The Gaming Den

Tussock: We don't care about the distribution of monsters across the plane. We care about the distribution of monsters across the places on the plane the PCs will go.

If the Endless Maze is infinite in size, but the PCs are searching for a missing sage who entered it at a specific point and wandered away, then there is not a zero percent chance for the PCs to find them. The sage has been there for finite time and travels at finite speed, so there is a finite area within which the sage must be. The same is true for markings left on the walls by the sage, bits of string leftover by prior adventurers, and the location of those adventurer's corpses. You can meaningfully treat the position of all those items as binomially distributed around the entrance.

If the Endless Maze is infinite in size, but contains 666 minotaurs, the average number of minotaurs per 10' square is zero. But if one minotaur is selected at random by Baphomet each hour to be teleported to the current location of the PCs, then the PCs will encounter one minotaur per hour, despite there allegedly being no minotaurs anywhere in the maze. Further, you can meaningfully treat the odds of encountering any individual minotaur as a Bernoulli distribution.

Let's say that the Endless Maze contains an infinitely long tunnel, the center of which is the t-junction at which you must enter. There are a finite number of minotaurs distributed along this tunnel as per a normal distribution (the mean is the t-junction itself at zero and the standard deviation is one kilometer). How far into the tunnel must you travel before the probability of encountering another minotaur falls to exactly zero? If there is no distance you can travel such that this will happen, does it follow that you can encounter minotaurs throughout the entirety of the infinite tunnel? If you can encounter minotaurs throughout the entirety of the infinite tunnel, doesn't that necessitate an infinite number of minotaurs?

That's not really a paradox or anything. Normal distributions are real things that really exist in the really real world, and the universe hasn't exploded. The interesting part of this isn't the question, it's the answer. You have to travel an infinite distance before the probability of encountering a minotaur falls to zero, and because you can only ever travel a finite distance the probability of encountering a minotaur never falls to zero. Your solution has absolutely nothing to do with the normal distribution and everything to do with declaring that because individuals (in this case the PC's) can only ever travel finite distances it's easy to imagine that interesting things are explicitly contained in the finite subset they can travel to.

Your solution has so little to do with the normal distribution it works exactly as well using a uniform distribution (actually better, because the probability that the sage walked a light-year in a day is zero, not merely "slim"). Let's define M as the maximum distance anyone has ever travelled into the maze from entrance E. Let's distribute a bunch of things in that region of the maze uniformly along 0,M from E. Voila. A uniform distribution in which you can encounter things. You might not have realized it, but your solution is almost identical to declaring there are finite regions anyone cares about (those anyone can reach by modes of transport available to them) and the rest is 'here be dragons.'

But isn't that still technically infinite in size?

I think there's a place for an infinite geographical thing. It could be an infinite chaos plane with mud slaads randomly popping up all over or an infinite maze with 666 demon minotaurs walking around, but it deserves a place. I don't think it needs to be nearly as common as the infinity infinite planes of the abyss or whatever, but some things should be like that.

Grek wrote:But if one minotaur is selected at random by Baphomet each hour to be teleported to the current location of the PCs, then the PCs will encounter one minotaur per hour, despite there allegedly being no minotaurs anywhere in the maze.

This sort of thing is the only argument so far that successfully combines "non-zero percent chance of an encounter per unit area", "infinite area", and "finite potential encounters", and it only works with an agent intentionally modifying the chance where the PCs are from 0 to not 0.

If you have a 20% chance of encountering someone per hex, that means (barring the situation above where their movements respond to yours) that there is, on average, one someone to encounter per five hexes, and thus infinite someones in infinite hexes. Assuming probabilities independent of your movement (e.g. no Baphomet dicking with you), the only way to have non-0 probabilities and finite population is to have different probabilities in different areas, which is, I assume, not how the Endless Maze works, and which boils down to finite populated area surrounded by infinite emptiness.

...and all I just said is that you can't have a uniform distribution over an infinite set, which isn't new, even to this thread. Ah well.

---

As for the use of infinite things, I'm personally fond of finite-but-borderless places with funky wrapping geometry.

