Groups and Symmetry Homework 1 Solutions

1.  
   1. Indicate whether each of the 5 shapes below is a fundamental domain for the translations of the square grid.
      1. No
      2. Yes
      3. No
      4. No
      5. Yes

      
      

   2. Now, instead of allowing all the translations of the square grid, we only allow translations up or down by any whole number of units, translations left or right by any even number of units or a combination of these moves. Decide whether each of the 5 shapes is a fundamental domain for this set of translations.
      1. No
      2. No
      3. Yes
      4. No
      5. No
2. Is it possible for a rectangle (four right angles) to be a fundamental domain for the translations of the triangle grid? If so, draw one on the triangle grid (you can use a photocopy of the grid in your book to do your drawings). If not, why not?

   
   Yes
   

3. Pictured below is the equilateral triangle grid, and outlined in red is the regular hexagon grid (6 equilateral triangles form 1 regular hexagon) superimposed on it. Draw arrows on the triangle grid labeled a and b to represent translations so that every translation of the triangle grid can be expressed as some integer times a plus some integer times b. Now draw arrows on the triangle grid labeled c and d to represent translations of the hexagon grid so that every translation of the hexagon grid can be expressed as some integer times c plus some integer times d. What is the relationship of a and b to c and d -- that is, write equations expressing c and d in terms of a and b. You can do this problem on either a print of the picture below or a photocopy of the triangle grid in your book (you'll have to draw in the hexagon grid).

   
   

   c=2a-b
   d=a+b
    

4. When the "legal moves" for the square grid consist of all translations, we can use one of the squares as a fundamental domain. If we include all rotations of the square grid so that it looks the same before and after the rotation in the "legal moves" for the square grid, what can we now use for a fundamental domain? Draw a fundamental domain for these legal moves on the grid below or on a photocopy of the square grid from your book.

   
   [other solutions are possible]