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44 45 eulers Formula a proof by pruning A cut diamond is a solid without indentations all of whose faces are flat polygons. Leonhard Euler 1707 1783 discovered the neat formula that relates the numbers of vertices, edges and faces of such a solid Vertices Faces Edges 2 For example, in the case of a cube we count 8 vertices, 6 faces and 12 edges and, indeed, V F E 8 6 12 2. Proof Start by opening out the network of vertices and edges to get a plane picture of the solid opposite, top in the form of a map with the same numbers of vertices, edges and faces the outside counts as one face. We notice that inserting a diagonal into a face yields a map with the same V F E second row, opposite and so insert diagonals until a map consisting solely of triangles is produced. Finally, working around the outer border of the map and eliminating one triangle at a time lower two rows, opposite we are left with a map consisting of just one triangle 3 vertices, 2 faces, 3 edges. Since at every step the value of V F E does not change, then V F E 3 2 3 2. Q.E.D. It is also not hard to show that Eulers formula holds true for any connected plane network of vertices and curve segments below.
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