hi can you explain more 4th dimension stuff? I think I *kinda* get it but I'm hung up on the directions thing because wouldn't a third dimension have three planes/directions in which an object could move? Like if there's 6 possible directions, wouldn't that correspond with the object's ability to move (or be moved) in either direction on an xyz, but it's curved instead for a 4th? Or is it more like object remains the same, but we see it differently according to the directions in which we view it?
I intentionally left out some information here because I didn't want to overwhelm. But now I have an excuse to explain in full!!
First let's define our coordinates:
X = North/South
Y = Up/Down
Z = East/West
W = Ana/Kata
4th dimension adds a 4th coordinate, W. We'll call W's cardinal directions Ana/Kata since that's what 4D Golf uses and it's got a nice ring to it. The key to understanding 4D is to demystify W and treat it just like you would any other coordinate.
XYZ = 3D
WXYZ = 4D
Here's what our compasses look like. Up/Down is treated separately from the 6 cardinal directions. When we turn our head side to side, our view is "locked" horizontally:
Try as we might, our brains are only able to see 3 coordinates at a time. So if we want to "see" W, the only way to do it is to substitute it for one of the other 3.
We can swap either X or Z out for W. This "rotates" our compass, putting Ana/Kata within our line of sight in place of North/South or East/West respectively. Here's what turning around in WYZ and WXY look like:
Alternatively, we can swap Y out for W; Ana/Kata take the place of Up/Down. This "unlocks" our view, enabling us to move the compass freely. The result looks like we're floating through space. This is because Up/Down is not necessarily the same as the pull of gravity. Here's what turning around in WXZ looks like:
(Worth noting that in 4D Golf, when the Up/Down arrow is turned on its side, that just means that Up/Down has been "disabled" temporarily in place of Ana/Kata. The arrows are not actually pointing at anything anymore.)
WXZ view is also called "volumetric" view. It gives us a new perspective to work with.
4D objects don't just magically morph without our input. Just like 3D objects, a stationary object is always going to be one whole unchanging shape, it's just that different viewing angles produce different appearances. All of those gifs/videos you see of 4D objects swirling around and whatnot are inherently confusing because they tend not to point this out.
Here's my friend the Klein Bottle demonstrating this with its unique topology.
Where volumetric view comes really in handy though is navigation. 4D Golf makes great use of this "coordinate swapping" mechanic in its early game to teach its importance:
Later courses expect the player to use all of these tools simultaneously in order to navigate comfortably. The key to learning is to move slowly and deliberately, and pay close attention to the compasses. The creator mercifully aligned the most challenging obstacles to cardinal directions, making it possible to normalize your movement with a bit of practice.
5D is a whole different can of worms though. Swapping coordinates becomes a compounding problem due to the sheer number of possible combinations, but it's still our only option. Remember, we can only see 3 coordinates at once. VYZ, VWZ, XVZ, XYV, VXY, etc etc......
