The first time someone explained the fourth dimension to me (referring to four dimentional euclidean space-not spacetime) they drew a picture of a tesseract and used the analogy of starships traveling through subspace at warp speed.
When ships are traveling at warp, are they traveling through euclidean space in the fourth dimension?
No.
Within the Trek universe, ships travelling at warp speed are still capable of interacting with normal three-dimensional matter (with potentially disastrous consequences). This is why ships use a series of nested static shields known as navigational deflectors to move particulate matter out of the way, tractor beams to move medium-sized objects and sensors to detect and avoid larger objects that could prove a threat.
In TNG: Best of Both Worlds, Part II, Riker orders a collision course at warp, implying that this would have a more devastating effect than a collision at sublight speeds:
RIKER : Mister Crusher, ready a collision course with the Borg ship
Wesley reacts, turns and looks for confirmation...Repeating --
RIKER You heard me. A collision course.
WESLEY : Yessir.
RIKER : Mister La Forge, prepare to go to warp power...
and in Star Trek : The Motion Picture the crew have to take evasive action to avoid hitting an asteroid while travelling at warp:
DECKER : Negative control from inertial lag will continue 22 point five seconds before forward velocity slows to sub-light speed.
ILIA Unidentified small object has been pulled into the wormhole with us, Captain! Directly ahead...!
KIRK : Forcefields up full! Put object on viewer...!
The picture is switched through two further levels of magnification, enlarging the object: an elongated, distorted, pitted asteroid, tumbling toward the Enter- prise on a collision course.
CHEKOV (punching button) Torpedoes away...!
EXT. SPACE - PAST THE ENTERPRISE AND THE ASTEROID
as the starship's photon torpedo tubes EJECT GLOWING BALLS OF LIGHT ENERGY, which seems to float toward the oncoming asteroid, almost too slowly. And in these brief seconds, the asteroid hurtles at the Enterprise, the huge pitted rock growing even larger than the ship itself. It FILLS THE SCREEN, as the photon torpedoes hit, disintegrating the asteroid into thousands of fragments. Instantly, these fragments pulverize them- selves on the ship's forward forcefield and deflector screens. The smaller pieces burn up on impact, clearly outlining the ship's forcefield barriers.INT. BRIDGE - INCLUDING MAIN VIEWER
The asteroid fragments still smashing into the force field screen; the smaller bits like SPARKLERS as im- pact heat consumes them. The larger sections bouncing away, the bridge QUIVERING as they hit. Then one final gigantic fragment strikes, the bridge SHUDDERS. And then the viewer shows only the normal SUB-WARP EFFECT: The stars ahead, relatively station- ary; a feeling of motion, but smooth, visually normal.
As regards the usefulness of the Navigational Deflector, n at least one episode of Voyager (Year of Hell, Part II) do we see the effects of it becoming inoperative (albeit in this case it's the objects that are moving at speed, not the ship)
KIM: Captain, with the deflector down those micrometeoroids are beginning to erode the hull.
JANEWAY: Emergency power to the deflector.
TUVOK: None available.
Within Star Trek on-screen canon, a vessel travelling at warp speed never leaves normal space, but manipulates "subspace fields" to warp normal space around it. There are examples of non-Federation space vehicles travelling "through" subspace as a mode of superluminal travel distinctly different from "warp drive", further supporting the notion that warp travel does not occur outside of normal space. The precise notion of what subspace is, and its role in warp travel is never really defined in on-screen canon.
For a somewhat...questionably canon answer:
In the second Star Trek TNG/XMen crossover novel (god help me) Geordi examines Nightcrawler in order to learn about how his teleportation abilities work. He eventually figures out:
Nightcrawler access a different subspace than the one that the Enterprise does when it travels through Warp.
Subspace is a separate universe from our own which has smaller dimensions even while otherwise corresponding to our own, which allows them to sort-of partially access Subspace, cruise along, and then pop back into regular space, having traveled a particular multiplier of the distance (the multiplier being the dimensional difference between the two universes). Note that when I say dimensional, I mean X,Y,Z dimensions.
As an aside, I always thought that this was a reasonable explanation for the warp speed cap. There are always claims about what a particular warp speed allows you to do, which is then broken and forgotten. To my mind, it may well be that the warp speed cap has to do with the maximum transversable speed in Subspace- perhaps the lightspeed limit in subspace is different.
The Nightcrawler Subspace is in fact a tiny fraction of the size of Regular Subspace. Geordi estimates it to be something on the order of a few kilometers in size, which is why Nightcrawler can transition more or less any distance almost instantaneously. It's not that he's directly teleporting, it's that he only needs to move a fraction of an inch in his Subspace for it to correspond to almost whatever distance he likes in Regular Space.
Nightcrawler Subspace mainly contains a lot of sulfur ions, some of which swap with his place when he accesses Subspace, leaving behind the characteristic brimstone smell. When he returns, some of it sticks to him, which brings the smell back again.
Geordi, as you may imagine, is very interested in studying the effect, with the idea of some day building a warp core around the particular effect and having ludicrously powerful starships.
