2009-26.md

course	course_year	question_number	tags	title	year
Geometry	IB	26	IB 2009 Geometry	Paper 3, Section II, G	2009

Consider a tessellation of the two-dimensional sphere, that is to say a decomposition of the sphere into polygons each of which has at least three sides. Let $E, V$ and $F$ denote the numbers of edges, vertices and faces in the tessellation, respectively. State Euler's formula. Prove that $2 E \geqslant 3 F$. Deduce that not all the vertices of the tessellation have valence $\geqslant 6$.

By considering the plane ${z=1} \subset \mathbb{R}^{3}$, or otherwise, deduce the following: if $\Sigma$ is a finite set of straight lines in the plane $\mathbb{R}^{2}$ with the property that every intersection point of two lines is an intersection point of at least three, then all the lines in $\Sigma$ meet at a single point.