Non-regular tilings of Hyperbolic plane
We add a bit to https://mathoverflow.net/questions/398191/which-polygons-tessellate-the-hyperbolic-plane.
Background: It is known that there is a tiling of the hyperbolic plane by regular hyperbolic n-gons (n-gons with all angles and sides equal) with k polygons meeting at each vertex iff 1/n + 1/k < 2. So, for any n, there are regular n-gons that can tile the hyperbolic plane. Further, if the internal angles of a hyperbolic n-gon are equal, then the lengths of its sides are also equal.
**Question:** Given a number n, how does one find some non-regular hyperbolic n-gon that tiles the hyperbolic plane? One obvious way is to find any regular (2n-4)-gon that tiles the plane and cut it in half. What one would want to ask is if there are non-regular hyperbolic tiles which cannot be got in this way. For example, is there any hyperbolic quadrilateral tile with all edge lengths different?
Background: It is known that there is a tiling of the hyperbolic plane by regular hyperbolic n-gons (n-gons with all angles and sides equal) with k polygons meeting at each vertex iff 1/n + 1/k < 2. So, for any n, there are regular n-gons that can tile the hyperbolic plane. Further, if the internal angles of a hyperbolic n-gon are equal, then the lengths of its sides are also equal.
**Question:** Given a number n, how does one find some non-regular hyperbolic n-gon that tiles the hyperbolic plane? One obvious way is to find any regular (2n-4)-gon that tiles the plane and cut it in half. What one would want to ask is if there are non-regular hyperbolic tiles which cannot be got in this way. For example, is there any hyperbolic quadrilateral tile with all edge lengths different?
posted by R.Nandakumar @ 12:30 PM
0 Comments:
Post a Comment
<< Home