Triangulacije ploskev
magistrsko delo
Abstract
Definicija simplicialnih kompleksov in mnogoterosti nam omogoči razumevanje dveh osnovnih lastnosti ploskev, ki sta Eulerjeva karakteristika in orientabilnost. S pomočjo Dehn-Sommervillovih enačb zapisanih s f- in h-vektorjem definiramo Eulerjevo karakteristiko. Vidimo, da ta, poleg orientabilnosti, natanko določa vse sklenjene ploskve. Do homeomorfizma natančno dobimo eno izmed ploskev: sfero, povezano vsoto n torusov ali povezano vsoto n projektivnih ravnin. Ploskve predstavimo na dva različna načina, s pomočjo mnogokotnika skupaj z njegovo notranjostjo in s pomočjo Heffterjeve prezentacije. Glavna ugotovitev magistrske naloge je minimalna triangulacija ploskev. Oceno za to nam pove Heawoodova domneva, ki je prikazana na primerih sfere, torusa, dvojnega torusa, projektivne ravnine in Kleinove steklenice.
Keywords
simplicialni kompleks;Eulerjeva karakteristika;orientabilnost ploskev;minimalne triangulacije;
Data
| Language: | Slovenian |
|---|---|
| Year of publishing: | 2022 |
| Typology: | 2.09 - Master's Thesis |
| Organization: | UL FMF - Faculty of Mathematics and Physics |
| Publisher: | [T. Pevec] |
| UDC: | 515.16 |
| COBISS: |
100846083
|
| Views: | 873 |
| Downloads: | 102 |
| Average score: | 0 (0 votes) |
| Metadata: |
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Other data
| Secondary language: | English |
|---|---|
| Secondary title: | Triangulations of surfaces |
| Secondary abstract: | The definition of simplicial complex and manifolds allows us to understand two general properties of surfaces, which are Euler characteristic and orientability. Using the Dehn-Sommerville equations written with f- and h-vectors we define the Euler characteristic. We see that this, in addition to orientability, leads us to classification theorem of surfaces. Up to a homeomorphism we get one of the surfaces: a sphere, a connected sum of n tori or a connected sum of n projective planes. We present the surfaces in two different ways: with polygon and with Heffter presentation. The main finding in this thesis is the minimal triangulation of surfaces. We show Heawood conjecture on examples of the sphere, torus, double torus, projective plane and Klein bottle. |
| Secondary keywords: | simplicial complex;Euler characteristic;orientability of surfaces;minimal triangulations; |
| Type (COBISS): | Master's thesis/paper |
| Study programme: | 0 |
| Thesis comment: | Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Pedagoška matematika |
| Pages: | IX, 35 str. |
| ID: | 14742770 |
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