That got covered in the other thread from which this conversation has been half-ported.

DSMatticus wrote:Now, I do think you can use infinite to describe planes, but you have to use it like you're saying "here be dragons!" Demogorgon's domain borders the abyssal sea, and "everyone knows" that the sea itself stretches on without end. If you travel past the farthest documented islands, you will either turn back empty-handed or be torn apart by fiendish sea beasts. People have tried to explore it, and the ones that come back have yet to find anything of interest or strategic value, and a bunch don't come back at all, so everyone assumes that that boring uselessness stretches on forever. Whether or not it's actually infinite, who knows or cares? The point is that it's a blank space on the map that the DM can add things to mid-campaign without people asking "how come we didn't know that was there before?" And when you're talking about individuals with the logistical mobility of demons/devils, your explanation for uncharted territory needs to be something like "no one's ever found anything of strategic value outside the borders of the known world, so everyone just assumes it's an infinite expanse of boring, useless man-eating trees or whatever." And at that point, you've made the world effectively finite and the question of whether it's truly infinite is a question for philosophers, not adventurers.

You can declare that the regions anyone cares about are finite and the borders of the known world are either uncharted (because they are large and either no one wants to or it's a work in progress) or unchartable (because they are infinite and you can't do it even if you wanted to). But once you've declared that the playable space is finite, there's no functional difference between having the remainder be uncharted (finite) and unchartable (infinite); it's just a blank space on the map for the DM to pull back the curtains on, and the DM is only ever going to do so with a finite amount of new material regardless.

The Endless Maze isn't a finite region plus infinite blank space. It's the infinite domain of a single ruler. That's pretty batshit. The Endless Maze should just be a finite area with magic geometry such that you can walk a long hallway and end up back at the start of it and then walk that hallway backwards and end up in a different hallway entirely and everyone gets around using portals and teleportation unless they're a fiendish minotaur (who have the unique racial ability to navigate that shit).

My fault for not reading. That makes perfect sense.

DSMatticus wrote:Let's say that the Endless Maze contains an infinitely long tunnel, the center of which is the t-junction at which you must enter. There are a finite number of minotaurs distributed along this tunnel as per a normal distribution (the mean is the t-junction itself at zero and the standard deviation is one kilometer). How far into the tunnel must you travel before the probability of encountering another minotaur falls to exactly zero? If there is no distance you can travel such that this will happen, does it follow that you can encounter minotaurs throughout the entirety of the infinite tunnel? If you can encounter minotaurs throughout the entirety of the infinite tunnel, doesn't that necessitate an infinite number of minotaurs?

That's not really a paradox or anything. Normal distributions are real things that really exist in the really real world, and the universe hasn't exploded. The interesting part of this isn't the question, it's the answer. You have to travel an infinite distance before the probability of encountering a minotaur falls to zero, and because you can only ever travel a finite distance the probability of encountering a minotaur never falls to zero. Your solution has absolutely nothing to do with the normal distribution and everything to do with declaring that because individuals (in this case the PC's) can only ever travel finite distances it's easy to imagine that interesting things are explicitly contained in the finite subset they can travel to.

Your solution has so little to do with the normal distribution it works exactly as well using a uniform distribution (actually better, because the probability that the sage walked a light-year in a day is zero, not merely "slim"). Let's define M as the maximum distance anyone has ever travelled into the maze from entrance E. Let's distribute a bunch of things in that region of the maze uniformly along 0,M from E. Voila. A uniform distribution in which you can encounter things. You might not have realized it, but your solution is almost identical to declaring there are finite regions anyone cares about (those anyone can reach by modes of transport available to them) and the rest is 'here be dragons.'

Pictured above: DSM not understanding statistics.

First off, nothing here should be distributed normally. I said 'binomially' for a reason. If you say that the location of people lost in the maze is a normal distribution around the entrance, then you are saying that there is a chance (however small) that someone lost in the maze wandered 10 sextillion kilometers from the entrance in the six hours they've been lost for. That's stupid and that's why you should use the binomial distribution which covers only a finite area and actually does give probability 0 for places far enough away that your missing person literally couldn't have gone that far. All of your ranting about normally distributed anything is a giant fucking strawman.

