Subdivision of the Poincare Circle
ala Escher
M.C. Escher (1898-1972) was a Dutch artist who used geometric ideas, particularly those of Euclidean, hyperbolic, and elliptic geometries, extensively in his work.  He became familiar with non-Euclidean geometry through the work of the great geometer H.M.S. Coxeter.  One of Escher's favorite subjects was that of subdividing the Poincare disk into congruent shapes, i.e. tiling the space.

His work Circle Limit I  illustrates a tiling of the hyperbolic disk by regular hexagons meeting four at a vertex.  To construct this tiling we will need to construct a point at the origin (0,0) of the Poincare disk. To construct a point at the origin go to the "HyperMisc" menu and choose the  "Point at Origin" sub-menu. A point at the origin of the Poincare disk will be created.  This is point "A" in the figure below.

At point A draw a vertical line from A to another point B. Now, set point A as a center of rotation and rotate the vertical line by 60 degrees yielding line AC.  Construct the perpendiculars to AB and AC through the points B and C.  Find the intersection point D of these two perpendiculars.  Next, select points A, B, and D (in that order) and measure the Ratio of AB to BD by choosing "Ratio" under the "Measure" menu.  Move point B until this ratio is approximately 1.0  Then,  the quadrilateral ABDC will form one-sixth of the hexagon that we will use to tile the Poincare disk.

Next, we will use reflections to generate congruent quadrilaterals to ABDC in the sector defined by angle CAB.  First, select perpendicular BD and choose "Mirror" under the "Mark"menu. Then, select lines CD, AC, and AB and hit the reflection button in the Transform Panel. These three lines will be reflected across line BD yielding a new quadrilateral BDEF. Likewise relfect ABDC across line CD getting CDGH.

Now we will rotate the construction in the sector dfeined by angle CAB five times around point A getting a start at a tiling of the poincare disk. Using the selection box, select all of the objects in the sector.  The rotation of 60 degrees about point A should still be stored in Geometry Explorer, so we just need to click on the rotate tool five times to get the figure shown.  (Labels are hidden)

The Escher tiling is the tiling you get by continuing to reflect the quads in the picture across lines, thus generating new quads, and then reflecting these again, and so on.  No new quad generated by reflection will intersect any of the lines in the picture. Thus, we get a regular tiling of the Poincare disk by congruent quadrilaterals.