Relations Between the Riemann Sphere and Models of the Hyperbolic Plane

 

Mathematik  |  Informatik

 

Luna Addi, 2004 | Zeiningen, AG

 

This paper explores relations between the Riemann sphere and models of the hyperbolic plane, particularly the Poincaré disk. It looks at the historical context of non-Euclidean geometry, outlines the properties of the models involved, and discusses the method used to determine relations between them. Using the hemisphere model and stereographic projection, relations are investigated between the models in the paper by differentiating between links and strong links, presenting both the values and limitations of the approach.

Introduction

This paper aims to examine if a relation between spherical geometry and hyperbolic geometry can be determined. To achieve this aim, one model of each of the two geometries was chosen, to form the research question: to what extent a link between the Riemann sphere and the Poincaré disk can be established?

Methods

Attempting a simpler approach to investigating non-Euclidean geometries, this paper uses projective geometry as the main method to attempt to answer the research question. Specifically, stereographic projection is used along with the hemisphere model to demonstrate links between different geometric models. The paper stays on a rather theoretical level, relying on mathematical reasoning and deductive approaches, rather than generating empirical data or relying on experimental procedures.

Results

The results of the investigation reveal that the Riemann sphere and the Poincaré Disk can be seen as an analogy which is a link. However, a strong link between the Riemann sphere and the Poincaré disk cannot be established as hyperbolic geometry and spherical geometry cannot be mixed. A model belonging to one geometrical space cannot be converted into a model belonging to a different geometrical space. Even though one cannot establish a strong link, links that provide insights into the geometric properties and transformations of the models hinting at the similarities between the two can be demonstrated. Furthermore, the use of projective geometry results in a simpler explanation as to why a strong link cannot be demonstrated.

Discussion

The results confirm that there exists a link between the Riemann sphere and the Poincaré disk model, but that there does not exist a strong link. While the chosen approach and method proved effective in answering the research question, there are different approaches one could take or combine with the chosen approach, such as exploring additional geometric models, extending the investigation to higher dimensions, and exploring the research question without the use of projective geometry.

Conclusions

In conclusion, this paper effectively achieved its objective of illustrating the relationship between spherical and hyperbolic geometry, revealing that a mapping between two models corresponding to each geometrical space is impossible as different geometrical spaces do not mix. However, through the exploration of the Riemann sphere and the Poincaré disk, with the help of projective geometry, an analogy can be made. Further investigations could consider alternative approaches or expand the analysis to higher dimensions.

 

 

Würdigung durch den Experten

Guillaume Buro

This is an interesting work on the Riemann Sphere and the Hyperbolic space. Luna developed a deep understanding of the concepts about classification of geometric spaces introduced in this work. Although the feeling that there exists common structures between different types of geometric spaces is right, the choice to try to construct such links is very bold. This choice led to some mathematical errors, but in this type of work, the most important is to understand new things, since it is almost impossible to do actual novel fundamental mathematical research right after gymnasium.

Prädikat:

gut

 

 

 

Gymnasium Bäumlihof, Basel
Lehrerin: Aline Sprunger