Primitivni tetraedri u trodimenzionalnoj cjelobrojnoj rešeci

Marin, Nataša (2015) Primitivni tetraedri u trodimenzionalnoj cjelobrojnoj rešeci. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

In this paper we study primitive tetrahedra in a three dimensional integer lattice. A tetrahedron is primitive if its vertices are lattice points but it does not contain any other lattice points in its interior or on its boundary. We use unimodular maps as a main mathematical tool to prove most of the results, and especially for classification of primitive tetrahedra, based on unimodular equivalence. Further, we investigate primitive polygons in $\mathbb{Z}^$2 so that a comparison could be made with the three dimensional case. A crucial part in this chapter belongs to the famous Pick’s theorem, which gives the relationship between the area of a lattice polygon and the number of lattice points on its boundary and in its interior. In the three dimensional integer lattice, first of all we state a characterization of primitive tetrahedra in terms of unimodular equivalence. Considering that there is no boundary on the volume of a primitive tetrahedron, a generalization of Pick’s theorem to dimension three is impossible, but there is a simple formula that counts the number of equivalence classes of primitive tetrahedra of a given volume.

Item Type: Thesis (Diploma thesis)
Supervisor: Šiftar, Juraj
Date: 2015
Number of Pages: 32
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 22 Oct 2015 10:00
Last Modified: 22 Oct 2015 10:00
URI: http://digre.pmf.unizg.hr/id/eprint/4164

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