course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Geometry |
IB |
26 |
|
Paper 3, Section II, E |
2019 |
Define a geodesic triangulation of an abstract closed smooth surface. Define the Euler number of a triangulation, and state the Gauss-Bonnet theorem for closed smooth surfaces. Given a vertex in a triangulation, its valency is defined to be the number of edges incident at that vertex.
(a) Given a triangulation of the torus, show that the average valency of a vertex of the triangulation is 6 .
(b) Consider a triangulation of the sphere.
(i) Show that the average valency of a vertex is strictly less than 6 .
(ii) A triangulation can be subdivided by replacing one triangle
Using this, or otherwise, show that there exist triangulations of the sphere with average vertex valency arbitrarily close to 6 .
(c) Suppose