Dehn-Hadwigerov teorem

master's thesis

• Author: Lorena Balošić
• Mentor: Sanja Varošanec (mentor)
• Committee member: Sanja Varošanec (predsjednik povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Committee member: Boris Guljaš (član povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ukratko, na početku svog diplomskog rada opisala sam povijesne okolnosti u kojima je nastao problem jednakosastavljivosti poliedara. Zatim sam opisala problem jednakosastavljivosti u ravnini te sam iskazala i dokazala Wallace-Bolyai-Gerwienov teorem koji pokazuje kako su poligoni jednakih površina ujedno i jednakosastavljivi. Nadalje, opisala sam analogni problem u prostoru i postavila pitanje vrijedi li sličan teorem i za poliedre. Na to pitanje pronalazimo odgovor u Dehn-Hadwigerovom teoremu, koji je ujedno i glavna tema ovog rada. Dokaz tog teorema pronašla sam u nekoliko različitih izvora. Na kraju sam pokazala primjenu Dehn-Hadwigerovog teorema na primjeru pravilnog tetraedra i kocke istog volumena te na primjeru dva tetraedra jednakih volumena.

Abstract (english)

At the beginning of my thesis, I described historical circumstances in which the problem of equidecomposable polyhedra has emerged. Then I described the problem of equidecomposable polygons and I had shown and proved the Wallace-Bolyai-Gerwien theorem that states that if planar polygons have the same area then they are equidecomposable. Furthermore, I described an analogous problem in three dimensional space and I asked a question if there is a similar theorem for polyhedra. The answer to this question we can find in Dehn-Hadwiger theorem, which is the main theme of this thesis. The proof of this theorem I found in several different sources. Finally, I have applied Dehn-Hadwiger theorem on the example of the regular tetrahedron and cube of the same volume and example of two tetrahedra of the same volume.

• Keywords: poliedri; problem jednakosastavljivosti poliedara; Wallace-Bolyai-Gerwienov teorem; Dehn-Hadwigerov teorem
• Keywords (english): polyhedra; problem of equidecomposable polyhedral; Wallace-Bolyai-Gerwien theorem; Dehn-Hadwiger theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:900436
• 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: 2019-03-01 13:56:01