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Mistborn physics anyone? My theory on Iron and Steel Allomancy.


Jofwu

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I've rejected this original theory, but I'm leaving it here for reference! The original post is below the line. :)

 


As an engineer, I've daydreamed about the physics of Iron Pulls and Steel Pushes a fair bit. There's some old threads, but they're pretty dead. And I don't think they provide very satisfying reads. There's a few problems I have and I'm curious what other people think. Yes, I realize that at some point we have to step back and say "it's magic, because obviously the world doesn't work this way". But humor me. Surely there's some other nerds out there like myself who think about these things. :)

I guess I should note that this topic may contain spoilers for all Mistborn books so far. (I made a Stormlight comment at one point, but it's not a spoiler in my book.)

The Basics

So let's talk about Allomantic force. It's usually assumed to be a body force (like gravity), because there's never any indication (that I can remember) of a point force inside your body (which would be weird) or a surface pressure and because the accelerations involved would seemingly give severe whiplash otherwise. Most people assume that it follows the inverse square law (also like gravity) because effectiveness obviously decreases with distance and that's a logical assumption. Lastly, there's some level of Allomantic Strength involved to account for [1] your inherent abilities, [2] your current burn rate or something like that (to account for flaring, Duralumin, etc.?), [3] the desired strength of the force (though this may be subtlety accounted for in 2?), and [4] the difficulty of Pushing/Pulling some object (as it's hard to push on something Invested, like a metalmind, a spike, or metal inside a person).

This gives you an equation something like:  F = A / r² where A accounts for the Allomantic Strength and r is your distance from the metal. Alternatively it could be something exponential, along the lines of F = A e^-r, so that the max force is finite at r=0. Both of these allow for non-zero force at infinite r, though I think we can get around this by arguing that the force would be incredibly weak (enough to be very negligible) at great distances. And there is an argument that, while the equation is always true, there's some mental element to the magic- so that even if you are physically capable of Pushing something from many miles away, you can't do so if you don't have a mental and/or spiritual awareness of (or connection to) the metal you want to Push. The exponential equation actually makes a bit more sense to me, because I feel like the 1/r² version puts way too much weight in the distances involved. Doesn't feel right to me that Wax's strength increases by 4x if he moves from 4 meters away to 2 meters away (from some metal object). So I'm going to go with F = C1 A (e^-r)^C2, where the Cs captures whatever constants are needed to scale our equation appropriately.

There also seems to be some debate on whether the Allomancer's own mass should be part of the equation as well, because there are a few instances where Keslier and Wax seem to suggest that heavier Allomancers are generally stronger. Those who disagree argue that they are only referring to how the extra mass makes you more resistant to the reaction forces and the pushes of others. I think it's debatable either way, due to unclear explanations.

Problem: It can't be a simple applied force.

I think there's a serious problem with assuming that Pushes and Pulls work like a simple force in the way I've described, and it's rooted in a simple example in The Final Empire (and pretty much every other book). When first learning her metals with Kelsier, Vin pushes a coin out of his hand and into a wall before she is suddenly thrown backwards. Newton's 3rd Law says that forces have to balance out, and people generally assume that this holds true for Allomancy. The text suggests it on many occasions. But if Vin is pushing this coin with a constant force it shouldn't matter that the coin stopped before she was thrown down.

It's easy to assume this sounds right, but it's not. If Vin is pushing with a constant force then nothing changes when the coin hits the wall. She should be feeling a reaction force at all times, and it should be equivalent to the force she Pushes with. But what we see is a sudden increase in force after the coin is restricted. It's like standing with a garden hose, water flowing. You don't get pushed back less when you spray water into open air and more when you spray it against a wall.

I assume this problem is rooted in a simple misunderstanding Brandon had when developing the magic, or maybe he just didn't care. But I've been trying to explain what we see in some other way.

Solution? Violation of Newton's 3rd Law.

The best solution I can see seems to require violation of conservation of momentum and energy. (Edit: Someone on Reddit has claimed that WoB says momentum is conserved, but I think the text suggests otherwise) But then, this really shouldn't come as a surprise considering Allomancy is "End-Positive" per the Ars Arcanum. So we're not really creating energy and momentum here, but taking it from Investiture. And this allows for an imbalanced force to exist. And this is where my solution comes in. I'm struggling to find the proper language to describe this, but what you basically have is more force going to the thing with less mass. Let's look at equations...

So the force we calculate above, F = C1 A (e^-r)^C2, is actually the sum of two forces being applied when a Push or Pull happens. There are two elements involved: the Allomancer and the "Object". By Object I'm referring to the metal being pushed, plus any other mass it is attached to. Can't think of an example, but I'm pretty sure a coin tied to a bag of sand is harder to push than a free coin. The extreme case is where the piece of metal is pressed against some massive object (like Scadrial itself). So let the Allomancer be noted with subscript 1 and let the Object be noted with subscript 2. What we have is F = F1 + F2. F1 is the force on the Allomancer and F2 is the force on the Object.

