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avdolian t1_iyd11w7 wrote

Reply to comment by its-octopeople in ELI5: How is that space is “flat” yet we are able to look around the universe (up, down, left, right, etc.,) as if it were not flat? by nhabz

>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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