Classifying Strictly Tessellating Polytopes
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Typ
Examensarbete för masterexamen
Program
Publicerad
2021
Författare
Blom, Max
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
Abstract
This thesis consists of a paper and additional results. The paper shows a connection
between the geometry of polytopal domains in Euclidean space and the eigenfunctions
of the Dirichlet Laplacian. The necessary and sufficient geometric properties of
a polytopal domain are shown for the first eigenfunction to extend to a real analytic
function on the whole space. Furthermore, alcoves are essential for the proof of the
main theorem. Additionally, the paper discusses how the results relate to crystallographic
restrictions and lattices. Strictly tessellating polytopes are defined and used
in connection to the main theorem. The paper concludes by formulating a conjecture
akin to Fuglede’s, replacing tessellation by translaton with strict tessellation.
In addition to the paper, results on the geometric properties of strictly tessellating
polytopes are presented, and bounds on the number of strictly tessellating polytopes
up to equivalence are shown.