Gaming: The Mathematical Principles of the Elements

 Therefore, I posit the following propositions:

First, that no elemental source may be created, nor destroyed, save by the action of elemental or physical forces in exponential proportion to the strength of the source created.

Second, that the force created upon an elemental source is inversely proportional to the distance of the actor.

Third, that the force exerted by an elemental source is exponentially proportional to the degree of purity of the source.

Fourth, that the force exerted by an elemental source is exponentially proportional to the degree of difference in kind between sources.

- Newton, Systema Mundi

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Previously, I discussed the evolution of a system parallel to modern science, of alchemical transformation based on the classical elements, up to Newton's publication of the Principia.

What practical differences does such a system make?

First, some terminology.  I've already discussed "binding" and "sources" without defining such elements.  In plain, unscientific, and frankly silly terms, a source is an approximation of the Platonic ideal of an element.  The closer to the pure Platonic form, the "purer" it is; as a corollary, the more dangerous and generally poisonous it is.  For instance, heavy water, in our world simply water where the hydrogen has a measurable fraction of deuterium or tritium, is moderately dangerous because of its differing properties; in this world, it is dangerous because it is closer to the "truth" of water and therefore warps reality slightly around it.

One of the effects of a source is that it turns objects placed in contact with it into... well, it.  The rate of doing so is dependent on size and purity, but drops off exponentially with distance, so that an object in immediate contact is consumed almost instantly, but an object even close to a low-strength, low-grade source is safe.

Binding a source involves creating a containment vessel at an arbitrarily safe distance from the source.  Arbitrarily safe is important terminology - in places where two different elements are bound, say, water and fire in an aeolipile, the binding should confine them closely enough to benefit from the repulsive force the two exert on each other.  Sources of the same type can be placed together without any risk except the growth of their area of effect; sources of differing type, especially opposing types, create a physical effect.

Most of my audience will be bored by the math that is involved in all of this, but what the above rules mean is that it is possible to figure out how much of an impact a source will have based on certain factors: its size, its purity, its likeness to the thing it is impacting, and its distance to the thing it is impacting.  Suppose for a moment, then, that two sources exist in an equilibrium, where they are balanced with their containment, and then something disrupts that equilibrium.  If the two are pushed farther apart, then the overall effect drops, but if the two are pushed closer together, then the force generated (in Newton's terms) increases exponentially.

So, we have additive operations - you take one candle flame and you add it to another, and now it's two, and its strength is greatest at the center of the fire but diminishing rapidly with distance.  Clearly, you would need a very strong source of a single type to have any great effect.  This is why there was a bonfire atop the Pharos, and why the three Great Fires of Persia were great fires.

We also have subtractive operations - you can, by unknown means, take a fraction of the original source, and take it somewhere else, so that you can light a source derived from the first in each new fire temple, or you can have a perpetual light derived from the Pharos aboard your galley.  The first source is diminished, but because the first source is probably huge to have a large-scale effect, the difference between 10001 and 10000 candles isn't that significant, and if this is being done on a continuous, maintained basis, we have to assume there is a way of feeding the source.

In the overwhelming majority of pre-1600 cases, the practical benefits to such a system compared to natural power are trivial enough compared to the potential costs and dangers of an anvil that slowly turns a chunk of iron into steel by itself, or a water clock that never needs refilling but might instead need emptying, aren't worth the trouble - just do it the old-fashioned way.  Where it starts to make a difference is industrialization.  Consider, for instance, the mid-1800s - a steam engine that requires no coal, or the race to the breech-loader because manufacturing a small bound couple of fire and air makes cannon power much more explosive when you suddenly smash them closer together.  You start seeing the use of bindings in new and interesting ways - Daimler attaching a water-fire couple to a flywheel, and then a series of water-fire couples driving a crankshaft; the development of the air-water-earth triple from an interesting tabletop model up into the basis for hydropower, and Tesla and Marconi feuding over who invented the bonded air-earth couple for wireless transmission.  All of them are about generating, containing, and manipulating energy, leading up to The Big One - the four-way bond, which is theoretically demonstrated in the first quarter of the 20th Century and is proven at lab scale in 1938 by Meissner and Hahn... and then in a collapsing-bond explosion at Trinity in 1945 and over Hiroshima and Nagasaki later that year.

How would one map this out with math? Well, I see two ways.

First is the physics of it.  I propose for my governing equation F = ke^(-(x^-b)).  Here, F is force, measured in units of pounds or newtons depending on system, k is an arbitrarily derived constant based on the size of the source, x is the distance between the center of the source and the center of the object affected, b is a factor based on the relationship of the elements involved, and e is the exponential function, the inverse of a natural logarithm.  This looks more complicated than it actually is.  Playing around with numbers will rapidly show several things.  First, the function is asymptotic - its boundaries are zero and k.  This is as it should be (and if we want to break that, we can, but the only example I've given where we would is tied to alternating current for a reason!).  Second, x^-b will change radically depending on what b is; if -b is -2, it's the square root of a number, if -b is -4, it's the square root of that.  This means that for high b-values, the behavior of F becomes less and less predictable and forgiving as x shrinks.  Since we are still assuming that Newtonian physics mostly works, this also means that there is a huge jump in energy the smaller that x gets, because work and energy are derived from force applied over a distance.

This is all a fun little party trick to show that I can make up numbers, but this post is tagged gaming for a reason, and that's that this all started as a way of addressing Magic Chernobyl.  So let's talk about a different kind of mechanics - gaming mechanics.