So how much force (F) gets directed to F1 and how much to F2? Let me introduce a new factor, X = m1 / (m1 + m2), which tells us how much force is directed into the Object. Then the amount of force directed into the Allomancer is 1-X. So F1 = (1-X) F = F - X F and F2 = X F. I'll illustrate the meaning with some examples below, so don't hurt your brain trying to interpret what I mean here.

Examples:

Let's consider the case where Vin pushes a coin in the open air. The coin's mass (m2) is much much less than Vin's mass (m1), so we essentially have X=m1/m1=1. The coin receives (almost) the full Allomantic force with F2 = XF = F while Vin receives almost no reaction force.

Suddenly the coin hits the wall and Vin is no longer just pushing against a small coin, but also against the building the coin is pressed against and the ground the building is attached to. So now we have m2 >> m1, which means we essentially have X = 0. Now Vin receives (almost) the full Allomantic force while the coin receives almost none. This explains why suddenly she is knocked back with a higher force. The reaction force she feels dropped from almost F1=0 to F1=F in an instant.

This also explains the "middle of the road" case where the Allomancer and the Object have a similar mass. If m1=m2 then X=0.5. Half of the Allomantic force goes into the Object while half goes into the Allomancer.

As you can see, this does a reasonably good job explaining how pushing a small, free object produces little to no reaction force while pushing against a great mass throws the Allomancer with their own full strength. But it also accounts for the cases, in between, where Newton's 3rd law seems to apply normally. It also might provide a bit of an answer for why an Allomancer's own weight/mass plays into his strength. It doesn't actually boost the forces they are capable of, but it doesn't just provide resistance to a reaction force either. It actually tips the scales of how the force is distributed, in non-extreme cases.

Other Problems...

This isn't quite perfect however...

It's pretty obvious in the case above, where the coin is pressed against the building. If the coin no longer experiences any force then it would drop to the ground. But we don't (apparently) observe this. You can imagine that the coin slipping a little means it isn't quite in full contact with the building, perhaps... And so then it's rapidly getting pushed back again every time slips. But this feels a little weird, and doesn't work great. My best solution is that maybe my X factor above isn't quite perfect. Perhaps there's some minimum amount of force that the Object will receive no matter what, for example.

This also makes me consider the case where Vin and Kelsier have their little Push battle, with the coin between them. There's a few weird things there as well...

First, we have the "center of mass problem" that I'll get to in a moment.

Second is what does this mean for the mass in my equation above? For Vin pushing a coin that Kelsier is also pushing... what is the m2 in that equation? The mass of coin plus the mass of Kelsier? Plus the mass of Scadrial if Kelsier is anchored? What if he's standing free and relying on friction with the ground? Perhaps the calculation of m1 and m2 is a bit more complicated and involves some Cosmere concepts? Maybe there's some kind of "effective mass" that we have to determine?

Third is explaining why the coin doesn't just fall down. If two equal and opposite forces are at work, why doesn't gravity take over? My best guess is that this plays into the "center of mass problem"... Kelsier and Vin are obviously keeping the coin balanced between them rather than letting it slip to one side. This configuration is unstable, so there has to be some means for them to balance the coin. And it's not a big leap from there to say that they're giving the coin some upward force as well, to balance gravity. The explanation deserves an eye roll maybe, but the scene is so awesome that letting it slide is easy. Then again, maybe gravity just does weird things when magic is at work. There's precedent... In Stormlight we see a bit of an oddity that's similar (perhaps) with the Surge of Gravitation... But I won't get into that. And maybe this also plays into why the coin against the wall doesn't fall, in my framework above.

Fourth, we have to explain why the coin flattened. It's tempting to just give up on this one and say it's a mistake. If Allomantic force is a body force, then this is hard to explain. Gravity can squish an object, but only because it presses you against the ground. Two body forces just cancel out and don't causes any stresses inside the body. Technically the outsides of the coin experience a bit more force because they are closer to the Allomancers than the center... But that's definitely not enough of an explanation. Perhaps Allomantic force is more of a pressure applied along a body's surface? Not sure if there's much support for this elsewhere in the text, but I'll have to keep an eye out on my next read.

The Center of Mass Problem

I have a strong opinion on this one... The characters are convinced that you can't really control the vector of an Allomantic force. It's just a two way force pointing directly through the centers of mass. I am of the opinion that they are wrong about this.

Wax proves that this isn't the case for the object itself, unless I'm mistaken. I seem to recall that he is able to break down a metal object into "smaller pieces" mentally and push on a single element out of a large piece of metal. (Edit: Someone on Reddit corrected my memory of this example and I no longer think it applies) Seems like Kelsier might also have done something like this by spinning spears (Edit: metal bars) midair? And of course there's evidence to suggest that Allomancy can be performed on trace metals. Logically, we know that any metal thing is ultimately made up of a bunch of smaller metal things. So it makes sense why it would work this way. It's likely that understanding how to manipulate metal this way is just not well understood. Might involve someone's power level. Definitely involves skill and understanding. Maybe a few more years down the road we'll see Allomancers who have a better understanding of this.

The same seems to be true for the point in your own body that you Push and Pull from. I don't have as much hard evidence that this is possible, though the ability to balance on a single small piece of metal is a great start. Again, I'm betting that an Allomancer has some level of control over where they Push and Pull from. I would expect that this "point" must be located within your own body somewhere, but that you can shift it with the right understanding/skill/practice/etc.

