Non-Euclidean Geography

This is a continuation of my non-Euclidean architecture posts (part 1, part 2) where I'm not going to postulate anything new, just expand on an idea in part 2 about multi-directional gravity.  See also, the three-sided square hallway in the Meal of Oshregaal.

So, here's a picture of the room you're in right now, viewed from the side.

It's got a chair in it, and doors at both sides.  (You should really think about getting some more furniture.)

The arrows represent the direction and strength of gravity.  As you can plainly see, the gravity in your room points straight down in all places with a normal amount of force (9.8 m/s^2).

But here's another room.  In this room, gravity is upside down!  The doors are in the same places (down), but the chair is now sitting on the ceiling.  The gravity is uniform throughout the room, with moderate strength throughout.

But what fresh new hell is this?  It looks like this third room has both types of gravity in it.  On the west, the gravity points in the normal direction, but in the east, the gravity is inverted.

But that's still nothing too fancy, as far as D&D is concerned.

(By the way, if you stand with one foot in a normal-gravity zone and the other foot in an upside-down-gravity zone, you'll start to do cartwheels in place as one side of your body falls up and the other side of the your body falls down.)

How about this room?  Notice the difference?

In that room above this paragraph, there is no abrupt reversal of gravity.  Instead, gravity gets smaller near the room until it eventually reverses.  It's a gradual inversion, instead of a rough curtain of gravity that you pass through

Let's see how complex we can make this, shall we?

Here's a room where all the gravity points to the center.  Anything not nailed down in this room will fall to the center of the room, where it will join the chair in a big Katamari pile.

If you fell on the chair, you could hang on to it.  Stand on it.  But trying to jump from the chair to the door in the (local) ceiling would be hard--if you jumped off the chair, you'd send the chair crashing against the opposite wall, while you wouldn't move very much at all.  (This is because you weigh a lot more than the chair.  Imagine trying to jump off a planet the size of a tennis ball.)

Here's a room that's the opposite of that one: one where the gravity falls away from the center of the room.

notice the 'B' at the top

In this room, you'd walk past the chair, walk over the door on to the wall, and keep walking across the ceiling, then down the other wall, until you returned to your (much abused) chair.

BUT HERE IS THE MOST IMPORTANT PART OF THIS ESSSAY

Your journey has ups and downs in the local sense.

Do you see what happens when you walk past the chair, going left to right?  The gravity goes from pointing down-left (where you were walking leaning forward) to pointing down-right (where you must start walking leaning back).  And when you are walking while leaning back relative to the floor, it is because you are descending a hill.  Likewise, when you are walking while leaning forward relative to the floor, it is because you are walking uphill.

Therefore, the chair sits atop A LOCAL HILL.

Sorry for the caps, but if you don't understand this section, the rest of this essay isn't going to make much sense.

To think about it another way, the corners of the room are lower (locally speaking) than the walls.  Imagine that there was an immovable rod fixed in the exact center of the room, and you had tied a rope to it, and you were swinging from the rope around and around the room, so that your feet dashed across all four surfaces (floor, wall, ceiling, other wall, repeat) like the hand of a clock.

While swinging from this rope, if you wanted to touch one of the corners with your feet, you would have to descend the rope, down to the corner, since the corner is farther from the center of the room, and therefore lower (locally). 

When I say locally, I mean that if a very tiny ant were passing through, that's how it would seem to the ant.  The corners would be downhill and the chair would be on the top of a hill.  (If you were a giant that filled the whole room, this whole thing goes out the window.)

So, here is another picture of the same room.

In this diagram, the gravity is pointing down, and the floor/walls/ceiling are the ones that are bent.  

THIS DIAGRAM IS LOCALLY IDENTICAL TO THE LAST DIAGRAM.  We have fixed the wonky gravity by making the space wonky instead.

Imagine an ant, walking counterclockwise from the 'B' on the ceiling.  He walks through a valley (the top-left corner), over a hill that includes the (left) door, into another valley (the bottom-left corner), and then over a hill that includes the chair (the actual, global floor of the room), and so on.

Anyway, once you recognize that the last two diagrams depict the same space (locally, from the surface of the walls), we can move on.  (Yes, larger walking-things like giants break this rule; that's literally what makes this space non-Euclidean.)

