Jutt
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Back on the topology issue. Say you'd move straight away from the centre of the Eighth along its surface, eventually passing the rim. Where will you end up? Can you keep going indefinitely without returning to where you started, or does the universe wrap around in this direction as well?

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Ignoring the possibility of getting burned up by the sun or falling into holes/running into mountains, I think it would be possible to keep going forward indefinitely, since the rim of the Eighth "wraps around" to the central axis (where the Sun is located). Once you wrapped around, you would find yourself at the edge of the central axis on Sun Island (not crushed or roasted as in real life) with the exact same header, and find yourself in an infinite loop.

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Syntax
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Hence the expression "to go one eighth of a moebius"

coppro
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If you go off the edge of the Eighth, you wrap around to the center, where the eight yous are compressed into a very, very tiny point. If you head to Sun Island, you'll see the eight other yous, and have a very hard time actually getting to the central axis because there would be seven other yous all simultaneously trying to get to the center. If you succeeded, you would wind up scattered across the edge of the Eighth, the exception being the Fegundo which has got the borders figured out.

kyevan
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So one surface of the Eighth would just be a torus, I think. (stretch it into a square for ease of mental manipulation, and it would connect top to bottom and left to right.) Obviously that's a terrible model for what actually happens, but it's closer than the pie slice metaphor.

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No - if it were a torus in the normal sense of the word, then the centre would have finite width and objects passing through the centre would appear at the rim unchanged. I'm pretty sure that holds if the Eighth is topologically equivalent to a torus as well. If you try to deform it into a plane torus then you're mapping a point (the centre) onto a line, which is highly dodgy mathematically speaking.

Plus it can't be fully embedded in three dimensional space the way a torus can, which I believe means it must be a 3-manifold or higher - can someone with more topology knowledge weigh in?

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coppro
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I don't think the Eighth can be defined in terms of an actual topology because of the way a single point is connected to the entirety of an edge (that and the fact that the eightfold replication makes it even more confusing)

Jutt
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The fact that the Eighth is only one eighth of a disk, is topologically not a problem (just stretch it around the centre to make a full disk; there is no replication).
Also, is there any official confirmation about the centre point being connected with the rim? I find this highly unlikely, and it would be impossible with a standard Euclidean topology.

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Some of the posts in this topic may be helpful. In particular, Erik and agaricus have a debate where Erik qualifies some of the Eighth's features and agaricus complains about the unscientificness of the whole thing

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coppro
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Jutt wrote:
The fact that the Eighth is only one eighth of a disk, is topologically not a problem (just stretch it around the centre to make a full disk; there is no replication).

I thought the same thing until I remembered that as you near the top you encounter eight versions of yourself, which is (presumably) how it was deduced that the Eighth was in fact one eighth.

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Chaco wrote:
Some of the posts in this topic may be helpful. In particular, Erik and agaricus have a debate where Erik qualifies some of the Eighth's features and agaricus complains about the unscientificness of the whole thing

Ah, thanks, That topic explains a lot.
If we assume that the outer rim is connected to a small circular edge in the centre, not a singular point, it's topologically not that bad at all. The sun would be actually be a toroid instead of a ball, but would indeed, as desribed in the other topic, be visible as a ring along the outer rim. And the entire universe would simply be a 3-torus .

coppro wrote:
I thought the same thing until I remembered that as you near the top you encounter eight versions of yourself, which is (presumably) how it was deduced that the Eighth was in fact one eighth.

There aren't actually eight versions of you. There's only one, you just see it eight times.



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This post continues a discussion from here.

Jutt wrote:

agaricus5 wrote:
The Eighth does not wrap from the top to the bottom [...].

If you like, read some of the info (secreted beacuse of spoilers) in this topic. I argued there that the Eighth's universe is actually a 3-torus.

Actually, I'm having some trouble visualising how that would work (I'm no good with topology). How does a 3-torus account for changes in vertical altitude? Surely an infinite stack of toruses, one for each surface plane, would be more accurate?

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Hmm... I'm thinking that the Eighth slice is like a piece of pizza symmetrically folded in half, but the piece of pizza is very distorted, so it's more like an inside out ball...

Oh! Imagine inside the Earth. Imagine you live on the inside of the crust, which faces the core, not the sun. All of the Eight's surface land mass exists on the inside of the crust, but when you delve into a dungeon, you delve towards the sun. When you reach
, which supposedly exists just a few meters away from the outer surface of the earth, the space bending theory bends space around so that you are now in the middle of the core of the Earth. Ascending upwards makes you travel away from the center of the Earth, until you reach the mantle of the Earth, which serves as the sky in this metaphor. When will gravity shift so that you're falling towards the crust of the Earth (the Eighther surface land mass) as opposed to the core? Gravity simply doesn't exist.

