Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Mon May 17, 2010 1:15 pm

It could be that I've got the math for the handwaving wrong. Let me see... (Update from the future: most of this post is stream-of-consciousness, so I change my mind a bunch)

If, say, an R-R-R arrangement and an R-S-Z arrangement are viewed in terms of symmetries, then R-S-Z has half as many, because it's not mirror-symmetric (a common theme here). However, things get really hairy from this point of view. It may be that I have to consider the initially available symmetries. In addition, this idea, while promising, doesn't address the issue of damping that I brought up... Hopefully, the two simply operate independently.

I think... weighting a state rotation to twice a state reflection makes sense in various intuitive fashions, and provides the behavior I want. So far, this has all been a highly simplified exercise in working out tartonic potentials between zero and infinity. However, I fear that the behaviors I predict could be somewhat... confusing.

So, to summarize: to calculate the tartonic potential of a given charge, from infinity to close to a group of charges... Count how many unlike charges there are, and subtract the number of like charges. Now, from that, subtract four times the product of the two numbers of unlike charges. The result is the amount of energy 'released' by taking the given charge from infinity to zero.

Now, back to that four, because it's the most arbitrary aspect of all that. I want a coefficient that works based on distance, and will keep three different charges on the points of an equilateral triangle precisely balanced. This means... two attractive forces, sixty degrees apart... Each thirty off the horizontal, equates to sqrt(3) attraction, which must be balanced by sqrt(3) repulsion from the midpoint, which is closer... In other words, four thirds as strong, per effective charge. As a result, the effective charge is just three-quarters of root three. Now, let me see... if I make it isosceles, with one vertex twice as charged... Huh. The implication I get is, that's only isosceles if I say it is. Buh? Or maybe not even then. It looks like doing things like that forces any distribution of unlike charges on the vertices of an equilateral triangle to balance. I mean, the charge groups themselves won't be stable, but the overall shape will be, if you ignore them...

(Sanity check: since most cirquons contain two charges, the addition of a charge in one place would repel the other cirqon that shares that charge. I... think that could work.) *scribble scribble scribble* Completely unworkable in that form.

Hold on, let me check if the potentials work at all, here...

Cirquons resist coming together, but that's to be expected... Now, what kind of behavior do I get if... Balance charges in a line, with altered repulsions confined to particles... Hm... let's give the equilateral triangle another shot... No... The real trick is getting a triangle with a really short leg to work nicely. Such an arrangement should cause minimal repulsion along the short side... Which implies that equilateral triangles are only stable at one length scale. I don't think "all of the necessary components and catalysts" are there for my reasoning, but this should have been obvious to me before.

This should mean that the 'ghost charges' in my way of viewing this manifest in proportion to the distribution of all appropriate charges, and that the ghost charge effect has to decrease faster than inverse square. This is because I want the effect to be strong when they're close together, and that means there needs to be very little far-away contribution... But I also want it to be finite at zero. I don't know what kind of distribution could fit those specifications. I mean, I know normal distribution would work for weighting the effect, but I'd like to look into the alternatives.

In any case, what I'd really like is for the ghosting to precisely cancel the attraction at some distance, be twice it at another, and approach zero at infinity.

(Hm... another thought about the symmetries idea is that it should be in terms of perimeter. So, twice as far apart would ghost half as much, which... doesn't work)

Anyway, I'm going to go walk for a bit, and think about this. Sorry for being all rambly.
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby Coda » Mon May 17, 2010 5:33 pm

f(x) = x/(k^x) has a curve with the desired properties -- f(0) = 0, f(inf) = 0, and there's a single maximum somewhere in between. If k = 2, f(1) = 0.5, f(4) = 0.25. The global maximum is in the neighborhood of f(1.4427). For k = e, the global maximum is at f(1) = 1/e, placing the half-power point in the neighborhood of f(2.6783).

EDIT: Updating for the sake of public record.

