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Separability Of Tilings, Nicholas Baeth, Jason Deblois, Lisa Powell
Separability Of Tilings, Nicholas Baeth, Jason Deblois, Lisa Powell
Mathematical Sciences Technical Reports (MSTR)
A tiling by triangles of an orientable surfaces is called kaleidoscopic if the local reflection in any edge of a triangle extends to a global isometry of the surface. Given such a global reflection the fixed point subset of the reflection consists of embedded circles (ovals) whose union is called the mirror of the reflection. The reflection is called separating if removal of the mirror disconnects the surface into two components. We consider surfaces such that the orientation preserving subgroup of the tiling group generated by the reflection is cyclic or abelian. A complete classification of those surfaces with separating …
Triangular Surface Tiling Groups For Low Genus, Sean A. Broughton, Robert M. Dirks, Maria Sloughter, C. Ryan Vinroot
Triangular Surface Tiling Groups For Low Genus, Sean A. Broughton, Robert M. Dirks, Maria Sloughter, C. Ryan Vinroot
Mathematical Sciences Technical Reports (MSTR)
Consider a surface, S, with a kaleidoscopic tiling by non-obtuse triangles (tiles), i.e., each local reflection in a side of a triangle extends to an isometry of the surface, preserving the tiling. The tiling is geodesic if the side of each triangle extends to a closed geodesic on the surface consisting of edges of tiles. The reflection group G*, generated by these reflections, is called the tiling group of the surface. This paper classifies, up to isometry, all geodesic, kaleidoscopic tilings by triangles, of hyperbolic surfaces of genus up to 13. As a part of this classification the tiling groups …
Lengths Of Systoles On Tileable Hyperbolic Surfaces, Kevin Woods
Lengths Of Systoles On Tileable Hyperbolic Surfaces, Kevin Woods
Mathematical Sciences Technical Reports (MSTR)
The same triangle may tile geometrically distinct surfaces of the same genus, and these tilings may determine isomorphic tiling groups. We determine if there are geometric differences in the surfaces that can be found using group theoretic methods. Specifically, we determine if the systole, the shortest closed geodesic on a surface, can distinguish a certain families of tilings. For example, there are three tilings of surfaces of genus 14 by the hyperbolic triangle with angles π/2 , π/3 , and π/7 whose tiling groups are all PSL2(13). These tilings can be distinguished by the lengths of their systoles.