Of course this raises the question of how to escape the sky and travel into space. Obviously, traveling high into the sky from either end of the Eighth will cause you to literally travel to the ground again. Stop imagining that the total Eighth is like a big ball and start imagining it as two sheets of paper. To escape the space bend, you have to travel between the two sheets until the sheets of paper no longer cover you on the top or bottom. This raises the question of where's the sun?

Jebus, Caravel! Why'd you make Beethro's home planets' physics so complicated?

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agaricus5 wrote:
Actually, I'm having some trouble visualising how that would work (I'm no good with topology). How does a 3-torus account for changes in vertical altitude? Surely an infinite stack of toruses, one for each surface plane, would be more accurate?

We have established that the surface of the Eighth is a torus, which leaves the question what happens in the vertical direction.
Now if you read this topic, it was revealed that if you travel from the underside of the Eighth downwards you'd eventually reach the surface of the Eighth again, i.e. the universe wraps around in the vertical direction. So all vertical paths in the universe are topologically equivalent to circles. Therefore the universes topology is the product topology of a torus and a circle, which is a 3-torus.

Now technically you could also think of it as an infinite stack of Eighths which are all exactly the same, just like you could imagine the surface of the Eighth as a full circle comprised of 8 identical pie shaped parts. It may be easier to imagine the universe this way, but it behaves not any differently from a 3-torus. In my opinion makes more sense to represent the universe such that everything exists once and only once.

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Jutt wrote:

agaricus5 wrote:
Actually, I'm having some trouble visualising how that would work (I'm no good with topology). How does a 3-torus account for changes in vertical altitude? Surely an infinite stack of toruses, one for each surface plane, would be more accurate?

We have established that the surface of the Eighth is a torus, which leaves the question what happens in the vertical direction.
Now if you read this topic, it was revealed that if you travel from the underside of the Eighth downwards you'd eventually reach the surface of the Eighth again, i.e. the universe wraps around in the vertical direction. So all vertical paths in the universe are topologically equivalent to circles. Therefore the universes topology is the product topology of a torus and a circle, which is a 3-torus.

Now technically you could also think of it as an infinite stack of Eighths which are all exactly the same, just like you could imagine the surface of the Eighth as a full circle comprised of 8 identical pie shaped parts. It may be easier to imagine the universe this way, but it behaves not any differently from a 3-torus. In my opinion makes more sense to represent the universe such that everything exists once and only once.

Ok; I see - that makes it a lot clearer.

As I said, I'm not very good at topology; I was having some trouble imagining what a 3-torus might look like.

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skell
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Jutt wrote:
In my opinion makes more sense to represent the universe such that everything exists once and only once.

Wasn't it Erik who stated that everything exists eight times? Unless I am missing something, it actually makes a difference if things exist in 'pairs of eight' or just as a single entity. Perhaps not for usual/any Eighth inhabitant (we can't know for sure) but maybe for outer beings?

And this makes me wonder. Is Eighth a universe or just a planet in Eighthian universe (not neccesarily Eighthual)? Because if the former then ok, but if the latter, what if an asteroid hit any part of eight, wouldn't that disrupt the 'balance'?

*scratches his head, confused*

Edit:

And this makes me think more - if you leave north when traveling south and vice-versa, how come you can possibly be able to see 8 of you when reaching the center? Behind the sun you will see the other side of the Pie-part, not the "top" of the opposite one.
To put it otherwise, if you are in Pie-part #1 and you leave it south, you should enter Pie-Part #4 from the south.

Doh, I just confused myself even more...

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[Last edited by skell at 05-08-2009 11:37 PM]

Jutt
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skell wrote:

Jutt wrote:
In my opinion makes more sense to represent the universe such that everything exists once and only once.

Wasn't it Erik who stated that everything exists eight times? Unless I am missing something, it actually makes a difference if things exist in 'pairs of eight' or just as a single entity. Perhaps not for usual/any Eighth inhabitant (we can't know for sure) but maybe for outer beings?

I don't have enough knowledge of forum history to know everything Erik has contributed to the debate.
The multiplicity could indeed make a difference for outer beings, but as we have only information about the Eighth itself, their existence would be entirely speculative. Plus they would have to be at least 4-dimensional beings to be able to observe the Eighth's structure, which is… freaky.

And this makes me wonder. Is Eighth a universe or just a planet in Eighthian universe (not neccesarily Eighthual)? Because if the former then ok, but if the latter, what if an asteroid hit any part of eight, wouldn't that disrupt the 'balance'?

As far as I can see the universe contains one 'planet', the Eighth, which incidentally spans the entire universe in two directions. And there is a sun, but how that works is a whole story of its own.

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Actually, while thinking about the nature of the 3-torus, I think the Eighth appears to be an example of a perpetual motion machine.

