1. Follow the steps in Section 7.7. Note that a (5,4) tiling means a tiling of regular pentagons, with 4 of
these pentagons meeting at each vertex. Be sure you understand why the angle
measurements of 72 degrees and 45 degrees are used in constructing this tiling.
2. Do Exercise 7.7.3. Carry out the steps to construct
the tiling just like you did in part (1).
3. Area and defect:
a. Construct a triangle in the
Poincare model.
b. Select the three vertices and use
“Defect” in the “Measure” menu to find the defect of
the triangle.
c. Again select the three vertices
and use the “Filled Polygon” button to fill in the triangle.
d. Use “Area” in the
“Measure” menu to measure the area of the triangle.
e. Drag the vertices of the triangle
around. What is the smallest defect you can get? What is the largest defect you
can get? What is the largest area? The smallest area?
Does the defect get larger or smaller as the area gets larger?
f. Again dragging the vertices of
the triangle around, calculate the ratio of defect to area for several
triangles (use a calculator). What pattern do you notice?
g. What constant does
Geometry Explorer use for k (as in
Theorem 7.22)?
h. How would your answers
to (e) and (f) be different in Euclidean space?