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Note before I begin: I am aware that this might be a Math-heavy question that might be better suited for the math.se

Goal: I'd like to use n-dimensional geometry as a method for effectively travelling/sending information with FTL speeds without technically violating the lightspeed limit in 4D Spacetime.

Assumptions

  1. The technology to manipulate space to works
  2. The necessary power can be generated on board of space vessels
  3. You can transfer mass into a higher and from a higher into a lower dimension

Question

How would perceived distances change in higher dimensional spaces? Can you reduce the perceived distances by using a dimensional transformation between our 3D-Space/4D Spacetime to higher dimensional space and back? If not, is there another (mathematically) feasible solution that would reduce the distances using some kind of transformation?

How would the dimensionality of the hyperspace affect the ability to shrink distances in normal space when shifting dimensions?

Notes

I prefer solutions that allow for a more or less physically different "hyperspace".

I am aware of only some spacial properties like geometrical transformation of appearance of objects that change from 4D-Space to 3D-Space to 2D-Space only by what little I can remember of geometry classes and the novels from Xishin Liu The Three-Body Problem

It is explicitly encouraged to provide the Maths for me to understand the principles, or to provide geometrical explanations.

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  • $\begingroup$ Warp Bubbles and Wormholes are well explored concepts that relate directly to what you're aiming for. $\endgroup$ – StephenG Mar 17 '17 at 15:30
  • $\begingroup$ @StephenG yes, I am aware of that. I wanted to explore something different, like basically leaving the normal spacetime, perhaps having a concept of a physical hyperspace. $\endgroup$ – Doomed Mind Mar 17 '17 at 15:32
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ONLY IF you manage to find the proper 4D distance to a parallel space to your own, where distances are naturally physically shorter. This necessitates your 4D axis being a bit warped, as otherwise you save no travel distance between the two 3D spaces.

2D demonstration

As an example, imagine traveling away from a 2D plane to a compressed mirror of the same. If you move exactly perpendicular to the plane, the distance you travel on the higher plane is exactly the same as if you had stayed on the lower. However, if your line is instead somewhat askew to allow you to meet a matching point on the higher plane, you can possibly save some travel distance by doing the bulk of your movement on said higher 2D plane, and gaining the difference between your two steps along the warped 4th axis.

Violating geometry

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  • $\begingroup$ Note that for this to work (h/(R-r)) must be less than 1, otherwise the distance is still greater than R. Second note: if the relationship you suggest here holds true across the universe, and if the relationship is linear (causing your depiction to create a cone with a tip at the end), then the maximum distance any point in the universe at all (not just observable universe) would be h where r=0. $\endgroup$ – Aaron Mar 17 '17 at 16:33
  • $\begingroup$ Note that when I said "maximum distance to any point" I meant "maximum distance to any point using this method," and my attempt was to stress that phenomenal distances could be traversed. $\endgroup$ – Aaron Mar 17 '17 at 16:36
  • $\begingroup$ Of course. The visual here is not to scale, and the particulars of how far this phenomenon holds true is left as an exercise to the writer. $\endgroup$ – thepizzaelemental Mar 17 '17 at 16:53
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    $\begingroup$ Although, it could make for a highly dramatic moment if the protagonist has to push his 4th axis drive further than anybody has ever attempted before at a time of great need and runs the risk of intersecting all of infinity at once when he hits the limit, after which no mortal soul can tell what will happen. $\endgroup$ – thepizzaelemental Mar 17 '17 at 18:42

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