If pi were a different number, then the definitions of what constitutes a circle would be different.
You're essentially talking about a universe where no circle can exist. You try to go around something, but you always end up in a different place. That would be. . . complicated.
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Uh, your friend needs his math addressed
The diagonal is 1, then 2 averages out to 1.5 which is only slightly off the actual value of 1.414 (sqrt of 2). It is admittedly considerably less accurate for 1 single square.
It is insanely better than what 4th edition does.
While the shape is off the area is pretty close to reality. Definitely close enough that even a math geek like me has no issues with it (well, using a hex grid would clearly be a lot better)
A "circle" of 15' radius when put onto the grid has 24 squares each of which are 25 square feet. So 600 square feet.
Actual circle of 15' radius has 705 square feet. A difference of less than 20% isn't at all bad.
Those 600 square feet would represent a real circle of just under 14 feet.
Alternatively, it would represent Pi being 2.7 or so.
Breaking news: an abstraction is abstract and doesn't accurately or precisely reflect reality.
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Yeah, the 5-10-5 rule approximates physical reality much more closely than the 4th edition rule. In 4th edition, a circle, if defined as the set of all points equally distant from some center point, looks like a square in our Euclidean geometry sense of things. And if you want to draw a square, it would look like four shallow U's, with the dip of all four of them facing the middle (like a diamond where the sides are all curved inward). It's kinda fun to think of the "shape equivalences" between our universe and the 4th edition's non-euclidean universe.
But I might be a nerd.
... 4th edition's non-euclidean universe.
>t4@oR]3& /$20-; `!bFc{ \q*%**********w ~?
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... 4th edition's non-euclidean universe.>t4@oR]3& /$20-; `!bFc{ \q*%**********w ~?
In a very real sense, of course, our own universe is non-Euclidean - but Euclidean geometry is the default geometry taught, as it is valid for portions of our universe where the gravitational field is very weak - and thus fits intuitively within our everyday frame of reference.
:P
Also...
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The ElementsMain article: Euclid's Elements
The Elements are mainly a systematization of earlier knowledge of geometry. Its superiority over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost.Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, e.g., If a triangle has two equal angles, then the sides subtended by the angles are equal. The Pythagorean theorem is proved.[5]
Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as prime numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved.
Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base.
I need to read books 5, 7-10! That sounds pretty cool!
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... 4th edition's non-euclidean universe.>t4@oR]3& /$20-; `!bFc{ \q*%**********w ~?In a very real sense, of course, our own universe is non-Euclidean - but Euclidean geometry is the default geometry taught, as it is valid for portions of our universe where the gravitational field is very weak - and thus fits intuitively within our everyday frame of reference.
:P
Also...
Thanks for clearing that up!
Er...is that RAW or RAI!
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... 4th edition's non-euclidean universe.>t4@oR]3& /$20-; `!bFc{ \q*%**********w ~?In a very real sense, of course, our own universe is non-Euclidean - but Euclidean geometry is the default geometry taught, as it is valid for portions of our universe where the gravitational field is very weak - and thus fits intuitively within our everyday frame of reference.
:P
Also...
Thanks for clearing that up!
Er...is that RAW or RAI!
RAW, we live in a non-Euclidean universe, as non-intuitive as that may be. RAI presupposes the existence one to do the intending, which opens another whole can of worms.
You're essentially talking about a universe where no circle can exist.Already been done.
A large enough "circle" in minecraft would still verify mathematically that pi = 3.14159265359. . . . For pi to not equal what it is now (like the OP says), no circle could ever even be simulated. You go to try to make one and something would simply mess it up. You try to make one based on a formula, and something would simply not work out right.
Basically, it would be a completely illogical universe. I think it's possible for something like that to exist -- depending on your definition of universe. For instance, in the Matrix movies the Matrix could potentially (incorrectly) return pi = 4 for all possible calculations. Of course, that would make the effectiveness of the Matrix program quite difficult to accept.
I'm really considering rather than using squares, just instead measure distances. It's a huge mess.
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Well, actually the ratio of c/r^2 would approach pi as the radius and circumference tended toward infinity.
But thie universe is stranger than that. It's postulated that our own universe is digital rather than analogue, with all measurement being a discrete number of Planck lengths. So the universe is pixelated too - but the pixel sizes make Retina Display look like an old Gameboy game where each pixel is the size of a planet.