In fact, it seems likely to me that my simple "point force" equation above is a simplification. More than likely what we're dealing with is a field, which emanates from the Allomancer. He has some kind of mental control over which bits of metal he is effecting (down to the molecular scale?) with his field, which is dependent on skill (etc.) and which is largely performed subconsciously/intuitively. (atium gives you understanding of how to process the data you receive, so it makes sense that other elements do something similar) And he also has some kind of control over that field, by shifting around Investiture (or whatever) within his body.

Thoughts?

What do you guys think? Got any passages that support this theory? Or any that disprove some part of it? Maybe some idea on how to refine this? I'm curious what other people think!

Edited by jofwu
updated based on input
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5 hours ago, jofwu said:

Wax proves that this isn't the case for the object itself, unless I'm mistaken. I seem to recall that he is able to break down a metal object into "smaller pieces" mentally and push on a single element out of a large piece of metal. (Edit: Someone on Reddit corrected my memory of this example and I no longer think it applies) Seems like Kelsier might also have done something like this by spinning spears (Edit: metal bars) midair?

I am re-reading Bands of Mourning right now so spoiler alert but here you go

Spoiler

In a flashback Wax does indeed push on one point of a bullet he and Forch are both pushing on to make it shoot ahead and kill Forch. I don't know if this counts as breaking the metal object into smaller pieces, as maybe the bullet was several metal parts and his natural ability (because it's too early for him in his life to be a Savant) allowed him to focus on that smaller bit. However, I'm still re-reading the book and if I see any more instances of Wax or someone else pushing on smaller part of a large piece of metal I'll let you know. 

Aside from that, I am not at all a science person so my head started hurting when I tried to follow your formulas, but I enjoyed the post a lot regardless. While there's a lot of science behind the magic in this series, more than any other fantasy series I've read, it's definitely still magic and Brandon has talked before about finding what science should allow him to do and then find what he is willing to break/limit to make it semi-functional scientifically while still interesting literary wise. I've only seen direct comments from him on this about Wax's weight-tapping and Wayne's bendalloy, both from the Alloy of Law annotations, but I haven't read the annotations from the original trilogy yet so there may be something in there about the science behind Pushing and Pulling.

Edited by Radiant Returned
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Hmm...with the example of Vin and the coin...aha. I've got it. Steelpushes or Ironpulls are a force that travels along a line, with the continuous force pushing it on all sides, not allowing it to fall. The metal sight granted by these metals points towards the center of mass, yes, but the force of the Push/Pull affects the entire object. And Vin was sent back because the force is an equal reaction based on weight of objects. The force of the push is a constant number. A coin with a weight of a few grams is going to go farther than a Coinshot/Lurcher with a weight of a hundred pounds. Really, Steelpushing yourself to fly is just an Allomancer riding the recoil of the push.

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I am not an engineer, but I did take physics in my undergrad. I also did a little bit of thinking about this. Here is my theory: Steelpushing and IronPulling are from your center of mass to any point on the metal that a person skilled/strong enough to pinpoint (i.e. Kelsier and the prison bars, Wax and the bullet). I believe that as far as your equations go, they seem solid. I like the second equation better for the purpose of discussion: F = C1 A (e^-r)^C2. I think this is helpful, however I do not think that your "hose" example is helpful. The way that I understand Allomancy is that it is a push from you, think if it like an arm, instead of a stream of water. If you push a coin, sure, technically, the coin pushes back on you, but so what? I weigh as much as 16,000 coins (give or take a few pounds) the force is nothing compared to the friction of my shoes on the ground, but if I push that same coin against a wall, now I am pushing myself back (assuming the wall is sturdy and such). So maybe think of the physics in a more "I'm pushing my arms against this" and not "I am using this magnetic force against it. 

 

Would that help keep Newton's Third Law in check?

 

-Will

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Hmm... Did you also post on Reddit? I'm not sure what comparison you're trying to make with pushing things with your arm. The point of the hose example is that it's an example where you are experiencing a constant reaction force without moving. I think you are bothered by the fact that you don't provide the energy in this example, but that doesn't matter. A steelpush could also be compared to shooting a gun. Same thing... But the hose served my purpose better because it's a constant and sustained force rather than a single impulsive force. Using your arm to push something directly works as well, but its harder to visualize my point in that case, I think.

Yes, you are directly exerting a force on some piece of metal. And there should be a reaction force that you experience. You feel the reaction even when pushing a small coin, it's just not very hard to brace yourself against it and you weigh enough that you generate plenty of friction to stay in place. The problem is that this doesn't change when the coin is suddenly pressed against the wall.

Say I'm pushing a block across the ground with 10 Newtons of force. I push the block into the wall, but continue pushing with the same force. What happens? Nothing at all. I was experiencing a 10N reaction before, and I continue to. There's not some sudden spike in force when I hit the wall that would knock me over.

Now... All of that said, a conversation on Reddit has actually changed my mind on some points. When I get my thoughts organized I'll have to come back in here and update everything. :)

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I could be wrong, but the comparison with the arm seems to me to be saying this; (I am not a physics major.  Feel free to disregard everything I say as gobbledegook.)

