In a crazy way, a cone is actually *flat* except at the cone point. Of course the cone's surface bends, but it's 'intrinsically' flat. This means that if you were a tiny ant living on the cone, studying geometry, you'd see angles of triangles add up to 180°, just like on a flat table!
To see this, just notice that you can take a flat piece of paper, cut a slice out of it, and glue together the two edges of the slice to get a cone. If you'd drawn a triangle on the flat piece of paper now you have a triangle on the cone, and no angles have changed! They still sum to 180°.
This is good because in 2d space, unlike in 3d space, the equations of general relativity say that spacetime is *flat* in a vacuum. So if we have massive point particles at rest in empty 2d space, space will be flat - except at the particles, which give cone points!
(2/n)
John Carlos Baez
npub1nf…3nqe4
2026-06-11 16:09:02 UTC
in reply to nevent1q…0qpe
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"content": "In a crazy way, a cone is actually *flat* except at the cone point. Of course the cone's surface bends, but it's 'intrinsically' flat. This means that if you were a tiny ant living on the cone, studying geometry, you'd see angles of triangles add up to 180°, just like on a flat table! \n\nTo see this, just notice that you can take a flat piece of paper, cut a slice out of it, and glue together the two edges of the slice to get a cone. If you'd drawn a triangle on the flat piece of paper now you have a triangle on the cone, and no angles have changed! They still sum to 180°.\n\nThis is good because in 2d space, unlike in 3d space, the equations of general relativity say that spacetime is *flat* in a vacuum. So if we have massive point particles at rest in empty 2d space, space will be flat - except at the particles, which give cone points! \n\n(2/n)",
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