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That would be a difficult game to play. You'd have to be quite a ways off, I think.
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Well, actually the ratio of c/r^2 would approach pi as the radius and circumference tended toward infinity.
But thie universe is stranger than that. It's postulated that our own universe is digital rather than analogue, with all measurement being a discrete number of Planck lengths. So the universe is pixelated too - but the pixel sizes make Retina Display look like an old Gameboy game where each pixel is the size of a planet.
...
That would be a difficult game to play. You'd have to be quite a ways off, I think.
We are God's SimCity.
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Well, actually the ratio of c/r^2 would approach pi as the radius and circumference tended toward infinity.
But thie universe is stranger than that. It's postulated that our own universe is digital rather than analogue, with all measurement being a discrete number of Planck lengths. So the universe is pixelated too - but the pixel sizes make Retina Display look like an old Gameboy game where each pixel is the size of a planet.
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That would be a difficult game to play. You'd have to be quite a ways off, I think.
Now I wonder if we could use the Planck length to model pi as a function of r, where the pixellation gets less important (and thus pi gets more accurate) with increasing radius.
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Now I wonder if we could use the Planck length to model pi as a function of r, where the pixellation gets less important (and thus pi gets more accurate) with increasing radius.Well, actually the ratio of c/r^2 would approach pi as the radius and circumference tended toward infinity.
But thie universe is stranger than that. It's postulated that our own universe is digital rather than analogue, with all measurement being a discrete number of Planck lengths. So the universe is pixelated too - but the pixel sizes make Retina Display look like an old Gameboy game where each pixel is the size of a planet.
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That would be a difficult game to play. You'd have to be quite a ways off, I think.
No, we couldn't. We could model pi as a function of radius and circumference, however. And as the radius gets larger, the granularity of distance measurements gets less important.
In fact, the Planck length is so small that it has pretty much no effect on our day to day lives, and we behave as if the universe were analog instead of discrete simply because at scales we can even measure, it is.
That seems a relatively inefficient way to design a universe.
That's what they probably think of ours, too.
I'm really considering rather than using squares, just instead measure distances. It's a huge mess.
I've considered methods for doing this that don't make me feel like I'm playing Warhammer. My current thought is a clear plexiglass table and blast templates on sticks to hold under it. The problem is that, no matter how you do it, it still takes more time on average than the square or hex translation.
I prefer hexes to squares, as I think many people do, but it would require some pretty significant changes to the magic system, in particular, as you redefine lines and cones. There comes a point, however, in which you start looking at things with measured lengths of string anyway.
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RAW, we live in a non-Euclidean universe, as non-intuitive as that may be. RAI presupposes the existence one to do the intending, which opens another whole can of worms.... 4th edition's non-euclidean universe.>t4@oR]3& /$20-; `!bFc{ \q*%**********w ~?In a very real sense, of course, our own universe is non-Euclidean - but Euclidean geometry is the default geometry taught, as it is valid for portions of our universe where the gravitational field is very weak - and thus fits intuitively within our everyday frame of reference.
:P
Also...
Thanks for clearing that up!
Er...is that RAW or RAI!
Can of wormholes?
Pi doesn't determine the nature of the circle, the nature of the circle determines pi. For pi to be anything else, a circle would have to be something else;
Oh.
We worked out something in our game for 3D diagonals.
To determine range to a target that is both distant and flying, Use the long side plus half the short. It's not Pythagoras, but it's something you can calculate in your head.
Example: Evil wizard is 60 feet away, and 40 feet in the air. 60 + 40/2 = 80 feet
Actual retail distance 72.11 feet
I plotted out the 5-10-5-10 distance in pixel a few years ago, it end up making an octogon (a very pixellated one) instead of a circle.
We are God's SimCity.Well, actually the ratio of c/r^2 would approach pi as the radius and circumference tended toward infinity.
But thie universe is stranger than that. It's postulated that our own universe is digital rather than analogue, with all measurement being a discrete number of Planck lengths. So the universe is pixelated too - but the pixel sizes make Retina Display look like an old Gameboy game where each pixel is the size of a planet.
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That would be a difficult game to play. You'd have to be quite a ways off, I think.
Can we leave God out of this Fantasy discussion? o_O
I vote we shift pathfinder over to hexagonal maps.
