Poliedri

master's thesis

• Title (english): Polyhedra
• Author: Dražen Lovrić
• Mentor: Tomislav Pejković (mentor)
• Committee member: Tomislav Pejković (predsjednik povjerenstva)
• Committee member: Igor Ciganović (član povjerenstva)
• Committee member: Matija Bašić (član povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj diplomski rad koji se bavi temom poliedara uz poglavlje sa osnovnim pojmovima sastoji se od još četiri značajna poglavlja. U drugom poglavlju konstruktivno pokazujemo da pravilni poliedri zaista postoje. Kako bi bili jedinstveni, popravljamo Euklidovu definiciju pravilnih poliedara tako što dodajemo uvjet konveksnosti te zahtjev da se u svakom vrhu sastaje jednak broj strana. Na kraju je dana ispravna definicija pravilnog poliedra. Dva najvažnija teorema dani su u trećem poglavlju. Eulerov teorem opisuje odnos broja vrhova, bridova i strana konveksnog poliedra, a Cauchyjev teorem otkriva kada su dva konveksna poliedra sukladna. Primjenom oba teorema klasificiramo deltaedre, konveksne poliedre čije su sve strane jednakostranični trokuti. U četvrtom poglavlju klasificirani su konveksni poliedri pravilnih strana gdje razlikujemo uniformne i neuniformne poliedre. Polupravilne poliedre, kao uniformne poliedre, čine pravilni poliedri, prizme, antiprizme i Arhimedova tijela. Takoder pokazujemo da postoji samo konačno mnogo neuniformnih konveksnih poliedara pravilnih strana. U završnom poglavlju istražujemo grupe simetrija pravilnih poliedara. Njihove grupe rotacija, zajedno s cikličkim i diedralnim grupama, jedine su konačne podgrupe grupe rotacija sfere.

Abstract (english)

This Master’s Thesis with the subject of polyhedra contains an introductory chapter with relevant definitions and the main part of the work consisting of four chapters. In the second chapter, we show constructively that regular or Platonic solids do exist. In order to make them unique, we amend Euclid’s definition of regular polyhedra adding the condition of convexity and the requirement that an equal number of sides meet in each vertex. The two most important results are given in the third chapter. Euler’s theorem describes the relationship between the number of vertices, edges and sides of a convex polyhedron, while Cauchy’s theorem explains when two convex polyhedra are congruent. Applying these theorems, we determine all convex deltahedra, i.e. polyhedra whose all sides are equilateral triangles. Convex polyhedra with sides that are regular polygons are classified in the fourth chapter where we distinguish between uniform and non-uniform polyhedra. Semiregular polyhedra, being uniform polyhedra, contain regular polyhedra, prisms, antiprisms and Archimedean solids. We also show that there are only finitely many non-uniform convex polyhedra with regular sides. In the final chapter, we investigate symmetry groups of regular polyhedra. Their rotation groups, together with cyclic and dihedral groups, are the only finite subgroups of the rotation group of a sphere.

• Keywords: Euklidova definicija pravilnih poliedara; uvjet konveksnosti; Eulerov teorem; konveksni poliedari; Cauchyjev teorem; deltaedri; uniformni poliedri; neuniformni poliedri
• Keywords (english): Euclid’s definition of regular polyhedra; condition of convexity; Euler’s theorem; convex polyhedra; Cauchy's theorem; deltahedra; uniform polyhedra; non-uniform polyhedra
• Language: croatian
• URN:NBN: urn:nbn:hr:217:448261
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-10-28 10:22:12