This is the writeup I promised would be interesting over a year ago, at the bottom of my flatland writeup. This writeup assumes you have read that. If you have not, you might get confused...

Flatland is a peculiar thing. it has two dimensions when we're used to having three, meaning we have a dimension left to do thought experiments with. This is what makes flatland such a good help for understanding dimensions. Flatland does have a third dimension, however, although it's not a spatial one (length, width, height), it's the dimension of time. But what if we were to imagine that as our third spatial dimension?

Take the flatland world standing upright1, like a piece of paper on its edge, and do a mental simulation of the world living its life. Now, instead of having everything move with time, try viewing the timeline as the third dimension, running into the background.

Not following me? Try this: imagine a square in flatland moving to the right for one second, then to the left for another second. At the start it's at position 1. A second later it's at position 2. Take the frame of position 2, and put it a few cm behind the frame of position one. Connect all the corners of the square with lines, this should give you a shape with 3 dimensions, resembling a cube. Place the frame of the next second, where the square was moving to the left, behind the preceding 2 seconds. Now connect the corners of frame 3 to those of frame 2. Then you should end up with roughly the next picture:

    |'-,  '-,
    |   '-,  '-,
____|   ,-'  ,-'|___Frame 3
     ;-'  ,-'   |
____|    |      |__________ Frame 2
    |    |   ,-'
____|____|,-'_______Frame 1

You can keep doing this for the whole of flatland, for every second. This would create a 3d universe. You could imagine walking around in there, looking at everything that moved. Things like a rolling wheel with spokes would create infinitely spiraling tubes on the inside. A collapsing bridge would be a marvelous sight to behold, standing straight at point one, lying in crumbles a few meters further, with stripes where small particles broke off and fell on the ground. It would seem as if nothing was moving, yet you could follow the entire life of a flatlander from beginning to end simply by following his 3d shape from start to end.

Things to think about:

  • Time doesn't actually work the same as spatial dimensions. Of course we can't go back in time, while we can go left and right, but there's another difference. A building cannot be supported by a pillar that stood there a minute ago or will stand there in a minute, while it is possible to place a pillar every meter and have your building remain solid in the spaces in-between.
  • Of course, flatlanders will have no use for spoked wheels, since their bicycles, if they have any, will probably consist of solid wheels which aren't held by an axle (since that's impossible). Instead, they'd be held by a shell which is around most of the wheel, so it can't fall out. There would be a lot of friction between the shell and the wheel though, so it might not pay off to use a bicycle in flatland. A better idea might be using a sled.
  • A bridge would probably not be like we imagine it is, either. Since there won't be any rivers to cross over , a bridge could only go over a chasm, either with or without water, or over a lake. If it bridges a chasm, it will probably just be a simple rope, since in 2 dimensions, you can't fall off of a rope. If it bridges over a lake, it will totally close off the lake, potentially bringing rain problems. This can be prevented by making a bridge that doesn't curve upward, but downward, and making a hole in the middle for water to drip through. Since by making a hole you lose the connection between both sides of the bridge, your bridge needs to be pretty strong.
  • A 2-dimensional world has lots of problems you generally don't think about, like the ones above, because they lack a third spatial dimension. You can think of lots of them, and devise solutions for them. If you want a mansion, for example, you will have a hard time keeping it from not breaking when you open the door, since you're actually removing the support of one wall. You can't do the trick I described in the flatland writeup, since you want a really large mansion. A solution to this is making a house with 4 doors, 2 on the sides and 2 at a third and two third of the house. Like that, the center of gravity of the house will always be between 2 walls when you open a door. You can even open certain combinations of 2 doors at once (sides, middles, or alternating). The roof would be in arm's reach, so you could climb through a hole to the next floor.
I'm getting a little distracted, there's so much to think about. Some more relevant things to note are these:
  • When you imagine yourself walking around in the three-dimensional world of space and time, you're actually creating another timeline for you to walk around, separate from the one the flatlanders are using.
  • If you try to move anything in the world, it gets a little complicated. If you were to move a part of the square mentioned in the beginning, you would move a part of the entire 3d shape, including in the future, but also in the past. To the flatlander, this would be like experiencing an unexplained force which moved them into a certain direction. This would change other events too. You'd actually change the timeline, and see the world 'in the future' change.
  • Interacting with the world would bring more problems, for example: can the flatlanders see you? When you're on their sheet of time, they should be able to see you, but the problem is you're not on that sheet of time forever. Other things are, that's why you don't see any movement in the world. If you imagine them being able to see you, you would change the timeline just by walking around, since you'd probably scare the flatlanders. You could imagine this similar to pointing a webcam at a computer screen with the live image of the webcam, seeing the image swirling off into the distance, constantly updating. When you move out of the timesheet into the future, things would revert back to the things they were before, since for them, you weren't there yet.
  • Personally, I imagine them as not being able to see me, since like that you can move things around, without it reverting to the way it was before.
  • Oolong adds that a 4-dimensional being would be able to walk around in our 3d world plus timeline just like we can do with flatland. That's quite hard to visualize though.

That's all I can tell you about 2 dimensions and a timeline. The rest has to come from your imagination. (Although you can always /msg me.)

1. The one described in my writeup in the flatland node.

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