Lets say I have a coin on the palm of my hand.  Let's also say I am on a very slippery surface, for the point of the experiment.  I push against the coin.  It moves back, I stay in the same spot.  Then I push the same coin against an unmoving mass.  When I do this, My arm flexes, moving me backwards (usually.) The coin does push me backwards, marginally, in my hand, but the force is extremely small, and does almost nothing.  Then, when I push against the wall, I slide back.  My own force cannot move the wall, so it moves me.  This is all a rather simplistic explanation.

When a coinshot shoots a coin at a wall, they are 'flexing' their 'arm'  The coin moves, they stay put.  Then, when the coin hits the wall, the 'arm' pushes the coinshot back with the amount of force the 'arm' was applying in relation to the wall, which will not be effected by the strength of the 'arm', however, you pushing against the wall will move you backwards.  Of course, a coinshot's strength is a good bit stronger than an arm, but I think you get the point.  This is not a great visual due to arm length, arm force, etc.

Another, possibly more accurate way of thinking about it (I am going to use metaphors, not equations, because I tend to be better with the former.) is that a coin shot's steel pushes are like a very long, very compact piston.  The piston starts out at a set length.  The coin is on the end of it.When the piston extends, the coin moves.  The piston's weight is equal to that of the coinshot.  When the coin hits a wall, it stops, then, depending on the weight of the wall, the piston is shoved backwards by it's own force.  As it extends, the metal tubes forming the piston's constantly extending arm get thinner, and weaker, until it can no longer push the coinshot.  If the coinshot pushes suddenly downwards, he gets shot up by the force of the piston, and has enough momentum to carry himself upwards.

I hope this is not confusing, and I would love for you guys to poke holes in it, or tell me what is wrong with it.  This does sound similar to what Jofwu is saying, but I had a little trouble understanding everything they said.  I think the problem they are having is that of the piston continuing to extend.  That force has to go somewhere.  Other than that, excellent show.  I agree with most of what I could understand of what you said.  I would say the coin pressed against the wall would not slip because of friction, but the coin between Kel and Vin should have slipped... Odd.  Tell me what you think.

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I see what you're saying, I think. But yeah, this is based on a simple misunderstanding of the physics. Perhaps one that Brandon himself made. I'll stick to the "arm" example. There's a bit of a disconnect with the example from one case to the next. You're imagining pushing the coin with a very light force... and then pushing the wall with a much greater force.

Imagine you've got a coin sitting on a table and you push it steadily along to the other side with your finger. You don't have to push very hard, and you certainly aren't knocked over by it. The table is pushed up against a wall and so the coin is stopped when it reaches the edge. You keep pushing the coin/wall with the same level of effort that you did before. Just a light, steady pressure. What happens? Absolutely nothing, right? In both of those cases you were pushing with a relatively low force, and so you didn't experience much of a reaction.

You can push against a wall hard enough to knock yourself over, of course. The reason you (probably) can't do this with a coin is just a practical issue. You're limited by how fast your arms are able to move themselves. The coin gets away from you before you are able to push on it with the same force. Throw a coin as hard as you can and you will probably barely notice the coin's own weight. The difficulty is in the throwing motion itself, because your arm can only move itself so fast. If this wasn't a problem, you could absolutely knock yourself over by throwing a coin. Imagine a robot who can push coins so fast that they accelerate to the speed of a bullet by the time the robot's arm is extended. Shooting a gun is essentially throwing a coin with a much stronger force, and is effectively the same thing as pushing the bullet with your hand. And the recoil can knock you right over if the force lasts longer than a fraction of a second.

Maybe that helps a bit.

Edited by jofwu
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Huh.  Now that I think about it, the physics of that does seem really weird.  I would only say that maybe Vin would push the coin at speed A, and then when it hits the wall, it pushes her back at speed B, a fraction of speed A?  But her speed seems much faster than it should.  I see where your coming from now, to some extent.

The piston analogy still makes sense to me, mostly.  If a piston is pushing down against a coin, at a certain speed, it will do nothing if the coin is not up against something.  But when the coin hits, a wall, either the wall or whatever is on the end of the piston has to move.  I think that mostly makes sense, but there might be something wrong with that as well.  Feel free to take this apart as well.  I'm actually enjoying this.

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Well, pistons are a little weird. The way you're thinking about it... You're imagining the piston extending at a constant rate, which is fine. But you have to realize that maintaining that rate implies different forces.

It's really the same story. Imagine a piston with a motor from a cheap toy car. Stick it between yourself and a coin, and yes it will push the coin with very little reaction against yourself. Stick it between yourself and a wall? It's not going to knock you over. It's going to push against you and the wall as hard as it can (which isn't much) and then it's just not going to be able to extend further.

Anyways, I'm going to see if I can't write up a new "theory" below sometime today...

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Hmmm.  I think I'm going to have to live with seeing this differently than you.  It was the speed Vin was pushing the coin that was enough to give her a bit of a lurch, IMO.  A pretty strong, pretty inflexible, pretty fast piston.

I look forwards to seeing your new "theory" when you make it.

