The best ways to display a map of non-Euclidean 3-D space?

• May 21, 2022

• #1

greswd

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You might have seen such a 3-D map of the stars before:

And I was wondering about a 3-D map if space was non-Euclidean, what would be the best ways to display it.

To keep it simple, if you're considering elliptic space, you can consider the entire "global" map. Just like we have world maps of our entire ellipsoidal globe of planet Earth.

And if you're considering hyperbolic space, you can also consider a closed seamless hyperbolic universe in addition to an "open" hyperbolic space.

Another factor to consider is whether the map is static or interactive. Though the region of space which we'll be displaying will be so large that the non-Euclidean geometry will be obvious, it won't be like zooming into a tiny portion of space which appears relatively "flat" and Euclidean.

 

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• May 22, 2022

• #2

greswd

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greswd said:

You might have seen such a 3-D map of the stars before:

View attachment 301762And I was wondering about a 3-D map if space was non-Euclidean, what would be the best ways to display it.

To keep it simple, if you're considering elliptic space, you can consider the entire "global" map. Just like we have world maps of our entire ellipsoidal globe of planet Earth.

And if you're considering hyperbolic space, you can also consider a closed seamless hyperbolic universe in addition to an "open" hyperbolic space.

Another factor to consider is whether the map is static or interactive. Though the region of space which we'll be displaying will be so large that the non-Euclidean geometry will be obvious, it won't be like zooming into a tiny portion of space which appears relatively "flat" and Euclidean.

and maybe the best way to teach non-Euclidean geometries is to use mock maps

 

• May 22, 2022

• #3

robphy

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greswd said:

And I was wondering about a 3-D map if space was non-Euclidean, what would be the best ways to display it.

In my opinion,
the answer for any visualization depends on what information you are trying to convey.
Each representation will have varying degrees of advantages and disadvantages
for various choices of the information to be emphasized.

 

• Jun 14, 2022

• #4

greswd

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robphy said:

In my opinion,
the answer for any visualization depends on what information you are trying to convey.
Each representation will have varying degrees of advantages and disadvantages
for various choices of the information to be emphasized.

let's say we're trying to convey the locations of all the features in the universe, their locations in 3-D space

kinda like a globe (although its a 2-D surface)

 

• Jun 14, 2022

• #5

robphy

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greswd said:

let's say we're trying to convey the locations of all the features in the universe, their locations in 3-D space

kinda like a globe (although its a 2-D surface)

I have no specific suggestions.
My comment was to suggest that your question was not specific enough
since the answer depends on what you want to display
(and what you hope your viewer understands from it)
and what distortions you are willing to tolerate
(and hope that your viewer doesn't misinterpret).

https://www.google.com/search?q=vis...ta&oq=visualization+of+three+dimensional+data

 

1. How do you display a map of non-Euclidean 3-D space?

There are a few different methods for displaying a map of non-Euclidean 3-D space, including using projections, tessellations, and 3-D modeling software. Each method has its own advantages and limitations, so it's important to consider the specific needs of your project before deciding on a method.

2. What is the most accurate way to display a map of non-Euclidean 3-D space?

The most accurate way to display a map of non-Euclidean 3-D space will depend on the specific type of non-Euclidean space you are working with. For example, if you are dealing with a spherical space, using a projection such as the Mercator projection may be the most accurate. However, if you are working with a hyperbolic space, a tessellation approach may be more accurate.

3. How can I make a map of non-Euclidean 3-D space visually appealing?

There are many ways to make a map of non-Euclidean 3-D space visually appealing. Some options include using vibrant colors and textures, incorporating interactive elements, and using different types of projections or tessellations to create interesting visual effects.

4. Are there any software programs specifically designed for displaying maps of non-Euclidean 3-D space?

Yes, there are several software programs that are specifically designed for displaying maps of non-Euclidean 3-D space. Some popular options include GeoGebra, Hypernom, and HyperRogue. These programs often have built-in tools and features that make it easier to create and manipulate non-Euclidean maps.

5. Can a map of non-Euclidean 3-D space be used for navigation?

In some cases, a map of non-Euclidean 3-D space can be used for navigation, but it will depend on the specific space and the accuracy of the map. For example, a map of a spherical space may be useful for navigation, but a map of a hyperbolic space may not be as practical. It's important to carefully consider the limitations and potential errors of any non-Euclidean map before using it for navigation purposes.