Pickova formula

master's thesis

• Title (english): Pick's formula
• Author: Stefan Hranj
• Mentor: Slaven Kožić (mentor)
• Committee member: Slaven Kožić (predsjednik povjerenstva)
• Committee member: Aleksandra Čižmešija (član povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Committee member: Matko Ljulj (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu proučavamo Pickovu formulu, formulu koja povezuje probleme iz dva potpuno različita svijeta matematike - problem prebrojavanja iz diskretne matematike i problem određivanja površine iz geometrije. U prvom poglavlju navodimo definicije, teoreme i rezultate iz geometrije i diskretne matematike koji su nužni za shvaćanje glavnog dijela rada. U drugom poglavlju promatramo probleme koji nas dovode do Pickove formule. U trećem poglavlju prikazana su dva dokaza Pickove formule koja se oslanjaju na elementarne geometrijske pojmove i rezultate. Nakon toga, u četvrtom poglavlju istražujemo proširenja Pickove formule i istražujemo odnos između Pickove formule i homotetije. Peto poglavlje proučava opću teoriju rešetki kojom također, kao posljedicu određenih općenitijih rezultata, dolazimo do Pickove formule. Rad završava šestim poglavljem, odnosno metodičkim pristupom otkrivanja Pickove formule u osnovnoškolskom obrazovanju.

Abstract (english)

In this thesis, we study Pick’s formula, a formula that connects problems from two entirely different realms of mathematics - counting problems from discrete mathematics and problem of calculating area from geometry. In the first chapter, we provide definitions, theorems, and results from geometry and discrete mathematics that are necessary for understanding the main part of the thesis. In the second chapter, we examine the problems that lead us to Pick’s formula. In the third chapter, two proofs of Pick’s formula are presented, relying on elementary geometric concepts and results. Next, in the fourth chapter, we explore extensions of Pick’s formula and investigate the relationship between Pick’s formula and homothety. The fifth chapter presents the general lattice theory which, as a consequence of certain more general results, leads us to Pick’s formula as well. The thesis concludes with the sixth chapter, which discusses a methodical approach to introducing Pick’s formula in elementary education.

• Keywords: problem prebrojavanja; problem određivanja površine iz geometrije; homotetija
• Keywords (english): counting problem; the problem of determining the area from geometry; homothety
• Language: croatian
• URN:NBN: urn:nbn:hr:217:904638
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-01 11:07:40