Second, the uniform distribution is even worse than normal or binomial for this case. People moving through a maze move basically stochastically. They start at the origin and then move randomly, backtracking toward the entrance or going sideways as often as they go away from the entrance. That means that people wandering the maze are going to be concentrated around the entrance, simply by virtue of the fact that most of them aren't going to have lucked into finding no dead ends. They are not, for example, just as likely to be at the entrance as they are to be 10 miles out as they are to be 10 feet out. That's not how mazes fucking work.

Finally, it is better for the game (or at least for this one adventure prompt) if we use a non-uniform distribution and an endless maze. It serves to make a soft level boundary around different parts of the maze. If getting lost means that you need Teleport to escape, then something that has been lost in the maze for over a decade is basically gone forever as far as anyone who can't teleport is concerned and potentially retrievable by anyone who can teleport. It means that if Baphomet's personal prison fortress is 1000km into the maze, you could walk there in a week with a guide but starve to death trying to find it without one. Or, as the case may be, starve to death trying to find the exit after having escaped.

Grek wrote:Finally, it is better for the game (or at least for this one adventure prompt) if we use a non-uniform distribution and an endless maze.

Wouldn't it be better for the game if we didn't use an endless maze at all and just used a "really big" maze? Literally the only things that an actually infinite size to the maze adds to the game are:

• Stories where you conquer, explore, destroy, or depopulate a meaningful portion of the maze become impossible to tell.
• You have to have stupid math discussions involving cardinalities to talk about any mundane or demographic property of the maze.

That's fucking it. Those are both bad. There is no advantage to having the endless maze be literally limitless.

-Username17

Read the thread, shitler. I've already covered ways to deal with #1 and nobody has done #2 at all.

Grek wrote:Read the thread, shitler. I've already covered ways to deal with #1 and nobody has done #2 at all.

No U.

You haven't covered shit. All you've done is point out that you can have measurable effects in a finite area of an infinite plane. You still haven't shown any way those effects could be significant to the plane as a whole. This is the argument equivalent of putting your fingers in your ears and yodeling while people try to talk to you.

Essentially you're making Sovereign Citizen arguments. Only it's more stupid than that, since you aren't just arguing for the import of one person's opinion against the entirety of the United States, you're arguing for the import of any finite number against the entirety of literal fucking infinity.

Name one positive effect from the Endless Maze being "infinite" rather than "very big" or "of unknown size." Just one.

-Username17

If the Endless Maze is infinite in size, but has a finite number of minotaurs guarding it, each of whom has the ability to teleport to intruders, then you can have a measurable effect on the entirety of the Endless Maze by interacting with those minotaurs. If there are 666 minotaurs guarding the Maze and you kill one, there are now 665 minotaurs guarding the Maze. For the entire maze, even the parts that you will never visit.
Likewise, if there are 30 Temples of Baphomet in your country and you raid all 30 temples and smash all the portals, there are no longer any entrances to the Endless Maze in all the land. If the minotaurs were mediating trade between your country and some other country with Baphomet worshipers, raiding those 30 temples will cut off that trade lane. No part of the Endless Maze will have access to your country, for good or for evil.

As for reasons why you'd want an infinite plane: Here's two:

1. Allows an unlimited number of subplanes. If the Endless Maze is infinite in size, and two portals are opened up, both to random locations inside the Endless Maze, there is no route across the Endless Maze from one to the other. Not "we don't know how to get from one to the other" or "it is to far for us to go on foot" but, "there is no route of finite distance to take in order to go between these two places". They are as separate as two different planes, but use up the conceptual space of a single plane. Instead of letting demiplanes clutter up the cosmology, you can have all of the extradimensional spaces and Tardis Fortresses just lead to some part of the Endless Maze that has been walled off and warded against being teleported into from the outside, saving you however much conceptual space explaining how demiplanes interact with planar travel took.

