Electronic Proceedings of the 12^th Asian Technology Conference in Mathematics

Abstract for 12731

Visualization of Gauss-Bonnet Theorem

Authors: Yoichi Maeda

Affiliations: Tokai University

Keywords: Intermediate, Advanced

The sum of external angles of a polygon is always constant, 2Pi .

There are several elemental proofs of this fact. In the similar way,

there is an invariant in polyhedron that is 4Pi. 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, PI. The

sum of these open angles is 4PI. As another example, let us consider

a cube. There are eight vertices and an open angle is PI/2 at each

vertex. The sum of open angles is also 4PI. 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.