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Well... I just spent a lot of time typing it up all nicely and then reloaded the page by accident when 90% done. Normally my life-saving browser add-on, Lazarus, pulls my butt out of the fire in these cases, but apparently it doesn't play nice with this forum. All gone. :(

Maybe later...

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Introduction

After a conversation on Reddit (here) with /u/Phantine I've had to rethink a lot of my points from above. So let me first explain the foundation of my change in thinking...

First, I now see how Newton's 3rd Law might not be violated after all. I was assuming that a typical Push/Pull involved a sustained force. If the force is not strong enough to knock you over while pushing a free coin then it is too weak to knock you over while pushing an anchored coin, and vice versa. But I’ve realized that Pushing a coin is not much different from firing a gun. We are dealing with a large force… It’s just typically applied for a very short duration. Long enough to get a coin moving fast, but not long enough to knock you over. A coin shot with the speed of a bullet is generally too far out of range before you get significant “recoil”. Vin was floored because the coin got stuck and she pushed too long- not because the physics suddenly changed. This doesn’t fit perfectly with how things are described in some cases. But it fits other situations much better, and it gives us the benefit of being able to trust and apply Conservation of Momentum (one of the most fundamental rules of physics).

I was also assuming that the masses involved, both of the Allomancer and the metal object(s), didn’t play into the force calculation. The references here changed my opinion on the former while I now see that the latter is necessary logically. I was resistant to the metal’s mass factoring in because, taken at face value, it implies I can get incredibly large forces by pushing on incredibly large metal objects. You could launch yourself into space by pushing on a big chunk of metal, and that’s a bit too outrageous for me to assume. But the opposite brings problems as well… For example, let’s say I can Push with 100N regardless of metal mass. I glue a 1kg object to a boulder. I take another 1kg object, break it into 1000 pieces and glue it to an identical boulder. If I can push each piece with 100N then this means I can accelerate the second boulder 1000 times faster than the first! This makes no sense. My solution to reconcile these two problems is below.

Lastly, I’ve changed my mind on how distance factors in. I definitely think the inverse square law doesn’t fit with the text. It implies that strength drops with distance off much much faster than we observe in the books. And while my e^-r concept above isn’t a bad thought, it is perhaps more complex than necessary given how little information we have. For now I’m going to assume that strength scales linearly with distance, up to a maximum range.

I’ll do my best to be brief… I’ve already typed this out once and lost it, so I’ve run out of patience to explain every thought and support every assertion!  That said, if you have any questions or issues with any point that I make then please reply and we can take the discussion further!

The Theory

Okay, so let’s talk about the new theory I’ve pieced together. I’ll share the equation first and name all of the elements that make it up. Then we’ll look at each of those variables in turn. At first we’ll assume we’re dealing with point forces on point masses, but at the end we’ll widen things up a bit.

So we’ve got Allomantic Force as:

F = A B m1 m2 X

where…

A = Allomantic Base Strength (Newtons per kilogram squared)

B = Burn Rate (dimensionless)

m1 = Mass of the Target Metal (kilograms)

m2 = Mass of the Allomancer (kilograms)

X = Distance Factor (dimensionless)

And further below we’ll see a few more variables:

r = Distance to the Target Metal (meters)

R = Range (meters)

M = Mass Capacity (kilograms)

So now I’ll start from the end of the equation and work back towards the start…

Distance Factor – X

So if force is related linearly to distance then we have to have a maximum range at which force is zeroed out, which is where R comes in. The Distance Factor is simply X = 1 – r / R, which gives zero force at r=R and max force at r=0m.  And of course X is limited to be greater than or equal to zero. So, with that in mind, we can actually rewrite our equation with this assumption as:

F = A B m1 m2 (1 – r / R)

Questions… Is there a max range involved? It’s hard to say for sure, from the text. But it isn’t hard to believe either. If not then we’d need another equation for X, and in that case my money is something like the e^-r concept mentioned in the first post. I’m also curious what affects an Allomancer’s max range. Is it the same for everyone? Does it vary with power level? Does it vary randomly? What level of variance is there? Does flaring increase it?

Mass of the Allomancer – m2

See above for the references on why I changed my mind here. I don’t think it has significant impact on things however, because people don’t vary in mass by orders of magnitude. A lightweight Coinshot will be able to accelerate coins to deadly speeds just as easily as someone heavier.

Mass of the Target Metal – m1

Again, see above for my logic on including this. But we need to talk about how to handle the two extremes. On one hand it doesn’t seem like I should be able to launch myself incredible distances with a simple Push on some large metal object. On the other hand, I shouldn’t be able to accelerate something faster just because I physically (or mentally) break up an object into smaller pieces. So what to do?

It’s not perfect, but my solution is to assume that Allomancers have a “Mass Capacity”- some maximum measure of mass that they are able to push against at a time. Let us say, for example, that my Mass Capacity is 100kg. This means I can use my full strength on one 100kg item, on two 50kg items, or on one hundred 1kg items. (that is, the mass of the metal that I am mushing on- not including any other mass that may be “attached”) So what happens if I want to push a 200kg item? I can only push on up to 100kg, so I can only apply the same force that I did to the 100kg item. How about two 100kg items? I have to choose… Do I apply half of my strength to each, all of my strength to one, or somewhere in between?

