Ravninske mreže in posplošitve Pickovega izreka

diplomsko delo

Anita Mandelj (Author), Matija Cencelj (Mentor), Boštjan Gabrovšek (Co-mentor)

Abstract

Obravnavamo enakomerno porazdeljene diskretne množice točk v ravnini, ki jim pravimo mreže. Najbolj znane in preučevane so kvadratne mreže, poseben predstavnik takih mrež je mreža vseh točk s celoštevilskimi koordinatami v ravnini R×R. Obravnavamo tudi pravokotne mreže, paralelogramske mreže in trikotniške mreže. Z raziskovanjem krožnic, postavljenih na različnih mrežah, ugotavljamo povezavo med številom mrežnih točk znotraj in na krožnici ter številom π. Ugotavljanje zgornje in spodnje meje za napako, ki pri tem nastane, imenujemo Gaußov problem s krožnicami, saj je prav on prvi raziskoval mreže in krožnice na njej. Pokažemo tudi zgornjo mejo za najkrajšo razdaljo med dvema mrežnima točkama. S pomočjo izreka iz teorije števil povežemo število mrežnih točk v in na krožnici z Leibnizevo vrsto. Obravnavamo vprašanje posplošitve Pickovega izreka na splošnejše mreže v ravnini in s protiprimerom pokažemo, da izrek ne velja za heksagonalne mreže. Posebej predstavimo dva programa: program za računanje števila mrežnih točk znotraj in na robu krožnice, ki obenem izračuna tudi približek za število π ter napako, ki pri tem nastane ter program za izračun ploščine večkotnika po Pickovem izreku.

Keywords

mreže v ravnini;Pickov izrek;

Data

Language:	Slovenian
Year of publishing:	2016
Typology:	2.11 - Undergraduate Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[A. Mandelj]
UDC:	51(043.2)
COBISS:	11142217
Views:	812
Downloads:	128
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Plane lattices and generalizations of Pick's theorem
Secondary abstract:	A uniformly distributed discrete set of points in the plane called lattices are considered. The most well-known and studied are square lattices, a special representative of such lattices is the lattice of all points with integer coefficients in the plane R×R. We are dealing with rectangular lattices, parallelogram lattices and triangle lattices. By exploring the circles, positioned on different lattices, we establish a link between the number of lattice points inside and on the edge of a circle and the number π. Determining the upper and lower bound of the error occurring, is called Gauss circle problem, since it was him who first explored lattices and circles on it. We also show the upper bound of the shortest distance between two lattice points. With the help of a theorem of number theory, we connect the number of lattice points inside and on the edge of a circle with Leibniz series. Generalizations of Pick's theorem on general lattices in the plane are considered and with a counterexample it is shown that the theorem does not apply to the hexagonal lattices. Separately we introduce two programs: a program for calculating the number of lattice points inside and on the edge of a circle, which also calculates an approximation for the number π and the error occurring, and a program for calculating the area of a polygon with Pick's theorem.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Matematika in računalništvo
Pages:	VIII, 56 str.
ID:	9166720