Subdivision of the Poincare Circle
ala Escher
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 the quadrilateral and hit the reflection button in the Transform Panel. The quadrilateral will be reflected across line BD yielding a new quadrilateral (colored blue in the figure below). Likewise reflect ABDC across line CD (the green quad).

escher2


Now we will rotate the construction in the sector defined by angle CAB five times around point A getting a start at a tiling of the Poincare disk. Using the selection box, select all three quads 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)

escher3


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.