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A topological proof of the modified Euler characteristic based on the orbifold concept
Bartosz Naskręcki1, Zbigniew Dauter2, Mariusz Jaskolski3
1Faculty of Mathematics and Computer Science, A. Mickiewicz University, Poznań, Poland.
Summary
This study proves a theorem on the vanishing Euler characteristic of topological orbifolds, explaining its connection to crystallographic space groups and honeycomb tessellations. The findings apply to asymmetric units and Dirichlet domains in crystallography.
Area of Science:
- Mathematics
- Crystallography
- Topology
Background:
- The Euler characteristic of polyhedra and tessellations is a well-studied concept.
- Previous work investigated the vanishing Euler characteristic in periodic tessellations under crystallographic space groups.
Purpose of the Study:
- To formally express the vanishing Euler characteristic phenomenon as a theorem for topological orbifolds.
- To demonstrate that this theorem arises from fundamental orbifold Euler characteristic properties.
- To re-prove Coxeter's theorem on honeycomb tessellations in a generalized context.
Main Methods:
- Formalizing the vanishing Euler characteristic as a theorem for topological orbifolds.
- Utilizing fundamental properties of the orbifold Euler characteristic.
- Applying the generalized formula to crystallographic objects like asymmetric units and Dirichlet domains.
Main Results:
- A theorem is presented regarding the vanishing Euler characteristic of topological orbifolds.
- The theorem is shown to be a consequence of the fundamental properties of the orbifold Euler characteristic.
- Coxeter's theorem on honeycomb tessellations is re-proven with relaxed assumptions.
Conclusions:
- The study provides a unified framework for understanding the vanishing Euler characteristic in crystallography.
- The proven formula has practical applications for analyzing crystallographic structures.
- The work connects abstract topological concepts with concrete examples from wallpaper and 3D space groups.
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