Getting lost in the dimensions

[Keiji]

( administrator note: this thread was split off from http://tetraspace.alkaline.org/forum/viewtopic.php?t=12 )

get lost - very likely in 4 dimensions.

I see your point. If Fred got lost, he would just walk in a straight line to the left or right and he would find his home. If Bob got lost, he would have to use the "always turn left" rule to find his home, which fails if there are any loops in his path. If Emily got lost, she would also have to follow the "always turn left" rule, but since there are 2 more directions to move in, there is a 3x bigger chance of there being loops and the rule failing.

If you lived in the 5th dimension, there would be a 6x bigger chance of you failing than in the 3rd dimension, and for the 6th, there would be a 10x bigger chance.

I think.

[alkaline]

If Emily got lost, she would also have to follow the "always turn left" rule, but since there are 2 more directions to move in, there is a 3x bigger chance of there being loops and the rule failing.

i think if Emily followed the always turn left rule, she'd be stuck in a plane, and if that plane didn't coincide with the her destination, she would never find it. She would need a "alternate turning left and lambda" rule or something like that.

there is something called the Random Walk, where an entity starts out and moves in a random direction at every point. There is a different probability in each dimension for whether the entity will return to its starting point. In dimensions 1 and 2, the probability is 100 percent. For the third dimension it is 34%, and the fourth dimension is 19%. This page on mathworld shows the calculations if anyone was interested (uses integral calculus):

http://mathworld.wolfram.com/PolyasRandomWalkConstants.html

Brownian motion is a type of Random Walk motion, and so is diffusion.

[Keiji]

If Emily got lost, she would also have to follow the "always turn left" rule, but since there are 2 more directions to move in, there is a 3x bigger chance of there being loops and the rule failing.

i think if Emily followed the always turn left rule, she'd be stuck in a plane, and if that plane didn't coincide with the her destination, she would never find it. She would need a "alternate turning left and lambda" rule or something like that.

No she wouldn't. Imagine the only route forward involved turning ... well I don't know how you would say this but if it involved turning such that the new direction was 1,0,0,0 in wxyz, then she would be in a different plane. But the chances of finding home again are small.

[alkaline]

Well let's assume we're working with a 3d lattice, which would represent some kind of street system in Emily's universe. At any intersection, Emily has the choice of turning left, right, up, down, or going forward. In this system, four lefts lands Emily where she started, and she travelled within the confines of a single plane.

I'm not sure what kind of restriction you're placing on the routes and the intersections. It's possible that if you're in the fourth dimension, you could be oriented in such a way where you come to an intersection and it's only possible to turn up or down, and you can't turn left or right because there aren't any roads going in that direction from the intersection. If you rotated yourself, then you could orient up and down to be your new left and right.

I guess i could understand what you were trying to say if you were a little more precise with the parameters of the situation.

[Keiji]

A 4d person travelling along the orange block towards the upper-right would have the choice of turning up or turning right. Following the "always turn left" rule, as there is no left, you would turn up.

In 3d it's left -> straight -> right -> back.
In 4d it's left -> up -> straight -> down -> right -> back.

[alkaline]

so for the fourth dimension, if a traveller were faced by two paths to take, take the path counterclockwise from the other if they are less than 180[sup]o[/sup] apart.

In the directions you are giving, i'm not sure what kind of route you are describing. what are the possible choices at each point? It would help if you created an image showing the possible paths, and the paths actually taken in order to reach the destination.

At any particular intersection, if you make a turn and are now on a path, wouldn't you be forced to go straight until the next intersection, thus making the "straight" part not really a choice? Also, you can't turn right and then procede to go backwards - there is no path this direction.

[Keiji]

In 3d it's left -> straight -> right -> back.
In 4d it's left -> up -> straight -> down -> right -> back.

These are priority lists. IE:

In 3d:
If you can turn left, do so.
If you can't, then go straight on.
If you can't, then turn right.
If you can't, then turn back.

In 4d:
If you can turn left, do so.
If you can't, then turn up.
If you can't, then go straight on.
If you can't, then turn down.
If you can't, then turn right.
If you can't, then turn back.

Get it?

[alkaline]

oh! that makes so much more sense now. So, basically you're dealing with a connected graph and the entity searches through the graph using this heuristic at every node/intersection. For the image you showed, it was only one example of such an intersection - and since the "turn left" option wasn't available, it fell to the "turn up" option. It would have to turn up even if there was the option of going straight, because turning up takes precedence.

Even though i understand it now, it might be nice for others to see an example image of this heuristic being used on a graph to see an entity trying to find its way somewhere.

so how did you calculate that there is a 3x bigger chance of finding loops while in tetraspace (on a 3d surface)? It seems to me that the chance is up to the construction of the graph, not necessarily the number of dimensions - although loops are easier in some dimensions than others.

[Keiji]

There is only one way a loop can exist in the 3rd dimension, and it is this:



But, in the 4th dimension, there are 3 ways loops can exist, and these are:



So, it is 3 times more likely to encounter a loop in the 4th dimension than in the 3rd dimension.

[alkaline]

i can see the point that there are three times as many ways to make a loop in realmspace compared to planespace. But, does that mean that a randomly generated path would create three times as many loops in realmspace compared to planespace? I think there would actually be less loops, because it is less likely for a random path to coincide with itself in realmspace. This is because there are more dimensions to travel into - this is the Random Walk phenomenon, exemplified by Brownian Motion.

[Keiji]

i can see the point that there are three times as many ways to make a loop in realmspace compared to planespace.

In planespace you can't have loops - you can only move backward and forward because of gravity.

