Question

every quadrilateral tessellates the plane. However, can an arbitrary quadrilateral such as the one shown below have all its sides altered and still tessellate the plane? Decide which methods described in this activity set you can use to alter the sides of this quadrilateral and tessellate the plane. In the pictured quadrilateral, no sides are of equal length and no sides are parallel. For each method you use, make a template for your figure, and determine whether or not it will tessellate the plane. Describe your results and include any clarifying diagrams.

Answer 1

There is a Conway Thurston orbifold, being " 2 2 2 2 ", which corresponds to rotations in the four sides of a quadralateral.  

That is, any quadralateral has a notation 2/ 2/ 2/ 2/ in the krieger decorations of the CT diagram. This corresponds to a digonal rotation around the four edge centres.

One can suppose that one edge goes to zero, eg 2. 2/ 2/ 2/, whereon, the group is still formed in the same way, but one of the digonal rotations is at the cornner, and one gets a tiling of triangles, where the cycle is repeated twice over.

The other kind of rotation is to use a swallowed edge (%), which is formed by removing an edge (rather than setting it to zero, so there is a group "2/ 2/ 2/ 2%" formed by any hexagon, whose three opposite pairs of edges are parallel.

Likewise, there is a CT orbifold "2 2 2" which leads to a covering of a sphere with three arbitatory trianges, and the resulting tetrahedron 2/ 2/ 2/.

In hyperbolic space, the CT group allows "2 2 2 2 2" etc, so that any pentagon, hexagon, etc whose angles add to a circle will tile space. Likewise, it follows from setting one edge to zero, that there is a tiling of polygons whose angles add to a half-circle, and with a single %, a tiling of polygons 2p with a central digonal symmetry, whose vertices add to a half or whole cirlce.