2. Allows something to be lost forever. If you drop the One Ring off the side of a ship, it is lost. You will probably never find it again. But that's not the same thing as saying noone will find it again. Eventually, the ring will hit bottom and if you were desperate enough to find it, you could hire some Sahuagin to scour the bottom of the ocean near the place you lost it and bring it up. And if someone else were desperate enough, they could retrace your route and do the same thing. With an infinite plane, that is impossible. If the One Ring is cast into the Endless Maze and you don't enter through the same portal they threw it through, there is zero chance to find it no matter how many peasants you conscript to search for it for how long. Without unlimited range scrying and unlimited range teleporting, you literally can't find it. Ever. And that's a good "You must be this tall to ride" sign.

The hell are Grek and DSMatticus even arguing about? They seem to be in violent agreement that infinite spaces don't really matter as long as the things you care about are finite.

Edit:

Grek wrote:As for reasons why you'd want an infinite plane: Here's two:

1. Allows an unlimited number of subplanes. If the Endless Maze is infinite in size, and two portals are opened up, both to random locations inside the Endless Maze, there is no route across the Endless Maze from one to the other. Not "we don't know how to get from one to the other" or "it is to far for us to go on foot" but, "there is no route of finite distance to take in order to go between these two places".

And what definition of infinite are you using? You can totally have a boundless space where there's a finite distance between every two fixed points.

I was thinking about this infinite space weird math logic.

So... I get it (sort of), statistically, in order to have any real chance of finding a thing in an infinite location, the thing needs to be infinite.

However, as someone who approaches gaming as a writer and artist, rather than a mathematician, I would say that in order to have any chance of finding something in an infinite location, the plot needs to put you in the area where it is.

Like, if there is a single efreet in an infinite multiverse, sure, statistically efreets don't exist. But in actuality, there is precisely one efreet. And while wandering around your chances of running into them are basically 0, if you have magic that says "bring efreet to your location" or "go to location of efreet," it doesn't matter whether there are infinite efreets or just one, for the purposes of that magic, you just care about whether they exist at all, not exist statistically.

I mean, the same weird math logic implies that our real universe is uninhabited to our knowledge because only a single planet in all of not-infinite-but-enormously-gigantic space is inhabited.

But while you can say "oh, weird math logic, whatever, standard plot contrivances say we only care about the parts of infinity which are inhabited/have what we care about," that just makes infinite spaces doubly stupid and meaningless. If you don't care about most of infinity, then why have infinity?

Prak wrote:So... I get it (sort of), statistically, in order to have any real chance of finding a thing in an infinite location, the thing needs to be infinite.

Well, you don't need an infinite number of things to have a chance of finding a thing in an infinitely large space, as long as the distribution of those things is clustered near where you are. But if you want all points in that space to have a chance of containing a thing, then yeah, you need infinite things.

Also, in your efreet example, statistically there's one efreet, but the chance of finding it if you search in a random place is zero. But if you teleport into its harem, then the chance of finding it is just %(time efreet spends in harem).

Basically, the main? advantage of infinite space is you never find what you're looking for if you don't know where to look randomly search the entire space.

Grek wrote: If the One Ring is cast into the Endless Maze and you don't enter through the same portal they threw it through, there is zero chance to find it no matter how many peasants you conscript to search for it for how long.

That's dumb. Because the Maze itself has occupants. Infinite occupants, because the population density of the maze isn't zero. So the One Ring is going to be found by some Gelugon who lives in the maze. And the amount of time that takes requires you to do fucking cardinality math, which is totally fucked.

shiritai wrote:Basically, the main? advantage of infinite space is you never find what you're looking for if you don't know where to look randomly search the entire space.

That's dumb. A space could simply be "really big" and you'd never be able to manually search it all yourself. Anything you dump into an infinite space with a non-zero population density is being infinitely searched all the time. And then you compare orders of infinity to try to figure out whether it takes infinite time for the item to be found or not. That's stupid. You get literally nothing by bringing infinity into the business.

All we've done is confused Grek and Prak. Just like infinity confuses most other people, because infinity is in fact very confusing.

-Username17

At least I wound up on the side of "yeah, no, you don't actually want infinite planes, that's dumb."