Again, it’s not perfect. But it does prevent severely overpowered strength when working with large objects while explaining what happens when we break items up into smaller pieces. This is even more important when we realize metal can be broken down into microscopic elements, but more on this later. Lastly, it explains why there might be a limit on how many things an Allomancer can deal with at a given time. While that can partly be explained by how many things he can pay attention to at once, I think Wax’s “Steel Bubbles” demand some kind of discussion on this point.

While this is the best solution I’ve come up with so far, I’m not perfectly happy with it. I’ve explored the possibility that the metal mass is involved, but not directly proportional. For example, we could use something along the lines of -e^-m1 and set it such that m1=0 produces zero force. And then there is no maximum mass capacity while strength is still limited to some value. Once again, however, I feel compelled to take the simple option without further evidence.

Questions… If there is some kind of mass capacity, a lot of the same questions arise here as we saw with Range (R). Does it vary? Is it related to strength? Does it change with flaring? You get the idea. I’m also curious to get an idea of what a typical capacity might be.

Burn Rate – B

Brandon has said that strength is directly proportional to burn rate. While it’s tempting to take this value as a measure of literal burn rate (in kilograms per second) I think it will make more sense to normalize things such that a dimensionless value of B=1 represents a person’s “natural” burn rate. Brandon’s comments suggest that more force means more metal burning, and that would require the actual mass burn rate to be a function of general strength as well as the distances and masses involved. And that goes beyond what we want here.

Kelsier teaches that everybody naturally Pushes/Pulls with a given force, but that training and skill can allow for some deviation. It appears that they do this by controlling their burn rate to some extent. This also explains how flaring and Duralumin/Nicrosil bursts play into things.

Questions… Once again, I’m curious how this changes between people. I’m also curious for an idea of how much it increases for flares and bursts. For example, does one person flare at twice their normal rate while another flares at 3x? I’m also curious about what the differences between Steel and Iron burn rates might be.

Allomantic Base Strength – A

I’ve normalized Burn Rate so that we can use this number to directly compare one Allomancer to another. We can directly test for this number. It would look something like this…

  1. Measure the Allomancer's own mass
  2. Test for the Allomancer's max range
  3. Set a 1kg metal sphere at one meter away
  4. Have the Allomancer Push/Pull and measure the force with a scale at their natural burn rate
  5. A = F / [m1 m2 (1 - 1/R)]

With this number we can directly rank Allomancers according to their power level.

From Point Forces to Bodies and Fields…

This is going to take things beyond a simple equation, but it’s worth discussing I think. The books often talk about the forces being directed through centers of mass, but that raises an interesting question about how the forces are applied.

It’s probably best to assume they are body forces, like gravity. As far as I can remember, there are no indications that high accelerations cause something like whiplash, and this is the easiest explanation. That said, there are also cases suggesting they might be surface forces (notably the coin that is flattened). Either way, we’re not dealing with a point force located at the thing’s center of mass- as if we are tying a t to the middle of the thing and yanking it around. The forces are distributed.

Another important point is Kelsier’s fascinating feat in his battle against the Lord Ruler. He Pushes and Pulls on opposite ends of a single metal bar. This proves beyond a doubt that Allomancers have some level of ability to control what metals they are working with. They don’t just Push on whole objects. They can push on part of a metal object. And there’s no reason to assume that we can’t take this to the nth degree. Scientifically speaking, this makes sense. With our modern knowledge, you’d think it can be taken down to the level of individual metal crystals at least. While skill, strength, training, understanding, and mental capacity may all be involved in this, it doesn’t take away from the point.

The point is that Allomancy doesn’t just work on whole “objects”. It acts on smaller elements. It’s basically the same thing as the story with gravitation or electro-magnetism.  We can essentially imagine that an Allomancer has a field around himself, and that field interacts with bits of metal. The most notable oddity, of course, is how he has the mental ability to control which bits of metal that the field interacts with. But the mechanics are the same.

So we’re not actually dealing with a point forces and point masses. Coinshots don’t apply a point force at the coin’s center of mass- they apply a field of forces across the entire body.

Another interesting point is that the same must be true of the person’s own body. We see plenty of examples where an Allomancer seems to balance over a tiny point of metal. Zane’s balancing act is a particularly striking example. I think this implies that an Allomancer not only has the ability to determine which metal elements they affect, but also how the field extends around themselves. This might simply be a matter of “now utilizing” some of his own body mass. But it could also be a matter of something more obscure… like burning more metal “on behalf” of some body mass more than others. Or perhaps it’s even more abstract than that. In any case, it’s worth mentioning that this level of control exists, even if it isn’t easy to obtain or utilize for one reason or another.

Mass and Investiture

Lastly, I have to say something about Investiture. This absolutely must come up when we talk about Pushing/Pulling metal which is invested or metal which is imbedded inside of investiture. Unfortunately, I’m not sure how to address it exactly.