The way that I'd address this depends on system, but I'm going to focus on Savage Worlds because I like it as a system.  I'd say that to establish a stable, reliable system requires a skill roll at -2 for every element involved after the first, and then apply the system's size modifiers as a penalty as well.  So, for instance, if Bob wanted to establish from scratch an elemental source of fire the size of my office cube, in mini terms a "large" creature, he'd be rolling at a -2 - not impossible for a skilled character, especially if there are other things like equipment, circumstances, or an Edge that gives him a bonus.  If he wanted to establish a nuclear reactor out of that fire, of similar core footprint, he's looking at -2 for size, then -6 for earth, wind, and air - and suddenly that's a -8, before any ameliorating changes.  Even the most skilled characters in the system would be hard-pressed to do this regularly!

Now let's move on to the specifics of Chernobyl.  Without too much technical detail, the reactor was both cooled and moderated with water, and relied on control rods to keep the reaction within desired parameters.  The day of the incident, the reactor was near its lowest operating level, and therefore almost all of the control rods were withdrawn to keep the reaction going.  This brought the reactor to a near uncontrollable state, the water boiled off, and nothing was cooling the reactor.  Ironically, if it did not melt, at this point it was unlikely to become a runaway reactor, because the moderator was gone, but the risk of melting was high, so the operators dropped the control rods, all at once, in an attempt to scram the reactor - to kill the reaction, in other words.  The problem was that the control rods were tipped in six feet of graphite... a neutron moderator.  This caused a moment of uncontrolled supercriticality, and the rest is history.

So suppose Anatoli is conducting a safety test on our four-element bond - an automatic -6 penalty.  Well, he is using an established reactor, so there is no penalty for changing sizes, and he is an experienced technician, so we will give him a +2 for familiarity.  Still, he is in the Soviet Union; it would not do for everyone to know everything, else we would long ago have achieved Marx's dream and what would they need the State for? We decide that if he fails by 2 or more, we will have an Incident.  Anatoli is rolling a d8, and we will be generous and call him a Wild Card.  After all, we know Anatoli Dyatlov soaked the radiation damage later, so it's a safe assumption! So that is a d8 and a d6 just-in-case, at a net -4.  Anatoli rolls a 5 on the skill die.  Well, it could be worse.  He has a wild die still.  That... is a 2.  His high roll is, after modifiers, a 1.  This is very clearly not the standard target of 4.  It is in fact more than 2 under.  We have decided that there will be an Incident.  "Comrade Dyatlov, you feel a rumble through the floor..."

Well, how bad is this Incident?

Let's say that the size modifier for Chernobyl-4 is an 8, and say that there is one die per bound element, per size modifier (so Sz x Elem = D d6, or 8 x 4 = 32d6).  We will arbitrarily say that every 30 feet or 10 meters, this halves, because that's the easiest way to simulate our F=1/x concept.  We could just as easily use 100 meters, but we are going to be kind because this was not an intentional detonation and was not a true bomb but a major oops.  The direct blast inflicts 32d6, effectively total annihilation, within 30 feet, and scales down to 16d6 out to 60 feet, 8d6 at 90 feet, 4d6 at 120 feet, 2d6 at 150 feet, 1d6 at 180, and no noticeable blast effects at 210 feet.  I consider this a little unrealistic, but an acceptable deviation from reality in a world with elemental bonding and two-fisted adventure.

Nevertheless, it's bad.  We could do plume mapping, but it's an RPG, and Savage Worlds is about Fast, Furious, and Fun, so let's not do plume mapping.  Instead, we will say that every day for the first week, anyone within ten miles of the Incident takes 4d6 radiation damage, which they can overcome with protective gear, and which inflicts wounds which they cannot heal.  In the first week, every extra zero that goes on that distance goes down a damage die - 3d6 at 100 miles, 2d6 at 1000 miles; sub kilometers if you prefer metric and keep in mind that like all RPG numbers at the end of the day this is arbitrary and is meant to give a reasonable simulation, not to be map-accurate.  This is likely to deal long-term damage to all but the very, very frail or the very, very strong, who may die or be completely immune.  After the first month of the Incident, a die comes off those values.  After the first 90 days, another die.  Now radiation damage is a limited risk only to those working within ten miles of site, but people who have been injured already will feel its effects forever.  This will also be very swingy, and will also result in very different results for long and short exposure; Dyatlov, for instance, took a massive dose, but evacuated to Moscow the next day, while Schrebina and Lugasov, in the leadership team for the recovery effort, were routinely taking doses inside the hundred-mile mark, and many of the Liquidators were subjected to near-constant effects... all of which broadly maps to reality.

Now, what do those effects feel like?

Are you talking about real-world radiation poisoning, with its cellular destruction and obliteration of gene sequences, and thus the ability to repair cells? Or are you talking about Platonic ideals of elemental concepts, where a person is exposed not to fire, but to Fire; not to water, but to Water, et cetera? In both cases the answer is "a fundamental rewriting of the nature of the thing," but in the case of the world of elements, I'd expect that Chernobyl's red forest, where all the leaves turned red, the leaves turned to ever-burning, never-heating fire; that growths of crystal emerge from the bodies of those too-long exposed; that the sloughing off of flesh is because the flesh itself is turning to water, not because the cellular structure has disintegrated.  In either case, for the first 90-plus days, the danger zone should be absolutely terrifying, not least because no one knows what the effects truly are.

Okay, so we know what it looks like when Chernobyl happens, why Magic Chernobyl happens, and a broad view of the physics of this world.  Next time, I'll talk about the Picatinny Arsenal explosion of 1926... when lightning struck the US military's largest single stockpile anywhere of munitions, and struck a munitions factory while it was at it.