I misread; mwchase wanted it to be finite at f(0), not trivially zero. I propose f(x) = k/c^x; f(0) = k, f(inf) = 0, and the function is strictly decreasing with the interesting property of f(n) = c*f(n-1).

mwchase and I have discussed the matter briefly in chat; we'll see what comes of it.
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Mon May 17, 2010 8:22 pm

Well, Coda and I were brainstorming over IM, and I just worked a few things out that should have this majorly simplified:

Tartonic charge experiences simple exponential dropoff with distance. The attractive force between two different tartons is half the repulsive force between two of the same tarton, in the same positions. The three-total repulsive force is still kind of complicated, but in the two-together, one in the distance case, it works out to the equivalent of two normal repulsions, balanced against two normal attractions, for a net force of one repulsion.

Since that doesn't incorporate 'ghosting', I'm going to check that equilateral triangles force some specific length.

Hmm... I may have been imprecise in the second paragraph. If I assume that potential experiences the dropoff...

Okay, here's what I'm trying to work out: particles exist in a virtual scalar field that is calculated from the number of state-changes per time they'll undergo at a given position. Particles will follow their gradients to local minima. Okay, glad I got that out of the way; hopefully I'll stop switching back and forth in a disorienting fashion. What this means is, I get to ignore vectors for a bit. When I finish, I can just take the gradient and BAM, conservative vector field, physics probably works, happy times. So, every set of three different particles experiences the whole fun-times for repulsion. But the potentials involved are so low and so hard to change, that it doesn't matter. So, to recap what I have worked out, in this context: the potential created from two of the same particle in close proximity is an exponential decay function of their distance: states have a fixed probability to 'travel' a specific length: add that length in travel time, and the expected number of state switches will be multiplied by that probability. To generalize from two to three: two effectively demands state travel of twice the net distance between the particles. Three demands state travel along the perimeter, but may allow transfer in either direction. Next up: proximity to a different particle will impede state transfer. The potential is equivalent to a number of transfers per time, meaning that obstruction will either come in subtractive form (potentially illogical) or proportional. I'm going with proportional for now.

(The bit that's making my intuition hurt is, the potential doesn't need to be zero for stability. It just needs to hit a local minimum.)

Now, this setup implies... some function of distance, taking on values between zero and one. Multiply the result of every particle in the universe... Now, this implies that if you take the logarithm of the function, and multiply by distance squared, you'll end up with something with a finite sum. Furthermore, excluding the distances near zero should result in nearly-zero. Note that these are negative numbers there. In any case...

After much scribbling and typing, I have my function: ke^(-a/x^2). So, the overall potential comes out to
([sum over all particles the same as](ke^(-a2r))+[sum over all pairs of different particles](ke^(-a*[1 or 2]*p)))*[product over all different particles](e^(-b/r^[1 or 2]))

So, that's the scalar field. It shouldn't be too hard (really) for me to work out the form of the partial derivatives... In the morning. There's some coordinate transform stuff that I have to get exactly right. Anyway, hopefully this works.

EDITed the formula some: might be more consistent, and added some stuff I'm not sure about.

EDIT: one of the constants in the formula had to be 1. So I made it 1.
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Thu May 20, 2010 8:52 pm

Ugh... Getting this to work requires arguments from probability, but probability works differently in some respects, in Kloxkras. Well... sort of... Probability makes predictions from incomplete knowledge... The trick is, I think... Events in Kloxkras depend more on past events than events in Mundis. What I mean is... in Mundis, things usually start with a blank slate if you don't know anything. In Kloxkras, certain things have 'memories'. I need to have a way of setting things up so that these memories are relevant at higher levels instead of just lower levels.

This means that I'll be revising the formula once I figure out what all that implies. I'll be keeping the exponentials, and I thank Coda because they're a sensible primitive, but I need to work out the exact details of the combining.
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Wed May 26, 2010 9:28 pm

Okay, I may have POTENTIALLY worked out the math behind memory.

The basic idea of thermodynamics is that particles have 'states' that correspond to stuff like energy and magnetic fields and junk. In Mundis, the mechanisms of state change operate independently of each other, so the large-scale probabilities experience relatively little variation from reality. In Kloxkras, the large-scale probabilities work much differently, to the point where I must ask myself "why? Are there any laws governing this?"

