Gaming philosophy conundra

[Anthony]

Originally Posted by whswhs

It seems as if it still must be the case that if you represent every number on the apeirohedron as a decimal (which is how Lovewyrm chose to represent them), there must be many numbers whose representation has infinitely many digits and therefore counts as a "sword" of infinite length.

You are guaranteed arbitrary length but not infinite length. I'm actually curious what the difference between a polyhedron with aleph-0 sides and a sphere (same cardinality as the reals; if the continuum hypothesis is true this would be aleph-1) would be.

[whswhs]

Originally Posted by Anthony

You are guaranteed arbitrary length but not infinite length. I'm actually curious what the difference between a polyhedron with aleph-0 sides and a sphere (same cardinality as the reals; if the continuum hypothesis is true this would be aleph-1) would be.

If you are using a decimal representation, then so simple a fraction as 1/3 equates to 0.333 ..., which will have infinite length.

As for the difference between an apeirohedron and a sphere, I was wondering that myself. The Wikipedia article on apeirohedra describes some really odd configurations, but doesn't even mention anything that sounds spherelike.

[Anthony]

Originally Posted by whswhs

If you are using a decimal representation, then so simple a fraction as 1/3 equates to 0.333 ..., which will have infinite length.

As for the difference between an apeirohedron and a sphere, I was wondering that myself. The Wikipedia article on apeirohedra describes some really odd configurations, but doesn't even mention anything that sounds spherelike.

Well, an apeirohedron isn't a closed surface, so it can't be similar to a sphere. I'm not actually sure it's possible to define a fair (all sides equal) polyhedron with countably infinite sides (you can define an unfair one by selecting an infinite series that adds up to a finite value, such as (r^n) where r is a number between 0 and 1).

[whswhs]

Originally Posted by Anthony

Well, an apeirohedron isn't a closed surface, so it can't be similar to a sphere. I'm not actually sure it's possible to define a fair (all sides equal) polyhedron with countably infinite sides (you can define an unfair one by selecting an infinite series that adds up to a finite value, such as (r^n) where r is a number between 0 and 1).

My geometric intuition is that as the number of sides of the polyhedron goes to aleph sub null, the area of each side goes to 0, and so does the length of the boundary of each side. So it seems as if all the sides would be geometrically identical. In effect, each would be a geometric point. That seems to be a sort of regular polyhedron; you would lose the irregularities in taking the sidedness to aleph sub null.

[khorboth]

The PCs are in a trolley on a railroad designed by the GM. The players can pull a lever which move it from track A to track B. The GM has carefully planned out the railroad on track A, but track B results in chaos.

On Track A, The GM will have a great time, and the players will have a little fun.

On Track B, everyone will have a random amount of fun, but the GM will likely have less fun than on track A and the players will likely have more fun than on track A. The GM will also have great stress because he really likes his trolley railroad.

Is it morally permissible to pull the lever?

Is it morally permissible to not pull the lever?

Does the answer change when taking into account future campaigns and less need of rails?

[Anthony]

Originally Posted by whswhs

My geometric intuition is that as the number of sides of the polyhedron goes to aleph sub null, the area of each side goes to 0, and so does the length of the boundary of each side. So it seems as if all the sides would be geometrically identical. In effect, each would be a geometric point. That seems to be a sort of regular polyhedron; you would lose the irregularities in taking the sidedness to aleph sub null.

The degenerate case of a point is not aleph-null -- the number of points on a surface is the same cardinality as the set of real numbers.

[Lovewyrm]

Originally Posted by whswhs

The number 1/9, in decimal representation, is 0.11111 ..., with an infinite repetition of the digit one. So it seems it would be infinitely long. That might be a challenge to wield as a sword.

Yes, if you can't wield it, you lose the contest.

Just comparing numbers when you have a magical d∞ seems a bit mundane to me.

But yeah a real die like this might very well just a potential grenade that impales things.
P.S.:
I assume the d∞ to start as a small die that has the infinity symbol on it, and only shows the result when rolled.
I can't even imagine a die that actually has the sides needed to accomodate all the faces 'as is', otherwise.
Even my simple one kind of boggles me. Lol.

[Daigoro]

Originally Posted by Lovewyrm

Just comparing numbers when you have a magical d∞ seems a bit mundane to me.

That is the point of rolling dice though. You could just as well throw d6's at your GM if you wanted that as a game mechanic.

P.S.:
I assume the d∞ to start as a small die that has the infinity symbol on it, and only shows the result when rolled.
I can't even imagine a die that actually has the sides needed to accomodate all the faces 'as is', otherwise.
Even my simple one kind of boggles me. Lol.

It might be a sphere which holographically displays its result above it when it comes to rest. I would also presume it to give integer results, so that two very close results could be compared in finite time.

But my original point was that rolling d∞'s would have the paradoxical result that the probability of an outcome was determined by the order of the dice rolls, whereas for finitely sided dice the probability is independent of the order.

[whswhs]

Originally Posted by Anthony

The degenerate case of a point is not aleph-null -- the number of points on a surface is the same cardinality as the set of real numbers.

That's true, but I'm not sure it's relevant.

Here, for example, is the interval [0, 1]. Now we look selectively at only the points in that interval that correspond to rational numbers (it could be algebraic numbers, but they're no more numerous than rational numbers, so let's stay with the simpler case). As we go through the ordered list of rational numbers that correspond to proper fractions (x/y, where x < y and both x and y are positive), the length of the line segment occupied by each point seems to go to zero as the number of points goes to infinity.

Similarly, on a spherical surface, the area occupied by each rationally numbered point seems to go to zero. The fact that those points do not make up the entirety of the spherical surface may not make a difference.

[Agemegos]

Originally Posted by whswhs

Can you treat the interval [0, 1] as isomorphic to the interval from 0 to infinity?

Yes. Use a tan function. Or 1/x - 1