It's not correct to think of an object as a collection of potential states from a lower dimension. An object could potentially be considered a collection of contiguous objects of a lower dimension. For instance, a line is not a collection of possible points. A line is a contiguous collection of actual points.

The difficulty in visualizing a hyperspatial object is partially due to the fact that the increase in dimensional complexity is exponential. A point is simple enough, and a line is not much more complex, but a 2D object can be a circle, a square, a triangle, a scribble, a letter, etc, etc, etc. Likewise, an object in three dimensions has exponentially more complexity than an object in only two. A 2D person couldn't live the way that we live. How does he breathe? There's nowhere for the air to go! If you make a cross-section of him so that he has lungs and such then his digestive system will essentially become a split that goes right through him and he'll fall to pieces. You might potentially make a jellyfish-like creature, but that's about as complex as a flatlander could get unless we consider some serious changes to biological functions. For 3D people these things aren't concerns. Our digestive systems don't cut us in half because the third dimension holds the sides together, and we can cover all the bits up by wrapping them with skin, which is quite convenient.

Now here's a real brain-bender for you.

Time is a fourth spatial dimension. Because of the way we perceive time it's very difficult to reconcile this, but consider the following:

If I draw a box on the ground around a flatlander they're trapped. If I draw a box on the ground around you then you can just step over it. Your third dimension of freedom allows you to bypass the barrier. Now what if I lock you in a room? If you were able to move freely through time then you could simply step into a time when the walls were passable (door is open, or even before the building was built) and just step out of the room.

We consider time differently because we can't move freely in time and because our perception does not allow us to see through time. We have an advantage in that we can remember certain things about the direction we call 'past', and we can make some inferences about the direction that we call 'future', but our inability to see directly into either means that we have to rely on these primitive tools that exist in the current point in time with us.

Now I know you're looking for a different take. Really what you're asking about is an additional dimension that we could interact with in a way more like the way we interact with width, depth, and height. Understand that doing so contiguously would be so disorienting for a 3D being that they would have no means of navigating. The 'laziest' means of doing this would be to have a small number of 3D planes that the person can switch between. A more dedicated means of simulating it would be allowing a person to 'rotate' into a different dimension. This would be obscenely intense in terms of environment creation, and it would extremely disorienting for the player. For instance, we can look at the 'lazy' implementation as being like Zelda's Link to the Past. A contiguous implementation would be one where Link could control how far he is between the light world and dark world with the same amount of granularity as he has in his normal motion. The problem there is that the 'in-between' space between the two places would be a gibberishy fuster-cluck that would probably not even really be navigable.

There's an indie game in production right now that does something a little between. It's called Miegakure (mee-A-gaw-koo-ray). That game works by having several planes similar to the light and dark world, and the player can rotate along an axis that makes the ana/kata dimension replace the dimension of width, so the player can rotate dimensions, then step onto another plane, then rotate back into the 'normal' 3D space of a different plane. The planes end up coming in slices of a few feet across while the player is rotated. This is closer to what it would be like, and probably the limit of what a human can really deal with, but it's not really a 'correct' simulation. In a 'correct' simulation the 'planes' would be infinite just like the width or height of the 3D universe is infinite (theoretically), and the width of each plane would be the Planck length. In short, the world would look entirely like gibberish, and even reliably picking a plane to rotate back into would be impossible.

Now here's a real brain-bender for you. Time is a fourth spatial dimension.

Not quite.

Time is orthogonal to space, and it is a valid dimension for representation purposes.

Time can be traveled. Normally we just go in a single dimension, but numerically and mentally nothing prevents us from examining past and current events or considering future events.

A game is certainly able to use in-game time as a dimension that can be traveled.

This can be an incredibly fun mechanic, as seen in games like Braid.



However, that does not make it a spatial dimension.

Time is not a spatial dimension. You cannot move an object in the spatial directions "forward, up, right, and future". That would break most of physics.

I'd argue that it wouldn't break the math. Only our understanding of it. Consider time for flatlanders. We can make a strip of 8mm film where each cell represents a single instance. If we clip them all out from the strip and lay them on top of one another in order then there's a spatial and even geometrical relationship within flatland. Their time dimension expressed in our 'extra' spatial dimension reveals this.

The only evidence we have of time being different from space is the way in which we experience time compared to the way in which we experience space. This most likely says more about us than it does about time.

Remember the twin paradox. Theoretically I can move an object forward, up, right, and future. I'd have to be really, really fast in order for it to be even slightly noticable, but it's within the realm of physics. Forward, up, right, and past would be significantly more difficult.

When you're doing special relativity, however, in certain contexts time gets a negative sign and spatial position doesn't, which suggests to me a fundamental distinction of some sort between time and space.

Isn't it that we tend to measure time in delta and we tend to measure space in absolute distances? T minus 5 seconds is 5 seconds ago, etc.

Note that it's possible to move through time without changing spatial position, but it's not possible to move through space without changing temporal position

Ah, you have to give in on one or the other of those. If we're talking about real life then it's not possible to retain a spatial dimension because space itself is moving (expanding). If we're talking theoretical then we have to allow that an object can exist in two places at the same time because of the possibility of time travel.

Isn't it that we tend to measure time in delta and we tend to measure space in absolute distances? T minus 5 seconds is 5 seconds ago, etc.

We measure space in deltas too - 5 km west of New York, etc. I'm getting at a different distinction, though. Mathematically, special relativity treats time and space slightly differently. For example, the 'pythagorean theorem' in special relativity is delta-s^2 = delta-x^2 + delta-y^2 + delta-z^2 - delta-t^2; note the sign difference between the spatial and temporal terms.

This equation does not emerge from a difference in time as a dimension, but as a difference in the way we treat time. The signage is different there because space is being measured in deltas and thus requires an offset. This is not requiring a negation of time, it's relating space to time through a negation. We could just as easily write an equation that determines the volume of a cube with a cut-out section 's' by L*W*H - sL*sW*sH. This is not a measurement of negative length, it's just a length being used in a negative mathematical relation.

If we're talking about real life then it's not possible to retain a spatial dimension because space itself is moving (expanding).

I don't know much about general relativity, but my understanding is that points in space are preserved, but the distance between them increases. In that case, there's no problem with talking about a spatial position.

But a point is not a real object. Even subatomic particles occupy more than a Planck length and are thus subject to movement through spatial expansion - at a bare minimum.

If we're talking theoretical then we have to allow that an object can exist in two places at the same time because of the possibility of time travel.

Leaving aside the issue of how this time travel works, I'd say that's not really movement. You have two instances of the same object, but they're not connected by a smooth transition - or rather, they are connected, but their smooth transition goes forward in time, then back in time through a wormhole or similar, not straight from position to position.

If you consider the dimension of time as being spatial in nature then the movement is contiguous. That's what I'm saying. If you're talking about moving backwards in time through a wormhole then the movement is still contiguous, but requires at least one additional dimension for the wormhole to function.