I think I may have worked out those laws. First: square waves. Square waves change periodically between two distinct levels. The changes are completely regular, unlike what you'd get if you confined random noise to those two levels. So, what I propose to do is, have state-change probabilities governed by a mechanism that attempts to impose that regularity.

Now, to do that linearly, the natural choice for something flipping between two values, one negative, one positive, would be to normalize everything down to 1 and -1 so this multiplication gets less silly, and then... For each point in the past, multiply the hum up to that point by the hum up to now, then integrate. However, this produces infinity. To get around that, a decay function is required. Since everything else so far is exponential, I'll go with that. One for how far back in time each of the values getting multiplied are. Then, multiply the integral by the value the past hum had at the latest point. This should give somewhat regular behavior, through weighted probabilities. Naturally, for higher dimensions, it's necessary to worry about light-cones and space-time metrics and whatnot.

It is also midnight, and I expended most of my not-sleepy in working this out.

Now that I've got that out there, and ready to be picked over, the next issue is finally getting the concept of state laid out. I'm thinking vectors, which actually would have saved me a lot of heartache if I'd realized the potential of inner product earlier, but oh well. I believe what I have now for what I alluded to works better than what I just implied, would have. Or something. (I mean, this whole thing is still a bit dicey because state change requires coordination to ensure the universe sums to zero, and I guess I just decided these are two-vectors.) Anyway, I really really liked the hex idea I may have alluded to earlier, so I'll go with vectors constrained like that.

Note: I know this must be really confusing, because I don't fully understand the implications of what I'm suggesting, so I've got to be doing a terrible job explaining them. Anyone want clarification of anything?
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby Coda » Wed May 26, 2010 9:38 pm

Just to shed a touch of context on the subject for other readers (or myself in the future): When he says "hum" he's referring to the oscillations of state, because he was using AC power as an analogy.

Anyway, I'm not quite sure what you're getting at here. Are you suggesting that this integral produces the probability that the particle will undergo a state change? And if so, under what time scale?
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Wed May 26, 2010 9:55 pm

The idea is, the integral is multiplied by the vector value of the past-cone's latest point. Then, the vector values are all summed together to give an expected vector that the state will tend toward. As to time scale, I'm still not really sure.
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Mon Jun 07, 2010 10:40 pm

Update: I still haven't worked up the energy to figure out the math behind everything I just said, so I'm going to provisionally scratch this maybe-works setup in favor of laying out stuff I do know, to see if it's possible to work from there.

A contiguous volume (requires some formal concept of 'surface') is made up of a single chemical species, which is in a single state at any given moment. I just remembered some useful concepts from... I think it was linear algebra, not sure. So, point is, there exist a pair of units, such that any pair of measurements taken in those units uniquely describes a state. Not sure if that's actually possible, tbh. Might need some extreme weirdness. But for now, two variables. Within the possible states, there are singularities. That is, states that nearby states approach, or avoid, or are pulled toward, only to move off in a different direction entirely. (In order for the concept of ensterity to hold any weight, this probably has to actually take place within a restricted facet of a higher-dimensional space. But I digress.) I'd better refresh myself on the math involved before I make more of a fool of myself.

Okay, what I'm trying to describe is, I think, a first-order ODE. Um... Oh dear. I think I need some help getting this all together. The idea is that, near a singularity, the direction vector is A<x-a, y-b>. This is supposed to be a linear approximation to something. Unfortunately, I can't remember what. (Just smushing the approximations together seems like it'd be very unhelpful.) So, I either need to remember what these are supposed to approximate, or figure out a good way to attenuate the contribution of a singularity to the vector with distance. (I suppose I could always multiply it by e^(-r), though I can't think of a reason in this context.) I think doing the second is probably better, since I have some stuff in mind that shouldn't be possible with, say, the gradient of some multidimensional polynomial (I think that's what it was...). (It would be really awesome if the singularities had zero direction vectors at them, but I currently can't figure out how that would work.)

Perhaps, at some point, one of us will tease something sensible out of this convoluted mass of competing requirements. But anyway, that's what's been on my mind.

