Tiling regular polygons on hyperbolic half space (Joint work with Tianhao Xian)
2016, April 24
In Riemannian Geometry class, we wanted to have a more intuitive understanding about hyperbolic halfspace (admitting negative constant Gauss curvature). At first we created a tool that enabled us to draw lines, circles, and polygons in a hyperbolic half space. You can also move, rotate and reflect shapes along geodesics.
Now we can draw regular polygons on hyperbolic half plane with negative constant curvature, and map it onto an open ball isometrically. Moreover, we can tile polygons over the whole place.
The following is an example of tiling pentagons on the half space. There are four pentagons around each point, and all the inner angles are right angles.
The header of my homepage is a skeleton of this. You can actually try to hold and drag to see the deck transformation.
The following is a picture of tiling octagons on the half space. Likewise all the angles are right angles.
Notice that by quotient map of the half plane dividing its deck transformation group, we obtain a torus with genus 2. Thus we obtained a Riemannian metric on genus two torus which admits constant curvature. The following gif is to show the deck transform.
We found that for any integer $n > 4$, one can always tile 4 regular $n$-gons with right angles around a point, and cover the entire plane accordingly. Moreover, if $n > 2$, $m > 2$, one can always tile $m$ regular $n$-gons with inner angle $\frac{2\pi}{m}$ around each vertex, if $n-2 > \frac{2n}{m}$.
