1. Do exercise 4.2.7 on page 42.  Essentially you need to draw border patterns using human feet.  Be sure to draw one pattern of each type.  Next try to move your feet in that border pattern and figure out which of the seven names fit your motion.
2. Explain why a border pattern that has both a horizontal reflection symmetry and a half-turn symmetry must also have a vertical reflection symmetry.

Answer each of the following problems as clearly and concisely as you can.  Label each problem clearly, and show your work or explain your reasoning.  This homework is due on October 30 at the beginning of class.
 

1. A regular polygon is a polygon where all the angles have the same size and all the sides have the same length.
1. How many symmetries does a regular triangle have?  Describe them precisely and concisely (in words).
2. How many symmetries does a regular pentagon have?  Describe them precisely and concisely.
3. Suppose n is an odd number.  Describe the symmetries of a regular n-gon (a regular polygon with n sides) precisely and concisely.
2. Using the multiplication table for the regular pentagon, express each of the  symmetries below as a single symmetry.  Assume that r is the first rotation and that m is the first mirror reflection.
1. rrmmrrr
2. rmrmrmrm
3. mrm
4. rmr
3. Sketch a finite figure whose symmetry type is C4.  Sketch another one whose symmetry type is D4.  Both of your figures must be different from the ones ins the text and the ones we used in class.
4. Find the multiplication table for the symmetries of a regular hexagon.