used? A few sites said it was something different (one said it was the distance from the surface of the objects, and few pictures on google image search said 'radius of the moon' (which would almost be correct for the moon's surface, but is still wrong).

What an embarrassing thread. Don't drink and math. I suppose it would have been a lot better if I didn't butcher it every time numbers tried coming out of my mouth

Glancing over the thread I didn't see anyone talking about anything illegal; I saw people mentioning the existence of criminal hackers, but that's not against the rules, it's going to come up just by being a part of the definition of what's being talked about. It'd be like trying to tell someone about the history of AC and DC current and not being able to bring up Edison electrocuting an Elephant; you need to let people tell the whole story, it's just good book-keeping.

Being able to have 15 different programs open and 20 different windows on your internet browser are important; the laptop I'm using right now can't handle it. Put it this way though; my desktop computer which me and my friend built on a budget of $700 in 2009 is still able to play games. The RAM is dated though; I only have 2 gigs of DDR3 (back when I built it though, DDR3 just came out and my friend was selling it to me as being as fast as 4 gigs of DDR2). It's rocking an AMD Phenom II quad core 3.2GHz, which is still a great processor today, really. Also, a 1gb I-don't-know-what video card that seems decent I guess. Cheapest MOBO on newegg at the time, had a few quirks over the years but still working more or less. Cheapest case on newegg, its basically in pieces at the moment. Should have invested in a better case. Gone through about 5-6 power supplies until my friend pointe out to me that I was buying a scam-brand power supply over and over which is why they kept blowing up and sparks were shooting out the back of my computer; bought a decent brand power supply then for a few extra bucks and works just fine. All in all, my desktop has given me so much more than its worth over the years and I love it. And I'm the type of person who never usually spends that much money on something. Remember, time is money; the extra money you spend building a decent rig will pay for itself in the time you save not having to wait for programs to load or having stuff lag/freeze up all the time. If you have a little extra money to spend on building a decent computer, I highly recommend it. Also, speaking of 'time is money', windows 8 has the most time-consuming GUI to navigate; it actually acts as a deterrent from getting stuff done on your computer too because you feel aversion to the clunky interface and only use it as much as is necessary. You can of course go to the windows 8 program store or whatever it is and install user-made add ons reverting the UI to a more traditional windows UI, but doing that takes - you guessed it - time, about an hour I'm guessing just to get all the touch-screen crap features out of the UI and make it tolerable again. Then an update probably breaks it every month and you have to do it again; you can't win - they wanted to put a nail in the whole 'freeware, net neutrality, actually having control over your computer' thing and they're being real jack-offs about it

Sorry for another thread about Zitterbewegung. I was wondering if Zitterbewegung, being the hypothetical trembling motion of elementary particles, could explain the uncertainty of an electron's location resulting in its orbital (with the electron orbital viewed simply as an electron's probable location around an atomic nucleus). My idea is basically that residual amounts of external radiation from the rest of the universe affect the electron, causing trembling motion. The electron still stays coupled with the atomic nucleus following a path of least resistance; if the 'residual Zitterbewegung-inducing radiation' were to cause the electron to escape the atomic nucleus, that would mean the path of least resistance had been greater in another direction, and would have actually been what we'd observe as an interaction of some sorts, such as a chemical reaction, and the electron would become coupled with a different atomic nucleus. Or in different words: the energy required for an electron to escape an atomic nucleus is equal to the energy of another proton coming within closer proximity to the electron than its current proton. The Zitterbewegung resulting from residual radiation throughout the universe causing a trembling motion in the electron makes its coupling to an atomic nucleus uncertain, allowing for non-permanent coupling between protons and electrons; instead of viewing particles as 'binding together', opposite electromagnetic charges could be viewed as coupling together momentarily/temporary out of a path of least resistance. For this hypothetical model to work properly, space/distance as well as time at this scale would have to be reconceptualized. Since time only has metric based on changes in distances, and distance only has metric based on the time it takes something to travel between two points, then space exists as the position of its contents relative to each other, and the particle contents of space only have distance between each other relative to other particles - thus, when two particles are closer in proximity to each other than any other particles, the distance between them is indefinite, and the time it takes for the particles to travel that distance is infinite. Although stationary relative to each other, a particle pair may be rotating through space relative to other particles - observed as 'coupling' between particles. Electromagnetic charge then, results from the direction of this rotation after point-particle pairs coil into structural formations with other point particle pairs into larger structures with a specific, intrinsic orientation to the rotation of pairs. Positives and negatives could then be thought of as gears spinning clockwise and counter-clockwise: gears spinning in the opposite direction mesh together and stay coupled - while gears spinning in the same direction kick off each other, exhibiting a 'repulsion'. The principles which this model relies on could be summed up with central principle: space does not exist independent of its contents. Whatever 'space' is, it has to be made of 'something', thereby being part of space's contents; this paradox is resolved by viewing space as the sum of its contents, without there being an underlying Euclidean grid (as such a grid would have to be made of 'something', returning to the paradox). With space conceptualized in this manner, it becomes possible for the macroscopic universe to take on an oblong, branch-like, or coiling shape.

