[dipl] Perustieteiden korkeakoulu / SCI

Permanent URI for this collectionhttps://aaltodoc.aalto.fi/handle/123456789/21

Browse

Browsing [dipl] Perustieteiden korkeakoulu / SCI by Subject "2-complexes"

Filter results by typing the first few letters
Now showing 1 - 1 of 1
  • Triangulations of the topological closed disk and circle packings
    (2016-06-14) Voutilainen, Marko
     Perustieteiden korkeakoulu | Master's thesis
    The main studies of this thesis are triangulations of the topological closed disk and circle packings as providers of embeddings in the hyperbolic disk for such triangulations. Triangulations are first introduced for a more general class of topological surfaces, before focusing on triangulations of the closed disk. The combinatorial nature of triangulations is revealed and it is used to identify triangulations. Construction of bijections between sets of triangulations leads to a recursive formula for the number of rooted triangulations with given number of boundary and interior vertices. After writing the recursion in terms of generating functions, an explicit formula for the number of rooted triangulations is derived. The methods used to derive the recursive formula are also used to uniform sampling of rooted triangulations. Circle packings are introduced at first in more general context, before concentrating on circle packings in the hyperbolic disk. The main result is that for every triangulation of the topological closed disk, there exists the maximal circle packing in the hyperbolic disk obeying the combinatorics of the triangulation. This maximal circle packing provides us with an embedding of the triangulation in the disk. These embeddings are used to visualize a collection of uniform random rooted triangulations. In the final chapter, a definition for uniform probability measures on classes of rooted triangulations with fixed number of vertices is provided. After that, random boundary length variables from the classes to natural numbers is defined and proved that the random boundary length converges in distribution to a non-degenerate random variable, as the number of vertices tends to infinity. Respectively, after defining probability measures on classes of rooted triangulations with fixed boundary length, it is shown that an appropriately renormalized random number of vertices converges in distribution to a non-degenerate random variable, as the boundary length tends to infinity.