Update: I think I've figured out approximately how I want this to work... A state can be divided into smaller states, such that the substates are convex polygons. (Note that this requires all phase boundaries to be linear, in these units). The polygons can further be divided into slices centered on each vertex. Then, each vertex can do ODE-vector-matrix eigen-fun, until the path crosses the internal boundary. I'm not happy with this, really, since it obviously introduces discontinuities, or at least breaks in differentiability. Maybe... hmm... Along the edges, there ought to be points where the ratio between eigenvalues relates properly to the ratio between lengths. Going anywhere from there, though, would require... hmmm. I admit to some confusion over whether the points that satisfy that would necessarily form lines. Let me see... Okay... too confused right now.

On a more positive note, I think I've thought of a story that would work in Kloxkras. At some point in the future, or an alternate history, or I don't care, a cold war breaks out, in the wake of the discovery of multiversal travel. Upon the discovery of Kloxkras, the two superpowers both attempt to stake a claim. Since pretty much all earth life is incapable of surviving in Kloxkras, scientists on each side develop artificial lifeforms that self-assemble under Kloxkras physics (they clearly have a better grasp of how it all works than I do). The story picks up with human consciousnesses copied into Kloxkras-capable bodies, superficially resembling classic sci-fi robots. The two 'rival' versions of these shells exist in different bands of atmosphere, and therefore don't take well to attempts to incite a proxy war. (Since there's no efficient way to cross the borders between bands, and such. They're not too interested in building airtight tanks solely to oppress people who can't compete for the same resources without airtight tanks of their own.)

Hmmm... I bet I could do better with that concept if I actually remembered any of the Cold War.
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Wed Oct 06, 2010 9:04 pm

Yay! I worked out a way to represent math that doesn't go as insanely overboard. I don't know if it precisely works, but it's more consistent and better-defined than what I had before. I said I was going to hold off on this, but I don't feel like sleeping yet, so...

Before, Coda and I were kind of assuming that the tartonic charges corresponded to complex numbers in some fashion. This turned out not to be the case. Complex numbers act as rotations, tartonic charges multiply by reflecting.

It turns out (due to matrix shenanigans), that it's perfectly sensible to represent charges as vectors, with the same coordinates as the cube roots of unity (for the elementary charges), add together every coordinate pairwise from every vector in a point, find the gradient of the resulting field, and multiply it by the charge of a particle at that point.

Doing all that gives a complex force, along a direction. The real part of the force is... force. The complex part is torque, around the axis of the direction vector.

One thing that's definitely different is, I've resigned myself to relying on quantum effects, because, well... part of the reason QM was developed is, Bad Things happen if subatomic particles behave classically, and some of them are inherent in the physics I've already decided on.

However, so long as quantization exists, and specifically ZPE, I'll mess with the other details as I see fit. (ZPE can be seen as something that keeps molecules from collapsing. It's kind of a big deal.) (Hopefully, I got that vague statement accurate...)

So, that was a very brief sketch of what I have in mind now. I should expand on things. Which bits need the most elaboration?

EDIT: Oops. This violates conservation of momentum. I'm kind of tempted to work out the symmetry violations I need to stick in the overall structure of space, to let me do this.

Basically, what happens is... Stick two particles of different charge next to each other, and they'll torque each other. Specifically, they'll torque each other in opposite directions around opposite axes. That is, the same direction.

I dunno, maybe imaginary forces cause particles to dump torque into their surroundings.
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Sat Oct 09, 2010 10:38 pm

There's been a bunch of stuff going on over IM recently. This would be, too, but Coda is sensible, and therefore not online at the moment.

My big conflict in getting this all to work is figuring out how to reduce the role of statistics without bringing about the ultraviolet catastrophe, or something equally ghastly. To stop that, requires wavefunctions, and wavefunctions would seem to imply inherent randomness, which screws this up. But then I realized that wavefunctions don't have to describe probabilities. A key feature of QM here is that observing a particle gives us a very good idea of where it is. Suppose, however, that instead of point particles, we have linear, planar, or space-filling wavefunctions over a limited range, and a particle is presumed to exist at every point on its function, though kind of... fuzzy. I'm most inclined towards using exclusively planar (curviplanar?) wavefunctions, because they can nest, and don't necessarily have sharp edges.