Perhaps mass and investiture are proportionally related in many cases. Perhaps we shouldn’t really be talking about mass at all, but about investiture? Or perhaps investiture negates the effective mass that may be utilized. This is probably the best solution. A metalmind is hard to push because some of its mass is invested, and invested bits of metal are not affected by an Allomantic force field. You are left Pulling on a small fraction of the metal that you see, which means relatively little force. Metal imbedded is presumably invested as well, in some strange sense related to Hemalurgy. And I guess that metal inside of a body is immune simply because it is within a bubble of the person’s own innate investiture.

Plenty of interesting questions to ask still… But I think this is a pretty solid theory to work with! I’m doing a Mistborn reread now, so I look forward to testing it out as best I can. And maybe digging up some details to fill things in a bit more and shed some light on the more vague aspects!

Please comment if you have questions or support!

 

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Preface to my statements: I am attempting to answer the questions, not critique the math concepts. I feel like waiting until you have more information so that you may refine the math first. That is why it is a fairly shortened quote

32 minutes ago, jofwu said:

Distance Factor – X

Is there a max range involved? Given that the blue lines eventually fade as they get farther away from things, I assume so
I’m also curious what affects an Allomancer’s max range. Is it the same for everyone? Does it vary with power level? Does it vary randomly? What level of variance is there? Does flaring increase it? Max range is probably the same, but most things aren't at an easily measurable distance, so I don't know. The max effective range probably varies by power level and/or by flaring. I see effective range for this the same way I feel like distance from Elantris works.

Mass of the Allomancer – m2

I don’t think it has significant impact on things however, because people don’t vary in mass by orders of magnitude. I agree with this.

Mass of the Target Metal – m1

It’s not perfect, but my solution is to assume that Allomancers have a “Mass Capacity”- some maximum measure of mass that they are able to push against at a time. I imagine max capacity has some correlation with available metal reserves.
Let us say, for example, that my Mass Capacity is 100kg. This means I can use my full strength on one 100kg item, on two 50kg items, or on one hundred 1kg items. (that is, the mass of the metal that I am mushing on- not including any other mass that may be “attached”) So what happens if I want to push a 200kg item? I can only push on up to 100kg, so I can only apply the same force that I did to the 100kg item. How about two 100kg items? I have to choose… Do I apply half of my strength to each, all of my strength to one, or somewhere in between? Same as prior statement. Heavier things take more metals to push.

If there is some kind of mass capacity, a lot of the same questions arise here as we saw with Range (R). Does it vary? Is it related to strength? Does it change with flaring? You get the idea. I’m also curious to get an idea of what a typical capacity might be. I feel like most variable aspects are changed by flaring.

Burn Rate – B

Brandon has said that strength is directly proportional to burn rate. Kelsier teaches that everybody naturally Pushes/Pulls with a given force, but that training and skill can allow for some deviation. It appears that they do this by controlling their burn rate to some extent. This also explains how flaring and Duralumin/Nicrosil bursts play into things. Flaring, Savanthood, and Duralumin/Nicrosil all increase the power of the push by accelerating the burn rate. Well, I'm not sure if that is what happens to a Savant..

Once again, I’m curious how this changes between people. I’m also curious for an idea of how much it increases for flares and bursts. For example, does one person flare at twice their normal rate while another flares at 3x? I’m also curious about what the differences between Steel and Iron burn rates might be. Maybe ignore that last sentence of mine. Savants seem to get more "bang for their buck," so perhaps that is an example of flaring 3x compared to the normal 2x.

From Point Forces to Bodies and Fields…

Another important point is Kelsier’s fascinating feat in his battle against the Lord Ruler Steel Inquisitor. He Pushes and Pulls on opposite ends of a single metal bar. This proves beyond a doubt that Allomancers have some level of ability to control what metals they are working with. They don’t just Push on whole objects. They can push on part of a metal object. And there’s no reason to assume that we can’t take this to the nth degree. Scientifically speaking, this makes sense. With our modern knowledge, you’d think it can be taken down to the level of individual metal crystals at least. While skill, strength, training, understanding, and mental capacity may all be involved in this, it doesn’t take away from the point. I think this had to do with Kelsier possibly being a Savant. Zane's spike granted him great precision, so it's not too big of a stretch.

We can essentially imagine that an Allomancer has a field around himself, and that field interacts with bits of metal. The most notable oddity, of course, is how he has the mental ability to control which bits of metal that the field interacts with. But the mechanics are the same. Perhaps the magic itself is why people can mentally direct the interactions. Deviation, not science. Sorry.

Another interesting point is that the same must be true of the person’s own body. We see plenty of examples where an Allomancer seems to balance over a tiny point of metal. Zane’s balancing act is a particularly striking example. I think this implies that an Allomancer not only has the ability to determine which metal elements they affect, but also how the field extends around themselves. This might simply be a matter of “now utilizing” some of his own body mass. But it could also be a matter of something more obscure… like burning more metal “on behalf” of some body mass more than others. Or perhaps it’s even more abstract than that. In any case, it’s worth mentioning that this level of control exists, even if it isn’t easy to obtain or utilize for one reason or another. As mentioned, Zane's spike gave him precision. Kelsier might have been a savant. Not sure about other examples in this "plenty" but my memory has failed me on details before.