This next room is subtly unlike the last.  See the difference?

notice the 'A' at the top

This time, the gravity is pointed toward the nearest wall instead of away from the center.

This makes a big difference, because now there are no more local hills and valleys--everything you are walking on is always going to be flat ground.

Here's another local map of the ant's journey across all four surfaces of room A.

And here's another local map of the ant's journey:

The ant doesn't notice the corners because he's really, really tiny.  Just a point in space, really.  If he had eyes as the approached the wall in the first diagram, he would see that the wall curved up at him, just as he would see that the ground was truly flat (non-Euclidean) in the second diagram.  But our and has no eyes, only six infinitesmal footsies that he puts in front of each other.

Even for a point moving along a line, the direction of gravity determines whether the point is going uphill or downhill.

Here's two more pictures of surfaces that are locally flat:

i forgot to add doors and a chair to this one

i also forgot to add doors and a chair to this one

Anyway, once you recognize that the last five images depict the same local journey for the ant, we can move on.  (In all five diagrams, the ant feels like he's walking on flat ground the whole time, since the gravity is always pointed straight towards the surface he is walking on--local down.)

By now, it should be simple to understand what this next diagram would feel like to a person walking across it.  

the slope, as it appears (gravity lines are invisible)

The gravity is normal, except for the middle, where it slants to the left.  If you were to walk across it, it would feel (but not appear) like this:

the slope, as it feels

Basically, an incline.

What would it feel like to walk across this, from the left to the right?  Well, it wouldn't look any different, because it looks like a flat plain.  But as you got into the diagonal-gravity section, it would suddenly feel like you started walking up a hill.  You might fall backwards if you were unprepared.  And once you left the diagonal-gravity section, it would be like walking on flat ground again.  And if you tripped while walking along the diagonal-gravity section and tumbling all the way to the bottom (the left edge of the diagonal-gravity zone).

You could even make the diagonal-gravity section into a sideways-gravity zone, and then you'd have to climb your way across flat ground.

How about this one:

the flat valley, globally (how it appears from far away)

Although it would look flat from a distance, once you actually walked across it, it would feel like this:

the flat valley, locally (how it feels to walk over it)

The gravity changes are gradual, rather than abrupt, so the inclines in the local valley likewise change slope gradually.  It's a rolling valley, rather than the abrupt ramp of the previous example.

Walking across the flat valley would feel exactly like walking down into a real valley; it just wouldn't look that way.  From the bottom of the valley, you'd have no trouble seeing out of the valley, as if your vision was curving.  (And this is an important point--is it the local or the global version of the valley that is accurate?  From a local standpoint, you cannot tell if it is space or gravity that is bent.)

But it's a still a valley, right?  What happens if we fill it with water?

the flat lake, locally

Locally, the flat lake feels like a normal lake.  You walk downhill, enter the water, and swim around.  But from a distance, the lake looks like an enormous water droplet standing on a flat surface.

the flat lake, globally

If you were to walk towards it, you would feel yourself going downhill as you approached it (even though the ground feels flat).  You could run 'downhill' and even jump into the wall of water (equivalent to doing a bellyflop, since gravity is propelling you into the lake surface in both cases).

One big difference would be the light.  Since the lake doesn't sit in an optical valley, it would be well illuminated from sunlight hitting the lake on the backside.  The water would be lit up.  You may see fish and freshwater whales swimming around in there.

Here's the counterpart to the flat lake:

the flat mountain, globally

the flat mountain, locally

 What if you built a building on top of the flat mountain?

Since the gravity lines are always parallel to local up and down, you'd have to build your walls parallel to them.  That means that the walls of your mountaintop castle would appear to be slanting inwards.

It would look like this from far away.

the flat mountain castle, globally

 But once you were actually inside the thing, the walls would feel vertical to you.

the flat mountain castle, locally

Who lives inside?

The Dyzantine Brothers

People will talk freely, when they are frightened.  They will tell you about the three Dyzantine brothers, who live in the castle atop the Flat Mountain.

The brothers are cursed, they say.

Only one brother ever appears at a time, since the other two are cursed to sleep.  The brothers are always caked in frost, despite the heavy jackets they wear, and their breath is always chill bellows, no matter how brightly the sun shines.

One brother is very young, another brother is a youth, and another is middle-aged and stricken with gangrene.

That is what they will tell you, but they are wrong.

