= Solution
Form the <dual graph> of the finite <triangulation>: its vertices are the triangles, and two vertices are joined when their triangles share an edge. Connectedness of the <surface> makes this graph connected. Choose a <spanning tree> and begin with one disjoint triangle. Attach the remaining triangles along the tree edges, one at a time, always joining a new triangle along one boundary edge. At every step the result is a topological disk, so the final union is a polygonal disk.
The edges not used for these attachments still occur twice on the boundary, once for each incident triangle. Identify each pair as in the original <triangulation>. The result recovers the original <surface>. This proves \b[a compact connected closed <surface> is a polygon with its boundary edges identified in pairs]. Closedness is the convention here; a boundary would leave some edges unpaired.
Back to article page