Although "gravity" as we might understand it doesn't exist, a force of some sort is clearly in action that pulls objects on the Eighth (although apparently not the Eighth slice as a whole) in a single direction - down. This force must accelerate objects to at least some degree (otherwise if you dropped anything, it wouldn't fall), so if you were to drill a hole directly through the Eighth, and drop something into it, it will fall for ever, continually accelerating (and in this case, dissipating energy into the air as it does so).

It's weird, but I wonder if there is a way to explain away the obvious issue of how energy is seemingly being created out of nowhere.

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skell
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agaricus5 wrote:
This force must accelerate objects to at least some degree (otherwise if you dropped anything, it wouldn't fall), so if you were to drill a hole directly through the Eighth, and drop something into it, it will fall for ever, continually accelerating (and in this case, dissipating energy into the air as it does so).

Perhaps its acceleration would stop once it fell below Eighth? Perhaps it is the Eighth's bottom/'outerns' which does the pulling here? But how and why then... I wonder... Nevermind that, I just read the whole topic and it This post is clearly mistaken.

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[Last edited by skell at 05-09-2009 12:45 AM]

Syntax
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I see it as a sphere with a hollow centre (also in the shape of a sphere). The "eighth" space is the one between the layers of the 2 spheres ie walking inside the football with the middle bit missing.

What differs from the norm though, is that each straight path follows a trajectory which has a centre halfway between each layer. Walking straight North would eventually lead back to where you first were, but changing course partway (left or right) would put you onto a new arc, leading to a new circular route which would lead to a new location once you were back at the "bottom layer" - every path is infinite and repeating, but each variation goes through different places.

That's how I imagine it.

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Can't we just agree the Eighth doesn't work in our universe. Stop trying to picture it. It works because the story says it works.

What the Eighth can't be (for fun):

A torus/doughnut shape: If you walk off the south, crunch, if you somehow walk off the north, atomized.

A sphere: See above

A cube: See above

What the Eighth could be:

A non-Eculdian shape: Think about it. The weird wrap around and north/south death makes this possible.

Encased in Space-time anomalies: Going through one send you out the other. However the north/south one is a bit weird. This also explains why no space objects reach the Eighth, they get rifted around it.

Something made to be novel: This is very probably it. It doesn't make sense because Erik never put much thought about how this would work in, he just made it because it was cool.

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Universe--likely not. The mathiverse on the other hand, has this as a possibility. Thus, ceasing the attempt to picture it should be frowned upon at best. Just because it's "different" doesn't mean it's not worth thinking about.

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hyperme wrote:
What the Eighth can't be (for fun):

A torus/doughnut shape: If you walk off the south, crunch, if you somehow walk off the north, atomized.

As I said, it could very well be (and is) a torus regardless... stretching does nothing topology-wise, and it has been proven that the north associates to the south. Fegundoes, sir, fegundoes.

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How about this?

Just as the Eighth has "reflections" to both sides, it should contain a very tiny reflection of itself inside Sun Island (hell, it could be inside the -sun- for all we know) and similarly be surrounded by a gigantic reflection of itself. If you had a good enough telescope and could see through the sun and fog, you'd be able to see the gigantic back of your own head.

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The Eighth may be a torus, but darn it, I want to say something else!

I got the idea that the Eighth might be a hollow cone, where Rooted Hold and Lowest Point are within the cone, looking at the other side of the cone and seeing the outside world.

Probably wrong, but it's just a few stray thoughts coming from the dark back of my mind. Feel free to comment on either this idea or the other ideas here.

EDIT: Just for clarity, the outside of the cone is the surface of the Eighth, the "Aboveground". The crust of the cone is the Beneath. The inside is empty space around the center of gravity.

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Here's mt theory, if the eighth is a planet that is perfectly normal besides the fact that it's shaped like a cone, the surfacers would live on the side of the cone and the "second sky" is on the bottom.

Spoiler in the secret text in case you think it's just some theory about going over the bottom edge.



Well who said the physics had to make perfect sense? It's one of those stories where the author makes some weird world, writes some history that may be interesting to us but might just be boring schoolwork to the characters, and then makes something happen that sends the whole world into some sort of danger. Think lord of the rings, or world of warcraft, or pretty much any story that takes place on an alien world,(possibly a paralell universe) with no knowledge of earth.

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Some thoughts from a lover of topology:

First off, I could have sworn that I had read Caravel-official stuff on The Eighth which suggested The Edge was topologically glued to The Tip. While this is topologically possible, The Eighth would necessarily be an unusual space. Let's discuss its properties.

Consider points right at The Edge. If such boundary points existed, they would have different neighborhoods. However, being glued to The Tip implies they would need to be topologically indistinguishable from The Tip, and thus from each other. This would make the space not only a non-metric space, but a non-metrizable space as well, and physics as we know it would be impossible. This implies that The Edge can have no boundary points.