I guess these would have to be able to intersect, because thinking about how to handle not letting them intersect makes my head hurt. (Or it's 1:30 am)

Not sure how they'd bond. Soap-bubble-style intersections?

(One thing that happened in chatting was, we may have invalidated most of the title. Depending on how things shake out, this could go from an Aeryx with spherical planets, to a Khex. No, really.)
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby Coda » Sun Oct 10, 2010 8:24 am

I still think it's going to be an Aeryx unless it ends up as an Arcaynia, but you may have something up your sleeve I've not picked up yet.
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Sun Oct 10, 2010 10:07 am

Well, that remains to be seen. I've been trying to guide it back, so it's likely to end up as an Aeryx once more, but until I figure out a satisfactory long-range force, that's up in the air. *loses several minutes pondering whether that counts as a pun*

I like the idea of setting up bubbles; it sort of reminds me of VSEPR. Seems like the sensible progression, arising from numbers, would be something like... 1, 2, 1-2, 1-3, 2-3, 1-2-3, 1-2-4, 1-3-4, 2-3-4, 1-2-3-4, 1-2-3-5, 1-2-4-5, 1-3-4-5, 2-3-4-5, 1-2-3-4-5... I guess the pattern there is, to go up a level, find a gap of two instead of one, and shift the gap down. If there's no 1, add a 1.

Not really satisfied with that, because it doesn't allow periodicity. Maybe use poor man's radial nodes, and just stuff extra bubbles in the center? I feel like I'm missing something in thinking about this.
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby Coda » Sun Oct 10, 2010 1:01 pm

You lost me. Not even sure what you're talking about.
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Sun Oct 10, 2010 1:37 pm

Not sure where I should start from...

The basic idea was to not have anything be just a point, because that tends to mess with physics at the atomic scale. So, instead of the way Mundis handles wave-particle duality, I figured, maybe particles could just be inherently diffuse, but within a well-defined region. In other words, these particles are two-dimensional surfaces, basically an extension of the "hollow shell" idea.

So, instead of smearing out for orbitals, particles would inflate.

What occurred to me was, instead of counting up by twos, under this system, orbitals could count up sub-orbitals one-by-one. Uh, I mean, like... They'd start with the equivalent of s, sp, sp2, sp3, and keep on adding on.

Unless there were multiple particles per orbital, this wouldn't really work for bonding, I don't think...

However... If we give them energy levels based on vibrations within the linked surfaces, then we can discard that weird hack from the end of my last post.

So, spherical would count up cosine frequencies, while... actually, the way I'm visualizing this, everything would have cosine boundary conditions, with some weird coordinate systems that wouldn't matter overly much. <- Confusing gibberish.

The point is, with the sort-of-arbitrary signs attached to everything with a number above zero, that's how bonds would form. Two bubbles with complementary energies in their valence bubbles would touch, and create an new interface connecting them.

Anyway, once I've communicated this in a sensible fashion (Makes sense to me. All you have to do is obtain a sword forged under the unlight of a lunar eclipse, dip it in the waters of the Styx, and hand it to a sword-swallower. Perfectly straightforward.) I'll move on. (I believe, no fooling this time, that this provides a workable foundation for most everything above a molecular level that I was speculating about earlier.)
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Re: Aeryx Kloxkras D(4) H(E/fb) S(?) E(-l) C{w/+Bam,Xam,Gmp}

Postby mwchase » Sat Aug 06, 2011 8:27 pm

Hey, so. On the one hand, I really want to get back to this. On the other, this thread has gotten to the point where it's hard even for me to follow it, so I honestly don't know what hope anyone else has. In an effort to make things a little easier, organization-wise, I'm moving content onto my wiki. If anyone's interested, they can take a look, poke around at my other pages, criticize my home-spun gee-whiz style (the image is my own, the trick to producing that effect was ganked from a site about random cool-looking things that can be done with CSS), etc.
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