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This is quote from Hawking's book:

The surface of the Earth is what is called a two-dimensional space. That is, you can move on the surface of the Earth in two directions at right angles to each other: you can move north–south or east–west. But of course there is a third direction at right angles to these two and that is up or down. In other words the surface of the Earth exists in three-dimensional space. The three-dimensional space is flat."

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closed as unclear what you're asking by Aaron Stevens, WillO, ahemmetter, Jon Custer, ZeroTheHero Nov 30 '18 at 0:31

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    $\begingroup$ Your quote literally says what it means by a flat three dimensional space. $\endgroup$ – Aaron Stevens Nov 29 '18 at 3:52
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    $\begingroup$ The three-dimensional space is flat. That is to say it obeys Euclidean geometry. That's already the definition, which you can also easily find by checking the Wikipedia article on Flatness. $\endgroup$ – ahemmetter Nov 29 '18 at 14:54
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After Hawking wrote "The three-dimensional space is flat", he explained what that means: "That is to say it obeys Euclidean geometry." And as an example of what it means to obey Euclidean geometry, he gives the example "The angles of a triangle add up to 180 degrees". In a non-flat space, the angles of a triangle can add up to more than 180 degrees, or less than 180 degrees!

When discussing flat and curved spaces, physicists often think in terms of the "metric" of the space, which determines the distance between infinitesimally close points. In a flat 3D space, the metric is

$$ds^2=dx^2+dy^2+dz^2,$$

which is just a 3D version of Pythagoras' Theorem. But in a curved 3D space, the metric cannot be put into this simple form.

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    $\begingroup$ The surface of Earth is two-dimensional; you can specify a point on it with two coordinates, such as latitude and longitude. Space is three-dimensional, so it’s a totally different thing. You can have a curved 2D surface in a flat 3D space, or a curved 2D surface in a curved 3D space. $\endgroup$ – G. Smith Nov 29 '18 at 5:12
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The term “flat” in this context does not mean 2 dimensional. It means “Euclidean”. A flat manifold is one where the interior angles of a triangle sum to 180 degrees, parallel lines never converge or diverge, and if you parallel transport a vector in a loop it lines back up on top of itself.

In contrast, the surface of a sphere is a curved 2 dimensional space. If you start at the equator and go due north to the pole and turn 90 degrees, then walk due south to the equator and turn 90 degrees and walk due west until you get back to your starting point you will wind up 90 degrees from the direction you started. So the triangle you walked had 270 degrees, not 180. This is what it means that he sphere is not flat.

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