A More Suitable Definition of Quadratic Surds
Orchidea Maria Lecian1, *
Identifiers and Pagination:Year: 2020
First Page: 8
Last Page: 20
Publisher Id: TOSPJ-10-8
Article History:Received Date: 21/12/2019
Revision Received Date: 02/05/2020
Acceptance Date: 20/08/2020
Electronic publication date: 23/10/2020
Collection year: 2020
open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: https://creativecommons.org/licenses/by/4.0/legalcode. This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
The properties of the group PGL(2,C) on the Upper Poincar´e Half Plane have been analyzed.
In particular, the classification of points and geodesics has been achieved by considering the solution to the free Hamiltonian associated problem.
The free Hamiltonian associated problem implies to discard the symmetry sl(2,Z) for the definition of reduced geodesics. By means of the new definition and classification of reduced geodesics, new construction for tori, punctured tori, and the tessellation of the Upper Poincar´e Half Plane is found.
A definition of quadratic surds is proposed, for which the folding group corresponds to the tiling group, (also) for Hamiltonian systems on the Hyperbolic Plane (also realized as the Upper Poincar´e Half Plane (UPHP)).
The initial conditions determine the result of the folding of the trajectories as tiling punctured tori and for tori.