DescriptionTake a convex polyhedron in three-dimensional Euclidean space, and start reflecting it with respect to its faces, continuing by reflecting the images, etc. Question: for which polyhedra the images will tessellate the space without overlapping? The answer to this question is well known, moreover, it is known for any dimension, there are very few such polyhedra.However, if you ask the same question in hyperbolic space, the answer is known in dimensions two and three only, and for higher dimensions the problem is open for many years already. The project offers you an opportunity to understand general properties of such polyhedra and, probably, to find new tessellations. The project starts with an introduction to a combinatorial way to describe "good" hyperbolic polyhedra, namely, those who are candidates to tessellate the space. Then you will continue with an introduction to combinatorics of polyhedra. Next step will be to understand the classification of all simplices tessellating the space -- polyhedra with the number of vertices exceeding the dimension by one. Combining the knowledge obtained, you will be able to look for new tilings. PrerequisitesAlgebra II, Geometry III.ResourcesThe project is based on original research articles listed here Basic information about combinatorics of polytopes can be found in books
The main sources on hyperbolic geometry and hyperbolic polyhedra are
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email: Pavel Tumarkin