I was talking of realmspace and tetraspace.

[alkaline]

heh whoops, i was referring to a "plane" and "realm" as the surfaces in realmspace and tetraspace. So, interpret my last message as referring to the surfaces, not the spaces that the surfaces reside in.

[Keiji]

Ok then. But, we are not talking random. People wouldn't build roads going all over the place completely randomly. They would make them into blocks for quickest and most efficient transport from one place to another. So, because they have 3 times as many directions they must build in, they will, on average, build 3 times as many loops.

[alkaline]

what method could we use then to simulate the construction of roads in tetraspace? Would it be some kind of heuristical system with some kind of randomness built in? I think this would be an interesting exercise. It seems that the amount of loops formed would depend on what heuristics you use.

[Keiji]

Hm. Here's the best analogy I can think of. There's a game called TTD, which is a transport simulation. It has a town-building AI in it, and if you study it for a while, here's what it does: it will build roads perpendicular to any existing road, provided there is no road parallel to the to-be-built road right next to it.

So, if this was in 4d, it would have many extra directions to choose from, and could build more around the place. But, it would build loops just as often, and so the 3x directions will make it build 3x more loops.

[alkaline]

I did some more thinking on this and i came to the conclusion that the factor of loops in tetraspace vs realmspace depends on how you are counting.

- Method 1: loops in an n-cube.

If you are counting the number of loops in an n-cube of order m, here is how it goes:

square = 1 loop, cube = 6 loops
square order 2 = 4 loops, cube order 2 = 36 loops
square order 3 = 9 loops, cube order 3 = 108 loops

The generalized equation for realmspace is m[sup]2[/sup], while the equation for tetraspace is 3m[sup]2[/sup](m+1). Thus, tetraspace has 3(m+1) times as many loops in an n-cube of order m. This is greater than three even when the order is 1.

- Method 2: number of line segments

Construct squares and cubes by taking line segments and adding more at the same rate. After four segments, you will have a square in both realmspace and tetraspace, so at this point there is an equal number of loops. After adding three more, both have two loops. After an additional two, realmspace doesn't have a new loop but tetraspace does. By the time realmspace has four loops (a 2x2 grid), tetraspace has six loops (the faces of a cube). Tetraspace has a factor of 1.5 more loops here.

Going into higher orders, if you use 144 line segments, a grid from realmspace would have 64 loops (an 8x8 grid), while a cube from tetraspace would have 108 loops (a 3x3 cube). This is a factor of 1.685. I don't know what this approaches as the number of line segments goes to infinity. In any case, it changes as the number of segments changes.

- Method 3: line segments in infinite space

If you take an infinite grid, there is one loop for every two line segments. In an infinite cube, there are three loops for each group of three line segments. So, for six line segments, you could get three loops in a grid and six loops in an infinite cube. This is a factor of 2 times as many. This suggests to me that the limit of the factor in method three is as m goes to infinity is 2.

[alkaline]

I went ahead and worked out the limit for method 2, and indeed it is 2. If anyone ones the mathmatical proof i can provide it. Here is another method:

- Method 4: choices at an intersection

An entity on a surface in realmspace has three choices at a full intersection, so there are three possibilities for looping back. An entity in tetraspace has five choices at a full intersection, so there are five possibilities of looping back. Thus there are 5/3 = 1.6667 times as many paths that have the possibility of looping back in tetraspace vs. realmspace.

[Keiji]

An entity on a surface in realmspace has three choices at a full intersection, so there are three possibilities for looping back. An entity in tetraspace has five choices at a full intersection, so there are five possibilities of looping back. Thus there are 5/3 = 1.6667 times as many paths that have the possibility of looping back in tetraspace vs. realmspace.

Wrong, one of the choices does not loop back, so it's 4/2 = 2 times as many paths.

PWN3D!!

[alkaline]

what do you mean? does your definition of "loop back" mean that you can't use the same segment in more than one loop, or more than two loops? if it's the first, then there is 1 loop in a realmspace surface and 2 loops in a tetraspace surface, and one segment is unused in each case (factor of 2). If it is the second, then there are 2 loops in a realmspace surface and 5 loops in a tetraspace surface (factor of 2.5).

In my method, i was counting the number of paths away from the intersection not including the path that the traveller arrives from. It's *always* possible to have any path that goes out also loop back, whether or not it re-uses a path that some other loop has already "claimed".

I just thought of a couple more methods:

- Method 5: Number of shortest-length loops reachable from an intersection, direction independent. realmspace: 4 loops; tetraspace: 12 loops. This is a factor of 3.

- Method 6: Number of distinct loop-paths from an intersection, without leaving along the segment used to arrive at the intersection; direction dependent - separate loops can overlap. realmspace: 6 loops; tetraspace: 20 loops. This is a factor of 3.333.

So, the big question is: which of these methods do you use to calculate the likelihood of a loop being present?

[Keiji]

If you continue in a straight line, that will not loop! DUH!

[alkaline]

i'm not sure how that obvious fact factors into your argument - if you choose to go straight at the intersection you are studying, that doesn't mean you'll go straight at the intersection right after that - you could turn left there, then turn left two more times after that and arrive at the original intersection again. Thus, you have made a loop after choosing to go straight.

Something i haven't calculated yet is the number of outward paths vs number of loop possibilities from any particular intersection. That would have to work under the assumption that any path exiting something that is a "minimal loop" would never return to the system and thus never return to the original intersection. For example, choosing to go straight twice - if you wanted to loop back, it wouldn't be a minimal loop (which is crossing three segments). Or, if you went left then straight.