Hex maps are messy for dungeon but they work so much better for combat. They really work well for out door encounters though. Personally no grid works best. You use a rulers and templates. So if you have movement of 30 you measure out the scale on ruler. Say each 5 feet is equal to 1 inch. Movement of 6 is 6 inches, measure it out with ruler and move to the location. Need to figure out area of affect use template. Place it on the point orgin and see who is covered by the effect.
Gridless is best if you have experienced players. Inexperienced players or players that have a hard time seeing distances need a grid.
- Gauss
Gridless is best if you have experienced players. Inexperienced players or players that have a hard time seeing distances need a grid.
- Gauss
Grab your folding rule/measure tape and play PF like Warhammer Fantasy/40K. Really good idea !? :)
TOZ nailed it.
The 5-10-5 mechanic is a mess, and creates a whole bunch of niche-issue problems.
Measuring distances is really the way to go.
Eridan, I used to play Warhammer Fantasy so I am perfectly fine with that. But that does not help new players who have a hard time visualizing distances without a grid.
New Warhammer players have the same problem. Placement of those cannonball distances without the experience to judge when you weren't allowed to measure before your declaration is difficult for new players.
In the case of new players it just bogs down gameplay until they learn distances. When I switched back to gridded play things went a lot faster.
- Gauss
Okay. I'm a little late. But I disapprove of this thread!
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New Warhammer players have the same problem. Placement of those cannonball distances without the experience to judge when you weren't allowed to measure before your declaration is difficult for new players.
But it is fun when the new players realize you don't need a tape measure to call every shot accurately.
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As soon as you can tell me what part of the 5 foot square your character starts his movement from and what part of the 5 foot square he ends his movement at, I will start to care about there being a difference in diagonal movement versus horizontal movement.
It's not the individual squares that bother me, it's the cumulative effect. I don't care if someone 5 foot steps diagonally, and the difference at a 10 foot move is minimal. But someone who move 6 squares at 1-for-1 diagonals has moved 42 feet diagonally, whereas they could only move 30 orthogonally. (if you care about ranges due to internal positioning,36-48 or 26-34ish) How does one even explain that in the context of the world? Do foot races end more quickly if the track is aligned to run Northeast?
The grid is an abstraction and characters should not be able to do simple experiments that would verify its existence as an in-game thing. If counting 5-10-5 is really super tough we need better math instruction. Should we also eliminate criticals because multiplying is hard?
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How does one even explain that in the context of the world? Do foot races end more quickly if the track is aligned to run Northeast?
No, because they don't follow an imaginary grid, they involve skill checks that are affected by base speed. (i.e. they use the chase rules)
You can't test the combat grid in-world because it doesn't exist in-world. Characters don't notice that moving one way makes them move quicker than another because to them it doesn't. You are looking at the snapshot of six seconds that helps you organize combat as being an objective representation of the game world reality when it is an abstraction.
A character who makes a move action will travel 30 feet every time. He will not suddenly be 36 feet from where he started because of an invisible grid. Any discrepancy a player notices in combat is due to the abstraction of the game and the fact that each round is blending into one another.
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How does one even explain that in the context of the world? Do foot races end more quickly if the track is aligned to run Northeast?No, because they don't follow an imaginary grid, they involve skill checks that are affected by base speed. (i.e. they use the chase rules)
You can't test the combat grid in-world because it doesn't exist in-world. Characters don't notice that moving one way makes them move quicker than another because to them it doesn't. You are looking at the snapshot of six seconds that helps you organize combat as being an objective representation of the game world reality when it is an abstraction.
A character who makes a move action will travel 30 feet every time. He will not suddenly be 36 feet from where he started because of an invisible grid. Any discrepancy a player notices in combat is due to the abstraction of the game and the fact that each round is blending into one another.
TOZ, I'm so totally saving this post of yours to reference anytime I get into this conversation. It pretty much sums this up perfectly.
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A character who makes a move action will travel 30 feet every time. He will not suddenly be 36 feet from where he started because of an invisible grid. Any discrepancy a player notices in combat is due to the abstraction of the game and the fact that each round is blending into one another.
So if I understand you, the character isn't really at the location represented by their figure on the map, because that location is certainly too far away to get to, but just somewhere near there. Do you also play that loosely with flanking and threatened squares?
