Rigid Motions and Symmetries

A rigid motion of the plane is a function such that for any two points p and q in the plane, the distance between f(p) and f(q) equals the distance between p and q. We will (I hope) prove that besides the identity map f(p)=p, there are only four different types of rigid motions: translations, rotations about points, reflections across lines, and glide reflections (a reflection across a line followed by a translation parallel to the line). Notice that a translation by a distance of 0 is the identity map, as is a rotation by an angle that is an integer multiple of . Also a glide reflection involving a translation by a distance of 0 is simply a reflection.

Given any subset S of the plane, a symmetry of S is a rigid motion f of such that f(S)=S. This doesn't mean that every point of the subset is fixed by f, but that ; i.e., every point in S is mapped to a point in S, and every point in S is the image of some point in S.

Consider the set . We will call this a strip. What are the possible symmetries of the strip?

• The identity motion, which we will denote I.
• Translations by any amount parallel to the x-axis. If we translate by an amount a (which may be positive, negative, or zero), we will denote this symmetry by .
• Rotations by about a point on the x-axis. If we rotate about the point (b,0) we will denote this symmetry by .
• Vertical reflection across any line perpendicular to the x-axis. If we reflect across the line x=c, we will denote this symmetry by .
• Horizontal reflection across the x-axis, which we will denote H.
• Glide reflection across the x-axis. If we reflect across the x-axis and then translate by an amount d (which may be positive, negative, or zero), we will denote this symmetry by .

A repeating strip pattern is a subset of the above strip such that each of its translational symmetries is a repetition (positive, negative, or zero) of one particular translational symmetry.

1. For each of the strip symmetries, write a formula for the function.
   1. I(x,y)=(x,y).
   2. .
   3. 
   4. 
   5. H(x,y)=
   6. 
2. For each of the following strip symmetries, write the formula:
   1. A translation to the right by 2 units. .
   2. A translation to the left by 3 units.
   3. A rotation by about the point (-5,0).
   4. A rotation by about the point (4,0).
   5. A reflection across the line x=10.
   6. A reflection across the line x=-7.
   7. A glide reflection involving a translation to the right by 6 units.
   8. A glide reflection involving a translation to the left by 8 units.
3. Identify the following strip symmetries:
   1. f(x,y)=(x,y). The identity: I.
   2. f(x,y)=(-x,y). Reflection about the vertical line x=0: .
   3. f(x,y)=(x,-y).
   4. f(x,y)=(-x,-y).
   5. f(x,y)=(x+1,y).
   6. f(x,y)=(x-2,y).
   7. f(x,y)=(-x+1,y).
   8. f(x,y)=(-x-2,y).
   9. f(x,y)=(x+1,-y).
   10. f(x,y)=(x-2,-y).
   11. f(x,y)=(-x+1,-y).
   12. f(x,y)=(-x-2,-y).
4. In the next set of exercise, means first perform g, then perform f. For the following, determine the formula, and then identify the symmetry:
   1. .
   2. .
   3. .
   4. .
   5. .
   6. .
   7. .
   8. . Explain why this makes sense in a sentence or two.
   9. . Explain why this makes sense in a sentence or two.
   10. Find f such that .
   11. Find f such that .
   12. Find f such that .
   13. Find f such that .
   14. Find f such that .
   15. Find f such that .
   16. Find f such that .
   17. Find f such that .
   18. Find f such that .
5. Fill in the following composition table, filling in each entry with and identifying what kinds of symmetries can result. (You may have to redraw this chart to make it larger.)

   

6. We will now try to classify repeating strip patterns by the types of symmetries they have. In the following table, R indicates the presence of some rotational symmetry about a point on the x-axis, V indicates the presence of a reflectional symmetry across some vertical line, H in indicates the presence of a reflectional symmetry across the x-axis, and G indicates the presence of some nontrivial glide reflectional symmetry across the x-axis (i.e., where the accompanying translation is by a non-zero amount, so it is not merely a horizontal reflection.) For a given row there are sixteen possible ways to place either a check mark or an x in each cell of the row (why?). Fill in the table with the sixteen possibilities. For each possibility, either draw a repeating strip pattern to the right of the row exhibiting precisely that combination of symmetries, or else use the results of the previous chart to briefly explain why that particular combination of of symmetries is impossible for any repeating strip pattern. For example, the presence of H and T (all patterns have T) forces the presence of G, and the presence of R and V forces the presence of either H or G or both.

   

7. Refer to the handout labeled ``Transparency 4'' for samples of the possible repeating strip patterns. Confirm that you have found the same number of patterns, and label the patterns you drew according to the scheme on the handout.