FrankTrollman wrote:

shiritai wrote:Basically, the main? advantage of infinite space is you never find what you're looking for if you don't know where to look randomly search the entire space.

That's dumb. A space could simply be "really big" and you'd never be able to manually search it all yourself. Anything you dump into an infinite space with a non-zero population density is being infinitely searched all the time. And then you compare orders of infinity to try to figure out whether it takes infinite time for the item to be found or not. That's stupid. You get literally nothing by bringing infinity into the business.
-Username17

Right, that too. Going from a really large space to an infinite one only changes the chance from "practically zero" to "as close to zero as you can get." I completely forgot to add that.

Just as an addendum to this conversation, infinite possibilities has a strong conceptual hook to it, but the realistic playspace has to practically be limited. So for example, when you're looking at GURPS Infinite Worlds or Nexus: the Infinite City or Sliders or Star Trek or Stargate, one of the strong hooks for those settings is that each week you can visit a different planet/dimension/whatever. However, if you look at how those settings work in practice, each of them is defined by the limitations on the playspace - they can only travel using star gate or space ship, for example, or they have a finite number of points accessible determined by the gate network, or they can never visit the same world twice.

And those limitations basically define their settings; to the point that the antagonists and plot developments for the series over time are determined largely by the impact in how they interact with and/or overcome the inherent limitations. In the beginning of Stargate, the aliens used the gates too - the gamechanger came when they were sending a space ship which bypassed the gate network entirely; so too, when the protagonists discover a new cache of gate coordinates, their playspace opened up tremendously...etc., etc. One of the big issues with Star Trek was the discovery of a stable wormhole - again, bypassing current limits to open up new playspace. In GURPS Infinite Worlds, there are theoretically infinite worlds, but you can only access a certain number of them with current technology - and so can the antagonists; their main area of conflict is where their zones of interest overlap.

While I don't remember all the details at the moment, it has been shown that every mathematical proof that depends only on arbitrarily large numbers can be mapped to a proof depending on infinite quantities, and vice versa. This equivalence is not trivial, and until fairly recently there was a dispute between the majority of mathematicians, who accept the validity of proofs that use infinities, and a small but vocal minority who insisted that infinities were suspect, and demanded that all such proofs only reference arbitrarily large quantities. Turns out there's no difference in what can be proved either way.

Smeelbo

Grek wrote:First off, nothing here should be distributed normally. I said 'binomially' for a reason. If you say that the location of people lost in the maze is a normal distribution around the entrance, then you are saying that there is a chance (however small) that someone lost in the maze wandered 10 sextillion kilometers from the entrance in the six hours they've been lost for. That's stupid and that's why you should use the binomial distribution which covers only a finite area and actually does give probability 0 for places far enough away that your missing person literally couldn't have gone that far. All of your ranting about normally distributed anything is a giant fucking strawman.

Grek wrote:This is only true given a uniform distribution. If you have a normal distribution centered around some point in the Endless Maze (for example: the entrance) then your odds are not zero even if there are finite monsters.

I don't know how you forgot that. You've seriously only made two posts on this topic. Look, no matter what distribution you pick, there are two possibilities.

1) The distance at which your probability of encountering something falls to zero is infinity. Your distribution places encounters throughout an infinite region. The normal distribution you originally mentioned does this, and it does so in a way such that (because any specific individual's distance from the center of the distribution must be finite) that despite placing finite things in an infinite maze the probability of encountering them is not zero.

2) The distance at which your probability of encountering something falls to zero is finite. Your distribution places encounters throughout a finite region. This is what you are trying to suggest now. You are literally literally (seriously, literally) talking about distributing things into a finite region outside of which there happens to be infinite emptiness. You know, the exact fucking thing the people you are trying to disagree with are also talking about. There's nothing infinite about what you're doing. You've carved out a finite area of the Endless Maze and dumped things into it.