The truth is this:

Once there was a wizard who sought to move in the fourth axis, and to move in a direction that was neither up, nor down, nor any of the cardinal directions.  His name was Dyzan.

But three dimensional flesh cannot move in a fourth-dimensional direction, and so he needed to give himself a four-dimensional body.

He found a way to do this involving his own lifespan.  Time would be the fourth dimension, and he would alloy it to his body.

And so Dyzan became a four-dimensional worm.

To three-dimensional human eyes, he looked exactly the same, but he had hijacked his timeline, and plucked all of his past and future selves and wedded them together.

To four-dimensional eyes, Dyzan was a worm.  He was a baby at one end, and an old man at the other.  In the middle, he was larger in circumference, being a full-grown man.  But he was most definitely a worm, being soft and pink and tapered at both ends.

Dyzan immediately had two problems.

The first is that his worm stretched from birth to death, which meant that one end of his four-dimensional body terminated in an old, dying man.

And the dying man did what dying men do, and died.

Dyzan's immune system didn't flow in the fourth dimension.  His blood and lymph were isolated in three-dimensional layers.  When dead old man at the end of Dyzan's bulk began to rot, there was nothing to slow it down.

Dyzan has been rotting from the fat end ever since.  Hence the gangrene that is rapidly consuming his four-dimensional body.

Dyzan's small end, the one that was made from the newborn Dyzan, is fine.  It's a bit larger now, having aged a couple of years.  This irks Dyzan, who prefers his baby self to remain a baby, but his fourth axis is not Time, but now Space in the fourth dimension.

The second problem is that our universe is paper-thin.  To move a short distance in the fourth-dimension is to thrust yourself into the cold void of the outer dark.  This is why Dyzan is always cold--most of his body is in the lightless, freezing nothingness that is outside our three-dimensional slice of the multiverse.

Dyzan is seeking a vehicle that will grant him passage through the outer dark, to a four-dimensional world where he can be a four-dimensional worm in peace.

He didn't really think this one through.

As an NPC, Dyzan will be more than happy to explain how dimensions and non-Euclidean spaces work.  He hopes to get people interested in such things, so that they will be more eager to help him find a way to distant, extra-dimensional shores.  (He can be a quest giver NPC.)

He doesn't have much time.  He's almost halfway rotted already, but he is a long worm, and the gangrene needs to rot through 86 years before it kills him.  The baby--baby Dyzan--will die last.  He'll probably be about four or five years old--too young to understand what is happening.

Among his treasures are an immovable rod and an unstoppable rod (whose velocity cannot be changed by anything while the switch is depressed).  (He can also be an opponent that the PCs attempt to rob.)

In his tower, Dyzan also keeps a gorbel.  He has been trying to coax the secrets of void ships out of the beast, with no success thus far.

Stats as a level 9 wizard.

*Wrap-around - Dyzan can flex his body in the fourth dimension, curling it around so that re-enters the material (three-dimensional) plane as we know it.  This functionally gives him three bodies.  These bodies share HP and spells but otherwise take actions independently.

When he uses his Wrap-around ability, it looks like a frost-bitten corpse has just teleported into the room.  The corpse then rapidly de-ages back to a living age and attacks as normally.  Conversely, if Dyzan puts his smaller, baby-sized end into our dimension, it appears as if a two-year old has just teleported into the room, who rapidly ages up to an appropriate age.

*Fourth-dimensional Shove - Dyzan can shove you into the fourth dimensional outer void that surrounds our dimension.  Treat this as a normal shove attempt.  If it is successful, you are now in the outer void, riding a four-dimensional worm made out of the entire lifespan of a desperate wizard.  It's very cold (1 hp damage per turn) and dark.  You can get back to your own dimension by climbing along the wizard until you reach the part that is currently passing through our dimension.  So if you were pushed out by a 19-year-old wizard and you know that the 23-year-old wizard is still back in the Flat Mountain castle, you would have to climb across the 20-year-old wizard, the 21-year-old wizard, and the 22-year-old wizard in order to reach the 23-year-old wizard who is in the warm, well-lit room trying to kill your friends.

In this case, a year equals 100'.  You'll be climbing along the wizard-worm in a lightless void, but you will not suffocate.  The outer dark is filled with stale air and a very low concentration of peracetic acid, you poor bastard.