This is so cool, but its driving me crazy. I was wondering if someone could provide me with more resources that help intuit about 4d space. For example, in Miegekure, he walks through the 4th dimension to get to the other side of the wall, but thats assuming that no part of the wall extends into the forth dimension (aside from rubble). What would happen if he switched back to the normal 3 dimensions in the middle of the wall. In miegekure everything is kind of discretized (grassy area to desert area), but in reality that would be continuous. What would that actually look like, for example, the area right next to the wall? How would this work at a subatomic level, would electrons be traveling in and out of the 4th dimension? Could this explain things like action at a distance or black holes? How does the explain shared surfaces in the 4th dimension? I can't even answer a basic question like, if I were sitting in an easy chair and started looking down the 4th dimension what would happen. Since it has to share one cross section of the easy chair would it have to be simply a fatter or skinnier easy chair? But that is true for any cross section of the chair correct? The easy chair is the 3d cross section of the 4d object then what (would/could) the 3d cross section exchanging one of our spatial dimensions for the hidden one look like? How does gravity work in those 3 dimensions (2 of our spatial dimensions + 1 of the hidden dimension). Supposing the world was like this, wouldn't it be obvious if any object was extending into the 4th dimension thus we can stand to reason our world must be strictly 3 dimensional? If there were 4 dimensions since we can't see or interact with it, does it stand to reason that the spatial extent of any object doesn't extend into the 4th dimension?
For example, since the three dimensional projection of a hypersphere changes diameter, does that mean the 4d dimensional analogues of earth are just different size earths?
I also notice that in one of the miegekure videos the windmill in the new 3d space is like a cross section of the windmill but extending for a distance, Im guessing this is a product of the way the 4th dimension is discretized but Im wondering, what would that really look like if the game weren't made that way?
I found this Matt Parker lecture provided some good tools for getting your head around the fact that extra dimensions really do let you go around things, not through: https://youtu.be/1wAaI_6b9JE?t=2299
(the whole thing is worth watching, but the bit about higher dimensional visualization starts there)
I'll save people the effort and point out that the only relevant things (to the given questions) the entertaining video shows are a proper klein bottle and the nature of projection into 3D. In 4D, klein bottles don't self-intersect, just as mobius strips don't self-intersect.
It does not cover the question of putting oneself inside a wall or how to go around 4d objects, and the projections show detail that a 4d person would not see.
There are some simplifying assumptions that may help with some of your questions:
> What would happen if he switched back to the normal 3 dimensions in the middle of the wall?
The 4D cell the player is in is clear, no matter how you rotate the world around them. The cross-sectional representation of the world can misleadingly show you things you (and light) can't reach, though.
> In miegekure everything is kind of discretized (grassy area to desert area), but in reality that would be continuous.
It's as discretized as Minecraft, but with an extra dimension. In Minecraft, you can't simply go through things, just as the Miegekure won't let you.
> What would that actually look like, for example, the area right next to the wall?
All common representations of 4D space show you a cross section or x-ray projection, which, as with such things in life, show you internal details you can't see with visible light. Unfortunately, there seem to be no realistic representations available.
> if I were sitting in an easy chair and started looking down the 4th dimension what would happen?
There's no way to answer that without knowing what you're actually sitting in. An easy chair is a 3d object and a 4D you can't sit in it.
> How does gravity work in those 3 dimensions (2 of our spatial dimensions + 1 of the hidden dimension)?
Locally, remember that gravity is pretty much just a constant force in a particular direction. That works in 4D, too. If you're looking for physics answers, the answer is that most such forces would be incredibly weakened if they dissipated in an extra dimension.
> For example, since the three dimensional projection of a hypersphere changes diameter, does that mean the 4d dimensional analogues of earth are just different size earths?
This is answered in the Matt Parker video linked to in another reply. The shape doesn't change diameter - that's just a feature of the physically unrealistic but pretty representation that's most common. It could be called a shadow or projection.
> in one of the miegekure videos the windmill in the new 3d space is like a cross section of the windmill but extending for a distance
Remember that for it to exist in miegekure, it has to be a 4d object. It is one that has been chosen to look like a windmill in a particular cross-section. In other directions, the game designer simply chose how it behaved. It is, in essence, not a windmill, but something carefully made to look like one when looked at in one particular way.
