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**Flat Earth Debate / What's the Gauss curvature of the earth?**

« **on:**November 07, 2011, 05:06:22 PM »

This question has been asked several times in other topics, but as FEers are constantly avoiding the question, I'll start a topic on its own on this. Consider the surface of the earth as 2-dimensional differentiable manifold, with metrics defined "mecanically" (think tape measure). As you might know, the Gauss curvature is an intrinsic invariant of the manifold, it can be defined without refering to any embedding space, e.g. by the formula

where C(r) is the circumference of a geodesic circle of radius r.

The question is :

This question is crucial for the discussion FET vs RET, because :

FEers, it's your turn.

* A ringworld, for example, would have a collection of isometric flat maps, but not a single one.

where C(r) is the circumference of a geodesic circle of radius r.

The question is :

**Is the Gauss curvature of the earth zero or non-zero?**This question is crucial for the discussion FET vs RET, because :

- If the Gauss curvature is zero, the metrics are euclidean. So, the earth can wholly be mapped with isometric flat maps. Given the supposed disc shape of the earth*, said flat maps can be merged to a single isometric flat map.
**Please provide such a map**. - If the Gauss curvature is not zero, as we REers claim, the Earth is not isometric to a plane. In other words,
**the earth is not flat**.

FEers, it's your turn.

* A ringworld, for example, would have a collection of isometric flat maps, but not a single one.