Visualization of Gauss-Bonnet Theorem
نویسنده
چکیده
The sum of external angles of a polygon is always constant, π 2 . There are several elementary proofs of this fact. In the similar way, there is an invariant in polyhedron that is π 4 . To see this, let us consider a regular tetrahedron as an example. Tetrahedron has four vertices. Three regular triangles gather at each vertex. Developing the tetrahedron around each vertex, there is an open angle, π . The sum of these open angles is π 4 . As another example, let us consider a cube. There are eight vertices and an open angle is 2 / π at each vertex. The sum of open angles is also π 4 . This fact is regarded as a discrete case of the famous Gauss-Bonnet theorem. Using dynamic geometry software Cabri 3D, we can easily understand a simple proof of this theorem. The key word is polar polygon in spherical geometry.
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