I don't know why this houserule gets me so worked up. Most of my groups don't even play with a grid/maps, and the one that does we use Maptools to compute movement. 1 for 1 diagonals just really bug me on a physics level. It makes me want to do a coordinate transformation so that I'm always moving 40% faster.
Gridless is best if you have experienced players. Inexperienced players or players that have a hard time seeing distances need a grid.
- Gauss
Guilty on the latter part. I'm plenty experienced. I just can't do distances and keep track of locations in my head, I need something visual that can tell me "Okay this is where so-and-so is". Love Maptool for that. And because it does all the distance measurements for me.
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How does one even explain that in the context of the world? Do foot races end more quickly if the track is aligned to run Northeast?No, because they don't follow an imaginary grid, they involve skill checks that are affected by base speed. (i.e. they use the chase rules)
You can't test the combat grid in-world because it doesn't exist in-world. Characters don't notice that moving one way makes them move quicker than another because to them it doesn't. You are looking at the snapshot of six seconds that helps you organize combat as being an objective representation of the game world reality when it is an abstraction.
A character who makes a move action will travel 30 feet every time. He will not suddenly be 36 feet from where he started because of an invisible grid. Any discrepancy a player notices in combat is due to the abstraction of the game and the fact that each round is blending into one another.
So you're saying a human being doesn't occupy a 5 foot square? That two people standing in adjacent squares could be 8 feet apart?! You keep talking like this and some people might think RPGs don't model reality, Toz. :P
But seriously, folks: I enjoy the ease of play a grid gets you, but if you find 5-10-5 a deal breaker, you can always just play without a grid. You could even play with a ruler and a grid and just measure the diagonals.
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So if I understand you, the character isn't really at the location represented by their figure on the map, because that location is certainly too far away to get to, but just somewhere near there. Do you also play that loosely with flanking and threatened squares?
Sometimes....SOMETIMES....I play without the map.
To answer your question, no, I don't think so. But those are also a lot simpler to rule on. You've either got someone on either side of you or you don't.
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It's not the individual squares that bother me, it's the cumulative effect. I don't care if someone 5 foot steps diagonally, and the difference at a 10 foot move is minimal. But someone who move 6 squares at 1-for-1 diagonals has moved 42 feet diagonally, whereas they could only move 30 orthogonally. (if you care about ranges due to internal positioning,36-48 or 26-34ish) How does one even explain that in the context of the world? Do foot races end more quickly if the track is aligned to run Northeast?
If you use 1-for-1 and you spend your entire round moving diagonally, there is a weird +40% of movement you get, as you pointed out. But I think it's worth also pointing out that the "extra movement" you get on average is much much less than that, mainly for the following reasons:
1) Most corridors, streets, etc. in maps (whether home-made or in Paizo APs) and map tiles are orthogonal rather than diagonal because they're much easier to draw and use. Narrow corridors in particular restrict your ability to use diagonals. In a 2-square wide corridor, for example, it usually does not make sense to take more than one diagonal per round unless you're tumbling past opponents. And...
2) The first diagonal you take in a round always costs 1 sq of movement regardless of whether you use 5-10-5-10 diagonals or not, so it's more likely than not that you get a few feet of extra movement anyway. Assuming a vast featureless plain, I guess you take 50% orthogonal moves and 50% diagonal moves on average . So, on average, with 1-for-1 diagonals, you get less than 25 feet of movement for each 20 ft. of movement you spend. So it's statistically less than 25% of extra movement, not 40%.
3) After the first few rounds of combat, the combatans tend to move a lot less (often just 5-foot steps) because melee happens.
Combine these factors and I'm guessing that on average, you only get 5-10% of extra movement. Assuming 12 sq of movement per round, that's only about 1 sq per round.
If counting 5-10-5 is really super tough we need better math instruction. Should we also eliminate criticals because multiplying is hard?
Usability, or ease of use, is a positive quality in all design, whether UI, user manuals or RPGs. The problem isn't that counting 5-10-5-10 is super tough, it isn't. It's just that if you can reduce the memory load the rules impose on you (and thus improve usability), the players can focus on more important stuff like combat tactics. Any kind of extra work is always bad design.
I still like 5-10-5-10 and I use it when I play PF, but if I were to design a new RPG, I'd consider using 1-for-1 because ease of use should never be underestimated and the need to use diagonals should not be overestimated.