Graphically represent (map) multiple spatial dimentions

Question

I would like to create a 3D map that can represent a universe with more than 3 spatial dimensions. The humans who use the map can only normally interact with 3 dimensions (and time) but in my world the universe has several more dimensions that we can travel between by use of an alien artefact. This inter-dimensional travel allows FTL travel because the other dimensions do not map 1:1 with the regular 3:

There are multiple 'sets' of other dimensions that are 'shaped' differently so that they each have advantages and disadvantages (mainly distance and rate of time) when used for getting from A to B. This is a bit like 'hyperspace' but parts of these extra dimensions are 'smaller' than realspace and other parts can be larger. The FTL works by jumping a spaceship from realspace to one set of dimensions where the distance between A and B is shorter than in realspace. The other dimensions are literally smaller as in actual real distance! To use a rubber sheet example: realspace is a 25m by 25m square, while another set of dimensions is 5m by 5m but is stretched over to 25x25. Travelling from one corner to the other in realspace would be $\sqrt{25^2 + 25^2}$ but in the other (smaller) set of dimensions it would be $\sqrt{5^2 + 5^2}$ ie: 5 times less. The other dimensions however are more complicated than this and one set of dimensions might be smaller in one place and larger in another, or time might pass at a different rate.

Specifically I would like to represent the different shapes of these other spaces in a way that would allow an average human (ie: me/my readers) to eyeball a path that crosses these dimensions without getting a headache. The map only has to display a few (undecided, 5 for now) sets of dimensions, preferably at the same time.

The map can be 3D (possibly holographic, but I prefer something that can be seen on a screen), interactive (I plan on making it interactive anyway) and/or colourful. I have some ideas, but am open to more:

• The map might show realspace with an overlay of coloured semi-transparent blobs. There are solid (but curved) lines in the blobs that show how space inside each blob is different to realspace. Like a literal rubber sheet.

• The map might show a 2 dimensional slice of each set of dimensions, with different dimensions layered on top of each other. Lines inside each sheet show the shape of space in that set of dimensions. Markers can be added to show entry and exit points and can show the relative distance between points.

So my question is: Are my 'map' schemes okay? Is there a better way of showing multidimensional hyperspace to regular people?

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I don't think this is a duplicate:

1. There are more dimensions, ie: more complicated.

2. There are different requirements, my map can be 3D and interactive and I need to show different data.

3. I'm trying to show the shape of space not the positions of stars. Stars are ridiculously trivial if you have 3 dimensions to work with, while the shape of space is not trivial.

4. The linked question inspired this one, but is different. That question concerns celestial motion, positions, etc in 2D and is not related to showing the shape of space in 3D.

Answer 1

It sounds like different floors of a building. The 2D floorplan can be supplimented by jumping to the "same" point on a different floor. This makes the 3rd dimension occur in discrete layers.

Given the normal floorplan, you can draw the grid from a different floor over it: the gridlines are unequally spaced and wavy, but you know they represent equal distances on the other floor. So if you find a floor where the lines appear far apart on the overlay in the region you are passing through, you know that floor offers a shorter travel distance.

To visualize 3D space, the overlays would be wires in 3D, all rendered using some 3D technology even if it's just a static block. Interactivelly, you could choose which plane to show.

I think it would not be used by people to navigate. The computer can find optimal paths and trade routes would be published. The 3d image that cycles through several sets of overlays would be something to amuse the passengers or teach the concept.

If I understand you correctly, what you meant by "sets of dimensions" is sets of metrics in the same space. You have one additional dimension to move between 3d spaces at any point.

To display several sets, use different colored wires. You can touch controls to make one stand out and the others fade. A movie showing your course that highlights the in-use metric and ticks off distance would make perfect sense. Now, you only have to show the in-use metric for each segment of the journey, so have the old one fade out and overlap a bit with the neighboring patch. You could see that e.g. you switch to yellow now because purple is getting closer together and yellow is farther apart here.

Answer 2

What you are trying to do is a basic challenge of topology, so I recommend looking there for techniques.

There are two major approaches I have seen to dealing with this:

• Only provide a "local" map, which is a projection of the topology onto real space (projection is a technical term in the study of topology). You can look up map-making techniques, and the difference between the different projections to see how this can fare. This would be effective for the "I'm just trying to make sense of the nearby landscape."
• Reduce the map along the lines of "what does someone care about?" For example, a travel map would only show the shortest distances between things, because why would you choose anything else? Consider the difference between a mercator projection and an "equal area" projection for an example of how this changes things.

Beyond that, consider that is is well known that humans can only visualize 3 dimensions. Going beyond that is terribly challenging and usually relies heavily on abstraction. If your humans stand a chance of intuitively grasping a higher dimensional space, your humans are going to have to learn something so utterly alien to us today that you wont grasp it, much less your readers. I'd either stick to relying on 3 space intuition, or give the characters enough "magic" skillsets that the reader doesn't mind that the characters fundamentally understand space better than they do.

Answer 3

Observations

• Our universe is (so far) known to be 4-dimensional. This universe creates alternate "copies" with variable space-time, meaning that this universe is 5-7 dimensions.

• Dimensions can be grouped together, no matter what magnitude into sets trivially comprehensible, by taking, for example, the first three dimensions and representing them as a point in a greater space.

Layers

4. Four dimensions are easily represented with a "weighted" 3D map. At given intervals, space is mapped by dots whose size represents a normalized vector of space-time.

5. A five dimensional space can be represented as a "timeline" of 4D weighted maps. An FTL drive may only be able to traverse to adjacent spaces at a time.

6. With grouping, six dimensions are marked by a 2D space with each point in that space representing a 4D space time. Each point is brighter based on the level of spacial compression in the position of the ship in that dimension.

Eventually we will hit a trend. No matter what level of dimensions are given, we will skip through the display of all in-between dimensions until the top 3 are shown with the plotted course throughout them.

Unfortunately I couldn't find any images suited to illustrate but we are talking about theoretical physics.

Answer 4

You could visualize six dimensions as "x, y, z, x impulse, y impulse, z impulse". Basically breaking down the time axis into separate changes.

But I think the main challenge here is that your "extra dimensions" are not working the way proper spaces work. Take the usual three-dimensional space with x, y, z dimensions. You can map it two-dimensionally as x, y plane, x, z plane, or y, z plane. Those planes all have the same properties.