Now consider The Tip. Is it a point in the space? If not, the space is incomplete: it is missing a location which should exist by its distance metric. This implies that it is not locally compact, and it does not form a complete metric. It would still be completely metrizable, however, so it's less problematic. However, this "gap" in space still makes physics very difficult to model properly and introduces other mathematical anomalies. For this reason I will assume The Tip is indeed a point in the space. Note: this assumption is not absolutely necessary, but it makes things much simpler.

To review: The Tip is a point in space which has The Edge in its entirety in every one of its neighborhoods, and there are no points near The Edge with this property. Points near The Edge get closer and closer to The Tip: The Edge is an open boundary, and The Tip is a closed boundary (at a single point).

With this topology, the space would be surprisingly regular, and in most ways no different from the Euclidean space you and I are familiar with. Assuming we model the space with unique points, then all points are topologically distinguishable. It is a regular space, meaning every point not in the neighborhood of a closed area has a neighborhood also not in that area. It is a perfectly normal space, meaning every two closed areas can be separated by a function that runs between them (since the world wraps around, the function would actually need to encircle one area without touching it or the other area). These things together make it a "perfectly normal Hausdorff" space, and the added metric makes it a complete metric space. This comports with what we know of mathematics in The Eighth.

The metric on The Eighth is non-Euclidean globally, but it is Euclidean locally except in two cases. The first case is any area which includes The Tip as a non-boundary point. The second case is any area which admits a looping path laterally. In both of these cases, the metric becomes the minimum of the possible paths assuming the glued connections exist.

In some cases, this is very weird and non-intuitive. If you're a dimensionless being who is located one meter from The Edge, and want to get to a spot that is located one meter from The Edge and ten meters west of you, the fastest way to get there is to head South to The Tip, then North to that spot (since you can travel from The Tip to any spot on the edge). This isn't possible physically, but it is a mathematical feature of the space.

So what about the physical implications? They're stranger than you might think...

First, let's see what someone sees at The Tip. As you approach The Tip, you see seven other copies of yourself also approaching. The Tip itself appears as a faint vertical line stretching up into the Heavens and down to the Earth, like the refraction of a prism.

Trying to reach The Tip is impossible, because you bump into yourself. It's like trying to squeeze yourself into a narrow corner with impenetrable walls...it can't be done. You could push your hand into the crook and pinch your fingers on themselves, but you can't go inside. Air molecules could not pass through either. Elementary particles could not pass through normally, not even light.

While quantum tunneling would allow particles to travel through the barrier, this would never occur (statistically speaking).

So what about The Edge? Can't stuff pass through there? Well, not exactly. Remember, The Tip is essentially a one-dimensional singularity stretching vertically. No particles can fit through it, except by quantum tunneling. But that's the main difference between The Tip and The Edge: particles can tunnel from The Edge to The Tip, but not vice-versa. However, this phenomenon would not lead to any observable phenomena if the environment at The Edge were at a pressure and temperature consistent with any Earth climate (or anything remotely near one).

What would The Edge look like? It would look like a bright, shimmering curtain of light. In actuality, The Edge is pure black, but as light "hits" it, it gets arbitrarily close to every particle near the wall, and with near-perfect certainty will be absorbed by a valence electron of a molecule in the air at The Edge. (On rare occasion it will tunnel through to The Tip, but this is so rare it will have no detectable effect.)

Matter would not be able to pass through The Edge, assuming the atmosphere is consistent there. Getting close to the Edge introduces, through the spatially-warped distance, an electromagnetic pressure force from all of the air against The Edge. It would feel like touching hard, warm steel.

If The Edge were surrounded by vacuum rather than air, light would still be reflected, but for different reasons. Without the air, the light-curtain would not be there and instead it would be a dark gray wall (blacker as ambient light fades). Matter would behave much the same way, as the pushback would not be from other matter, but from the matter pushing against itself as it approaches the singularity. Individual particles would be more likely to tunnel through, but this shouldn't likely produce any macroscopic effects.

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This is a very interesting discussion to me, Trickster. You describe how particles cannot actually travel from the Edge to the Tip, assuming it is a "one-dimensional singularity". What about waves, like photons?

What if the Tip were not a singularity, but an incredibly small space, possibly a millimeter in width?

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Few things survive the journey through the tip, anyway, as the Sun is there. It should also be noted that no, the tip is probably not a singularity, else there would be a measurable contraction of spacetime in Akandia, and probably as far south as Tueno.

The site page on fegundos seems to be inaccessible, or I would mention them.

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Jatopian wrote:
Few things survive the journey through the tip, anyway, as the Sun is there.

True, but the sun travels up and down next to the tip as it rises and sets. So, I guess if you were far enough away from the sun when you passed through, it might be possible to survive.

If you have no chance to survive...make your time. Ha ha ha.

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