Illustrations of nonEuclidean geometry in virtual reality
Abstract
Mathematical objects are generally abstract and not very approachable. Illustrations and interactive visualizations help both students and professionals to comprehend mathematical material and to work with it. This approach lends itself particularly well to geometrical objects. An example for this category of mathematical objects are hyperbolic geometric spaces. When Euclid lay down the foundations of mathematics, his formulation of geometry reflected the surrounding space, as humans perceive it. For about two millennia, it remained unclear whether there are alternative geometric spaces that carry their own, unique mathematical properties and that do not reflect human everyday perceptions. Finally, in the early 19th century, several mathematicians described such geometries, which do not follow Euclid's rules and which were at first interesting solely from a pure mathematical point of view. These descriptions were not very accessible as mathematicians approached the geometries via complicated collections of formulae. Within the following decades, visualization aided the new concepts and twodimensional versions of these illustrations even appeared in artistic works. Furthermore, certain aspects of Einstein's theory of relativity provided applications for nonEuclidean geometric spaces. With the rise of computer graphics towards the end of the twentieth century, threedimensional illustrations became available to explore these geometries and their nonintuitive properties. However, just as the canvas confines the twodimensional depictions, the computer monitor confines these threedimensional visualizations. Only virtual reality recently made it possible to present immersive experiences of nonEuclidean geometries. In virtual reality, users have completely new opportunities to encounter geometric properties and effects that are not present in their surrounding Euclidean world.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 arXiv:
 arXiv:2008.01363
 Bibcode:
 2020arXiv200801363S
 Keywords:

 Mathematics  History and Overview;
 Computer Science  Graphics;
 Mathematics  Differential Geometry;
 51M10;
 I.3.7;
 J.5
 EPrint:
 Submitted for publication in the Yearbook of Moving Image Studies, this is the reviewed version