Due Friday, December 20.
This assignment is about topological quantities of triangular meshes. Feel free to ask any questions by writing a blog post!
A topological disk is, roughly speaking, any shape you can get by deforming the region bounded by a circle without tearing it, puncturing it, or gluing its edges together. Some examples of shapes that are disks include a flag, a leaf, and a glove. Some examples of shapes that are not disks include a circle (why?), a ball, a sphere, a DVD, a donut, and a teapot. A polygonal disk is any disk constructed out of simple polygons. Similarly, a topological sphere is any shape resembling the standard sphere, and a polyhedron is a sphere made of polygons. More generally, a piecewise linear surface is any surface made by gluing together polygons along their edges; a simplicial surface is a special case of a piecewise linear surface where all the faces are triangles. The boundary of a piecewise linear surface is the set of edges that are contained in only a single face (all other edges are shared by exactly two faces). For example, a disk has a boundary whereas a polyhedron does not. As you will show in the first exercise, for any polygonal disk with $V$ vertices, $E$ edges, and $F$ faces, it holds that \[ V – E + F = 1 \] while for a polyhedron $V – E + F = 2$. You may assume that surfaces have no boundary unless otherwise stated.
Clearly not all surfaces look like disks or spheres. Some surfaces have additional handles that distinguish them topologically; the number of handles $g$ is known as the genus of the surface (see illustration above for examples). In fact, among all surfaces that have no boundary and are connected (meaning a single piece), compact (meaning closed and contained in a ball of finite size), and orientable (having two distinct sides), the genus is the only thing that distinguishes two surfaces. A more general formula applies to such surfaces, namely\[ V – E + F = 2 – 2g, \]which is known as the Euler-Poincaré formula.
1. ( Polyhedral Formula — 5 pts. ) Show that for any polygonal disk with $V$ vertices, $E$ edges, and $F$ faces, the following relationship holds:\[ V – E + F = 1 \]and conclude that for any polyhedron $V – E + F = 2$.
Hint: use induction. Note that induction is generally easier if you start with a given object and decompose it into smaller pieces rather than trying to make it larger, because there are fewer cases to think about.
2. ( Regular Valence — 5 pts. ) The valence of a vertex in a piecewise linear surface is the number of faces that contain that vertex. A vertex of a simplicial surface is said to be regular when its valence equals six. Show that the only (connected, orientable) simplicial surface without boundary for which every vertex has regular valence is a torus ($g=1$). You may assume that the surface has finitely many faces.
3. ( Minimally Irregular Valence — 5 pts. ) Show that the number of irregular valence vertices in a (connected, orientable) simplicial surface of genus $g$ is
- at least $4$ if $g=0$,
- at least $0$ if $g=1$, and
- at least $1$ for $g\geq 2$,
assuming that all vertices have valence at least three and that there are finitely many faces.
4. ( Mean Valence — 5 pts. ) Show that the mean valence approaches six as the number of vertices in a (connected, orientable) simplicial surface goes to infinity, and that the ratio of vertices to edges to faces hence approaches\[ V:E:F = 1:3:2. \]
Total: 20 pts.