SCISSORS CONGRUENCE AND HILBERT’S THIRD PROBLEM
DOI:
https://doi.org/10.5902/2179460X14375Keywords:
Área e Volume, Poliedros e Polítopos, Terceiro Problema de HilbertAbstract
http://dx.doi.org/10.5902/2179460X14375
Given two polygons with equal areas, one can decompose one of them into a finite number of polygons and rebuild another. This fact is known as the theorem of Bolyai-Gerwien. It is natural to ask whether this theorem is true for polyhedra with equal volumes. This question originally proposed by Bolyai and Gauss in 1844 and then by Hilbert as the Third Problem in his famous list of 23 problems, was negatively answered by Max Dehn in 1902 for polyhedra in three dimension. The main objective of this paper is to present the proof of Dehn. This article has two main parts. The first one is devoted to the concept of area: we will briefly review involving some historical facts to his formalism in geometry and prove the theorem of Bolyai-Gerwien. In the second part we will see how the concept of area and its properties can be interpreted in Euclidean space of dimension three. The main result is the theorem of Dehn-Hadwiger which is crucial to solve the problem of Hilbert for polyhedra.
Downloads
Downloads
Published
How to Cite
Issue
Section
License
To access the DECLARATION AND TRANSFER OF COPYRIGHT AUTHOR’S DECLARATION AND COPYRIGHT LICENSE click here.
Ethical Guidelines for Journal Publication
The Ciência e Natura journal is committed to ensuring ethics in publication and quality of articles.
Conformance to standards of ethical behavior is therefore expected of all parties involved: Authors, Editors, Reviewers, and the Publisher.
In particular,
Authors: Authors should present an objective discussion of the significance of research work as well as sufficient detail and references to permit others to replicate the experiments. Fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. Review Articles should also be objective, comprehensive, and accurate accounts of the state of the art. The Authors should ensure that their work is entirely original works, and if the work and/or words of others have been used, this has been appropriately acknowledged. Plagiarism in all its forms constitutes unethical publishing behavior and is unacceptable. Submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. Authors should not submit articles describing essentially the same research to more than one journal. The corresponding Author should ensure that there is a full consensus of all Co-authors in approving the final version of the paper and its submission for publication.
Editors: Editors should evaluate manuscripts exclusively on the basis of their academic merit. An Editor must not use unpublished information in the editor's own research without the express written consent of the Author. Editors should take reasonable responsive measures when ethical complaints have been presented concerning a submitted manuscript or published paper.
Reviewers: Any manuscripts received for review must be treated as confidential documents. Privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. Reviewers should be conducted objectively, and observations should be formulated clearly with supporting arguments, so that Authors can use them for improving the paper. Any selected Reviewer who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the Editor and excuse himself from the review process. Reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers.
