How is the 3-d space an approximation of Euclidean Geometry?

In summary: So if there was a universe with much more curvature than ours, then the approximation wouldn't be flat locally, right?The local approximation for space is always flat, just as the surface of Earth appears flat to us.
  • #1
ask_LXXXVI
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I would like to know the basic experimental observations or the logic which prove that the 3-d space which we inhabit is a close approximation of Euclidean Geometry. is it because parallel lines don't appear to converge or diverge? But how is this established, as we can't draw perfect straight lines. Do we observe behavior of parallel rays of light in vacuum or something similar. I am not concerned about the 4-d space-time . Just the good ol' 3-d space . :smile: .
I am a beginner so it would be good if you direct me to websites where related info is given .
 
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  • #2
Triangle angles don't sum up to 180 deg if geometry isn't flat. But I think that best evidence comes from here: http://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.html"
 
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  • #3
Ok , I can appreciate that the behaviour of light is used to measure the geometry of the universe.

Is it because of the curvature of our universe that the triangles which we draw on paper have angles which sum up to ~180 degree ?

Had our universe been of, let's say , elliptical curvature would angles of triangles drawn on paper sum to less than 180 degrees ?

Or is the curvature simply a measure of the way light travels through free space?
 
  • #4
ask_LXXXVI said:
Is it because of the curvature of our universe that the triangles which we draw on paper have angles which sum up to ~180 degree ?

No, it is because local approximation is always flat, just as surface of Earth appears flat to us.
 
  • #5
Calimero said:
No, it is because local approximation is always flat, just as surface of Earth appears flat to us.

How is it "always" flat ?

Would the local approximation be flat even in a universe with much high curvature than ours ?(I am assuming a hypothetical universe just for sake of understanding).

Or were you implying our particular situation?
 
  • #6
Well, we obviously live in sufficiently large universe that locally it appears flat to us. Now we know even more. We know that we can't detect any curvature even if look at the largest possible observable patch.
 
  • #7
okay.
 

What is Euclidean Geometry?

Euclidean Geometry is a branch of mathematics that deals with the properties and relationships of flat or two-dimensional shapes and figures.

How does Euclidean Geometry relate to 3-d space?

Euclidean Geometry is the foundation of 3-d space. It is used to describe the properties and relationships of three-dimensional shapes and figures.

Why is 3-d space an approximation of Euclidean Geometry?

3-d space is an approximation of Euclidean Geometry because it is impossible to perfectly represent three-dimensional shapes and figures in a two-dimensional space. Therefore, some of the properties and relationships may be slightly altered or distorted.

What are some limitations of using Euclidean Geometry to describe 3-d space?

Euclidean Geometry is limited in its ability to describe curved or irregular shapes in 3-d space. It also does not account for the effects of gravity or other physical forces.

Are there any situations where Euclidean Geometry does not apply to 3-d space?

Yes, Euclidean Geometry does not apply to non-Euclidean spaces, such as curved surfaces or spaces with a negative curvature. It also does not accurately describe the effects of relativity in physics.

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