You're right. I took 18.8^3 and asssumed that's how many "millions" there'd be; It's been too long since I've actually been in a math class . I'm a bit rusty. It's a good thing I'm not trying to be a physicist for a living; I'd get booed straight to the unemployment line.

Here was my thinking behind that: if the diameter of a hydrogen atom is indeed 5.2x10-11 meters, and placed side by side the length of a millimeter, would number something like 18.8 million, this number cubed would be ~6.6 billion. However, assuming that would be of unrealistic density, I guessed it would be much less. Now however, using avogadro's number instead of just putting numbers together, I'm not so sure about the math in my original post or the diameter of a hydrogen atom being 5.2x10-11 meters... Taking the volume of one gram of liquid hydrogen H2 which is roughly 0.011 cubic meters at room temperature, or 11,000,000 cubic millimeters, and roughly using avogadro's number 6*10^23, we get 50,000,000,000,000,000 atoms per cubic millimeter of liquid hydrogen -- quite a bit larger than 6.6 billion. And considering the volume of gaseous H2 hydrogen, 22.4 liters per gram at room temperature and 1atm, which would be 22.4 million cubic millimeters, that's still 26,000,000,000,000,000 atoms in a cubic millimeter.. I need to investigate how they calculated the diameter of a hydrogen atom then, and where the number 5.2x10-11 meters came from. I'm guessing they're somehow including the reach of the atom's electromagnetic field (but wouldn't that technically be infinite? Just negligible after a certain distance?) Edit: Found out where I got 5.2x10-11 meters is the Bohr radius. So the diameter would be twice that 10.4x10^-11 meters; I'm going to correct that in my original post. Also, how was Bohr radius calculated, and why does an estimate using Bohr radius differ so much from what Avogadro's number gives you?

