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its-octopeople t1_iyczn2m wrote

Flat doesn't mean 2-dimensional. Things can be 2-dimensional and not flat (the surface of a sphere, for example), or 3-dimensional and flat (like the space we live in appears to be). In a flat space, parallel lines stay parallel. In a non-flat space, they don't

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nhabz OP t1_iyczvps wrote

Thank you. The 2D non-flat/3D flat were good examples.

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LOSTandCONFUSEDinMAY t1_iyd6qt5 wrote

We use the word flat because we haven't got a specific word for 3D nor a general word to use so we use the closet one available which is 'flat' a word traditionally for 2D. Not curved/a curvature of 0, might be a less confusing way to describe the universe.

A similar issue arises when talking about 'size' in higher dimensions. An object in 1D is a line and its size is called length, in 2D you have a surface with an area, in 3D your object has a volume.

But in 4D we haven't got a unique name for its size and call it volume again. and it doesn't help that like every 3D object has a 2D surface area, a 4D object has a 3D volume. Therefore every 4D object has two measurements called volume (though its specified surface or interior volume).

And get gets worse as you go up in dimensions because every dimension you go up you get an additional type of volume to consider but the highest dimension measurement is usually considered its size.

TLDR: dimensions are confusing and we haven't got enough words in english to properly or easily talk about it.

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avdolian t1_iyd11w7 wrote

>In a non-flat space, they don't

You can draw parallel lines on a sphere and they stay parallel

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SoulWager t1_iyd1nvi wrote

>You can draw parallel lines on a sphere and they stay parallel.

From the perspective of two people walking on those lines, at least one of them will need to constantly be turning left or right. If you're both walking parallel and straight, your paths will intersect 1/4 the way around the sphere.

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avdolian t1_iyd26av wrote

If my friend stays one metre to my north and we both walk around the globe one of us will have walked further but they wouldn't have to constantly turn and we would never cross.

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stools_in_your_blood t1_iyd2ye5 wrote

The one who isn't walking on a great circle (i.e. an "equator" of the sphere) is constantly turning away from you, in the sense that if they imagine their path laid out in front of them, it appears to curve to one side.

Put another way, imagine doing this not with walking but with cars with the steering fixed dead ahead. Try it with toy cars and a basketball if you have those objects handy.

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SoulWager t1_iyd370q wrote

The only line of latitude that's not bending left or right is the equator. For an extreme example, imagine walking the line of latitude one meter from one of the poles.

A line of latitude one meter from the equator is still bending, just not as much.

Lines of longitude do not bend left or right, and they all intersect at the poles, even though they're all parallel at the equator.

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its-octopeople t1_iyd7xq4 wrote

Okay, not parallel lines but parallel geodesic curves. I don't know if I can ELI5 geodesics, but I'll have a go

Okay, you can't take a straight line on a sphere, obviously. But if you walked around the equator, most people would a agree you'd walked pretty much a straight path. However, if you walked a 1 meter circle around the North pole, no-one would recognise that as a straight path, even though they're both lines of latitude and they're both parallel

What's the difference? Pick any two points on the equator. The shortest path between them (staying on the sphere), also follows the equator. For the small circle you don't have that property - you can find a shorter curve that cuts through the interior of the circle. Curves that have this shortest distance property are called geodesics

So the statement about flatness should be; two geodesics - that is, two shortest distance curves - that are parallel at some point, stay parallel their whole lengths

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