Assignment 10 (Due Wed 12/7)

Problem 10.1: Two tessellation experiments

Cut out two "general" quadrilaterals from cardboard. Here are two examples, but yours do not (and for variety, probably should not) look too much like these. By "general", the main point is that the quadrilaterals should not have any symmetries besides the identity.

Then for each of the quadrilaterals separately, trace the quad on a piece of paper, then move the cardboard quad and trace again repeatedly so that you fill up the space on the paper completely with the quads, i.e., a tessellation. So your end result should be two pages more or less covered by quadrilateral tessellations (or at least about 16 quadrilaterals to give a very clear picture of the pattern).

Problem 10.2: Indicating the symmetries of your tessellations

a) For each of the tessellations that you created, mark the centers of rotation and indicate the angles of rotation. You can check your ideas about the rotations by placing the cardboard quad on one on the paper and then moving it to another, analyzing what you are doing.

b) For each tessellation, shade in one quad and then shade in the same way all the others that are translations of the shaded ones.

c) Finally, tell whether there are any line reflection or glide reflection symmetries.

Problem 10:3: Analyzing the symmetry group of your tessellation

Look in Chapter 2 for some theorems that will help you answer these questions.

a) Describe the pattern that the centers of rotation form on the plane.

b) How are the translations related to these centers?

c) If A, B, C are centers of half-turns, where will a nearby fourth half-turn center D be, according to a theorem in Brown?

d) What theorem about quadrilaterals from an earlier assignment appears in a new guise in this figure?

Problem 10.4: Midpoint Quadrilateral

a) If ABCD is any quadrilateral, prove that the midpoints of the sides form a parallelogram.

b) How is this related to the previous questions?

c) If WXYZ is a parallelogram and A is a point, show how to construct a quadrilateral ABCD with WXYZ as midpoints (warning: the construction may sometimes produce a "quadrilateral" with intersecting sides.