Edit: mixed up Bohr radius with diameter, fixed some numbers (sort of), as other posters have pointed out the estimate of number of atoms placed side by side the length of a millimeter was wrong, and was based on Bohr radius. If we keep the length of a meter in mind for comparison when looking at units expressed in scientific notation, it starts to dawn us that lengths such as the Bohr radius of a hydrogen atom, (5.2x10-11, which is 52pm) are really not that small.. not very small at all. In fact, it's large enough we can visualize it mentally: If put in a straight line side by side, there'd only be ~10 million hydrogen atom diameters in a millimeter. Looking at a millimeter, having in mind that it's only 10 million hydrogen atom diameters long, and 10 million not being that large of a number, everything starts to just seem.. simple -- and that's just hydrogen, with its 1 proton; every other element would have an even smaller number of diameter-lengths in one millimeter (lets forget for a moment about deuterium and whatever the name is for +2 helium without any electrons). Realistically however, in one cubic millimeter of the air in front of you, there'd be even less than 10 million atoms. [Edit: as other posters have pointed out, this is wrong ] (perhaps someone could provide some more accurate estimates of how many atoms there are in a cubic millimeter of air, or a wood desk or other common object). Let's have a paragraph just to conceptualize and point out that 10 million is not that large of a number; 1,000 can be verbally counted to in 5-10 minutes; 1,000 dots can be plotted on a piece of paper in 4; and there is only one thousand thousands in a million. To put the number one million in perspective visually, here's a picture of 1,000,000 pennies: https://c1.staticflickr.com/1/93/235878348_d72f315683.jpg Less than 10 million atoms in one millimeter is a small enough number you'd almost think you'd be able to visibly see them; however, you wouldn't be able to see individual atoms anyways for purely mechanical reasons: the photoreceptors in your eyes are, themselves, made of many atoms; to trigger an action potential in an optical nerve, the electromagnetic radiation required has to, in almost all cases, come collectively from many atoms (excluding hypothetical scenarios with extreme conditions). Action potentials do not clear and reset fast enough either to be able to differentiate individual atoms from one another - and even after that, the processing of visual stimuli done subconsciously in the visual cortex would lose track of the stimulus produced by a lone atom; the stimulus would be melded in with (or simply lost amidst) other activity taking place. Thinking of the figure "<10 million atomic diameters in a millimeter" a little bit more, one might then start to think about the size of a biological cell, how many atoms wide a cell is, and how then a large number of cells can fit in the length of a millimeter: assuming we'd have to knock at least few figures off 10 million, we're only in the thousands to hundreds range for number of cells that would fit placed side by side in one millimeter (which likely isn't far off from reality), which then begs the question "why can't we see individual cells?" I think part of the answer lies still in the mechanical limitations of the eye, such as there being a finite amount of photoreceptors and optical nerves, so a 'perfect image' of something is always an impossibility at any scale, and our vision is really nothing more than a quagmire of blurry nonsense splashing against our photoreceptors, which sputter out electrical noise down the axon of an optical nerve, where it then haphazardly blast electrochemicals at other neurons across synapses, adding collectively to a sum of neuronal noise in the visual cortex. Only with the aspect of 'mind' (arising out of the semantics in neural connections' structural organization), is the noise made coherent into a rough image of something. The subject of how the brain processes visual stimuli is actually quite interesting on its own; there are a lot of other mechanical limitations to the macroscopic structure of the eye as well, with visible consequences that aren't usually noticed unless specifically pointed out; I'll link to more about it in a follow up post. Anyways, the scale of the atomic world put in perspective sort of makes us lose that awing, unimaginably deep and complex feeling as an aspect of the microcosm. But what about the macrocosm? We should still be okay in having that, 'small, insignificant, mystifying feeling' about how big the universe is, right? I'd like to think we should, and with there even maybe being a whole multi-verse, I don't think we should lose that magic of being mystified by the cosmos. However, with the convenience of the metric system and being able to conceptualize scale with exponents of 10, the macroscopic cosmos is entirely fathomable mathematically, especially if you keep the numbers laid out in front of you: The estimated radius (according to the featured link by Google for the search 'diameter of known universe') of 46.5 billion light years is only 4.40×1026 meters. 4.46x1026 means we're only dealing with around 27 figures, when concerning our unit of measurement the 'meter'. In kilometers, just 24 figures; and with a thousand kilometers per unit, or 'megameters', we knock 3 more figures off to have just 21 figures. By itself, a number in the 20s is very easy to work with mentally, even toddlers are quite capable of counting to 20, or conceptualizing 20 of something; it's not as though we are dealing with some obscenely large number with 10 to the power of a several-hundred figure number; even 10999999999999999999, which is much larger than 4.46x1026, is easy to write out and work with using scientific notation. Given the potential of mathematics and how insanely tedious calculations can be, things like the size of the universe and the scale of the subatomic world start to seem kind of boring, really. 'Mathematics in practical measurements' compared to 'arithmetic which seeks to harness the potential of mathematics' makes the practical world seem small and perhaps even a little bit claustrophobic for a mathematician who dabbles in the philosophical.

Someone on reddit gave a solution to the problem finally; unfortunately they wrote out how they found it entirely with differential/integtal operators, which aren't readable unless you already know how to do the problem being solved, so I still have no clue how to find line integrals. Here's the solution he posted: A = int_S |ds/du X ds/dv| dudv (parameterized area integral) The surface is defined by x in [-3,0], y=2x, z in [0,x2+y2+2x+2y+2]. Let u=x, v=z, and write s(x,z) = [x,2x,z] in terms of u and v to get s(u,v) = [u,2u,v]. ds/du = [1,2,0], ds/dv = [0,0,1], ds/du X ds/dv = [2,-1,0], |ds/du X ds/dv| = sqrt(5) u in [-3,0], v in [0, 5u2+6u+2] A = int{-3:0}{0:5u2+6u+2} sqrt(5) dvdu = sqrt(5)int{-3:0} 5u2+6u+2 du