A Visual Model of the Lorentz Triangular Moduli Space

Abstract

We present a schematic construction of the triangularly-shaped space in the Lorentz plane, i.e., the moduli space of all Lorentzian triangles up to similarity. This space turns out to be homeomorphic to a band where the different types of triangles may be located according to the length and the causal character of their sides. In particular, several kinds of equilateral and isosceles triangles are identified.

MSC:

• Primary 51N05
• Secondary 51F99; 97G80; 97G10

ACKNOWLEDGMENT

This work was partially supported by Spanish MINECO and ERDF project MTM2013-47828-C2-2-P  as well as “FQM-193” of the 3rd Andalusian Research Plan (PAIDI).

Notes

1 Recall that a homothety is an affine transformation determined by a point C, called its “center,” and a real number λ, called its “ratio,” that sends .

Additional information

Funding

Ministerio de Economía y Competitividad(MTM2013-47828-C2-2-P, 10.13039/501100003329)

Notes on contributors

Daniel de la Fuente

DANIEL DE LA FUENTE is currently a postdoctoral researcher in the Department of Geometry and Topology of University of Granada (Spain). He obtained his Ph.D. degree in mathematics in 2016, and he graduated in mathematics and physics from the University of Valladolid (Spain). He teaches graduate courses in geometry and does research in both geometrical analysis and mathematical physics.

Rafael Ramírez-Uclés

RAFAEL RAMÍREZ-UCLÉS is an assistant professor in the Department of Didactics of Mathematics at the University of Granada (Spain). He is experienced in teaching secondary school mathematics as well as for the Degree of Primary Education and Master of Teacher Training. He completed his doctoral thesis on visualization skills of students with mathematical talent. His main line of research is attention to the high mathematical ability and the development of the spatial sense in students.

Juan F. Ruiz-Hidalgo

JUAN FRANCISCO RUIZ-HIDALGO is an assistant professor in the Department of Didactics of Mathematics at University of Granada (Spain). His career includes experience as a teacher in secondary education as well as in higher education. He has published in partial differential equations and currently he is interested in both advanced mathematical thinking and the meanings of school mathematics concepts.