Mass and Investiture

Perhaps mass and investiture are proportionally related in many cases. Perhaps we shouldn’t really be talking about mass at all, but about investiture? Or perhaps investiture negates the effective mass that may be utilized. This is probably the best solution. A metalmind is hard to push because some of its mass is invested, and invested bits of metal are not affected by an Allomantic force field. You are left Pulling on a small fraction of the metal that you see, which means relatively little force. Metal imbedded is presumably invested as well, in some strange sense related to Hemalurgy. And I guess that metal inside of a body is immune simply because it is within a bubble of the person’s own innate investiture. Brandon has said that large quantities of investiture make push/pulling difficult. Not impossible. TLR has done it, Brandon mentions duralumin a few times.. Perhaps it is just a matter of power.

I also leave it to those with better understanding of mathematics to make comments about your math.

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I still don't understand all the math you are doing, but where it takes you is much closer to what I was trying to say.  I would agree with almost everything in your theroy, RN.  So, nice job.

You should totally take this to Brandon some time.

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@The One Who Connects

It seems to me that a lot of your thoughts can be summarized by saying that "mass burn rate" affects LOTS of different aspects. That flaring and bursts affect mass capacity and range as well as strength. This may be true- I just don't think we have much proof either way. If it is true... If we can assume that all three of these factors are affected equally (and proportionately) then I'd slip in another B anywhere that M and R appear. So you'd have X = 1 - r / (B R), for example. And this is another good reason to normalize B. If they are all affected independently, then I guess you'd have to determine how they were for each person and have a different variable or each factor.

I disagree that savanthood (or other alterations) can allow you to get more strength/force per kilogram of metal burned. I take this response from Brandon to mean that you can't "get more efficient" with your metals in any way. Accelerating a coin at 5g for 1 meter is going to cost a savant just as much metal as a trainee. I would definitely agree that savanthood (and even just practice) might increase range and mass capacity. I could also see how it might increase your max strength. But utilizing that higher strength is going to come at the cost of higher metal usage. Though I may be mistaken about all this? Is there anything to indicate that Spook gets more efficient with his Tin burning?

Regarding the ability to balance on a small piece of metal I disagree that skill is enough to explain this ability. If you assume that forces are always directed through the centers of mass (of both Allomancer and metal object) then it's just plain impossible. Kelsier's explanation that you need to shift your weight around to balance is misguided. A person on a single stilt, which is always directed through the person's center of mass, cannot balance. If you can push on different parts of a metal object, as Kelsier demonstrates is possible, then this helps a little bit. But it's still not perfect. To balance on a coin you would have to keep your center of mass directly above the coin, within it's edges. This means very little wiggle room, and I don't think even supernatural balancing abilities are enough to explain it. It only makes sense if you can shift the force vector around so that it doesn't necessarily act through centers of mass. All of that said, I'm absolutely NOT arguing that Kelsier's skill (or possible savanthood) or Zane's spike are not involved with what they're doing. Skill/savanthood/spikes/etc.... these things all still matter. A new Coinshot isn't going to be able to balance like this right away. I'm just saying that we need a physical explanation for what they're doing that goes beyond the books' brief explanations.

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@jofwu I see what you are saying, my bad on some parts.
Paragraph 1: See, you're refining the math already.
Paragraph 2: I guess I mistakenly equated Savanthood with efficiency rather than with practice. Wrong mental leap to take. But I do think that Spook's base senses were enhanced by being a Savant; hence why he had to cover his eyes/ears, but I also think his tin was always on so I cannot be certain.
Paragraph 3: Zane But that's all I have on him. I imagine that the extra skill from Zane's spike/Kelsier's practice would make it easier to grab multiple parts of a metal object, but that might just be a matter of personal knowledge discerning what is/isn't the same item. People have mentioned more than once that Vin/Kel shouldn't be able to do as much as they do around the spires of Kredik Shaw using the center mass argument. Oh joy, I'm deviating into Realmatics.. give me a second.

Quote

Cognitive Realm, how people view things, relevant discussions. Basically, Zane and Kelsier can push on multiple parts of objects because they see the object as a combo of distinct parts. (probably because they understand that boxings/buldings aren't one solid piece of metal) Your average Coinshot/Lurcher sees buildings that way, but likely not coins, so their limited in that way.
In essence, a modern physics/chem teacher could practice pushing multiple parts on an even smaller scale because they can understand things on a smaller scale

Back to simple discussion, precision can be used in a few ways. I understand that precise on a single coin is unlikely, but could he not be precise enough to do a couple light pushes on nearby buildings so that he is stable, and then only be balancing on the coin? I'll reread the scene again to see if this holds any weight, but ponder it a moment.

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Regarding "paragraph 3"...

The quote you provided talks about how you can mentally break metal things down into smaller pieces. And with modern science in mind this totally makes sense. I just don't see how the same shouldn't be true for the Allomancer as well. Especially if you argue that the Allomancer's mass plays into strength.

To be clear, the notion that you might be able to "turn off" some of your mass could absolutely be wrong. The thing I'm defending here is the idea that individual "particles" of metal matter are drawn to individual "particles" of your own matter. Or something like that. Just the way that gravity and electromagnetism work. We treat them as point forces for simplicity in many cases. But the reality is a bit more complex, mathematically.

Edited by jofwu
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