# Exam I Study Guide - Spring 2010

**Topics:**

- Reflectional and Rotational Symmetry
- Symmetry Groups: Rozette groups, Border Patterns, Wallpaper Patterns
- Tessellations by Polygons
- Tessellations by Recognizable Figures.

## Part I of the Exam: Essay Question(s)

You will be asked to answer one or more of the following questions in detail:

(A) Explain how to determine the symmetry groups for rozettes, borders and wallpaper patterns. Explain which symmetries you would be looking for. I.e. what symmetries are you looking for when determining the group for rosettes? What symmetries do you look for in borders? What symmetries do you look for in wallpaper patterns? Be specific: Why do we not look for - say – 5 fold symmetry in a border pattern? Why would we never look for 36-fold symmetry in a wallpaper pattern? Which specific symmetries do we look for?

(B) Explain in detail why there are only 3 different regular tessellations of the plane.

(C) Explain in detail why all triangles and quadrilaterals tessellate the plane.

(D) Explain in detail how we can create tessellations by recognizable figures from an underlying tessellation by polygons. Be detailed and describe at least 2 different techniques using different isometries. The tessellations by recognizable figures should follow the examples set out in the online text and lead to Escher like tessellations. Give at least 1 example (sketch only) to illustrate a technique.

## Part II of the Exam

Questions may be taken from the homework or from the explorations we did in class.

These questions may require a short answer or they could be multiple choice or true/false.