Grek wrote:Second, the uniform distribution is even worse than normal or binomial for this case. People moving through a maze move basically stochastically. They start at the origin and then move randomly, backtracking toward the entrance or going sideways as often as they go away from the entrance. That means that people wandering the maze are going to be concentrated around the entrance, simply by virtue of the fact that most of them aren't going to have lucked into finding no dead ends. They are not, for example, just as likely to be at the entrance as they are to be 10 miles out as they are to be 10 feet out. That's not how mazes fucking work.

Given that the uniform distribution is actually physically possible, and the normal distribution is not, no, no it is not worse. It's obviously not good, but it is also obviously better.

FrankTrollman wrote:That's dumb. Because the Maze itself has occupants. Infinite occupants, because the population density of the maze isn't zero. So the One Ring is going to be found by some Gelugon who lives in the maze. And the amount of time that takes requires you to do fucking cardinality math, which is totally fucked.

This right here is proof you either haven't read what I've written. What part of there being finite minotaurs guarding an infinite area by way of non-random distribution did you not get? You do not have infinite occupants. You have finite occupants teleporting around to intercept the finite intruders.

@DSM: The difference between what I'm saying and what you're saying is what happens if you open up a portal to a random part of the Endless Maze. Under my suggestion, the portal always opens up into a vast, uninhabited labyrinth that nobody else has ever explored, but then stuff starts teleporting in as it detects you. Under your suggestion, the portal sometimes opens up in a region that has been explored and sometimes it doesn't, and if you keep spamming plane shift you will eventually end up somewhere you recognize. It should be obvious why having at least one plane that works like the first is a good thing, especially when every other plane works like the second.

"There are no minotaurs in the maze, you're just wearing Cursed Underwear of Hostile Minotaur Summoning"

Grek, you are playing a game of musical chairs with your own ideas. The thing where Baphomet knows where you are in the maze is and can teleport hitsquads to you has nothing to do with the distribution of stuff in the maze! Nothing at all! The problem is not a spatial one, the chain of events that allows Baphomet to do that is wholly space-independent.

Also, fun fact: if you select a random part of the infinite maze, there is a 100% chance that that random part you'll select is infinitely far from the center of the distribution(s) you used to populate the maze. This means if you populated the maze using a normal distribution, you will be an infinite distance from the center of that distribution and the probability of encountering anything finite will be exactly zero. This means if you populated the maze using a distribution with a finite range, you will be an infinite distance from the finite region and the probability of encountering anything finite will be exactly zero. Picking a random part of the infinite maze guarantees you will get an infinitely large region within the infinite maze and that that infinitely large region will contain zero finite things regardless of the distribution of things inside the maze.

The only story you're setting yourself up to tell is the story where someone is dropped into an infinitely large, infinitely empty subset of an infinite space (and the subset is infinitely far away from anything interesting at all) and then Baphomet detects them because magic and sends an endless parade of teleporting hitsquads after them until they leave or die. That is the only story you can tell under with the "random part of an infinite maze containing finite things but Baphomet is a magical dick" idea. That is a really boring fucking story.

There are not endless hitsquads. There is a finite number of minotaurs, and eventually Baphomet is going to stop sending then after you because obviously that plan isn't working. At which point you have a portal leading into as much extradimensional space as you care for, allowing you to build a demiplane to hide out in, or a hogwarts-style castle that's bigger on the inside than the outside, to set up a portal hub using whatever spell lets you place portals non-randomly, or even just to ward against teleporting and use as the inside of a bag of holding. And later, once you're dead and gone, adventurers can loot whatever the fuck it is you build in the bit of Maze that you carved out for yourself.

None of the things you are talking about require access to an infinite space. They only require access to a remote place. Specifically, a place you have made remote by threatening to stab everyone who enters it. You can do that anywhere. As a matter of fact, it's easier to do in a place where Baphomet can't teleport minotaurs to. Seriously, your bag of holding could just be empty one day because Baphomet got lucky and the minotaur he sent stole all your shit. Your Hogwarts is going to have a chronic minotaur abduction problem, because instead of linking together a bunch of different places on the prime to get the same effect you specifically chose to link a place Baphomet can teleport minotaurs into. "Right under Baphomet's nose" is not the best place for your fortress or your belongings.