Lab 9 Part 1
This lab can either be done during normal lab time or as a take-home lab. For those in the regular lab there will be a Sketchpad document to produce and turn in. For those with the take-home, the lab can either be turned in as a Sketchpad document or on paper. There will be more detailed in instructions in the lab.
Lab 9 Part 1: Dihedral groups and Rosette Groups
Experiment A: Two mirrors at a 36 degree angle.
Draw two lines that meet at point O at 36 degrees. Draw some polygonal region p inside the 36-degree angle such as the one here (but you can make your own shape). Try to avoid a shape p that already has a line of symmetry.
Sketchpad: Draw one line m1 through O; then rotate the line m1 by 36 degrees with center O to get m2. Then draw any polygon you like.
Paper: If you have a printed regular pentagon, you can use it to draw a 36 degree angle. Otherwise, you can use a protractor (or you can construct it, but this is not necessary). For the reflecting the region, you may wish to cut a copy from cardboard so that you can trace around it.
Now if we let R1 and R2 be the reflections in m1 and m2, the idea of this experiment is to draw the image of p by reflecting in these two lines repeatedly in all possible ways.
The Construction: Reflect p in m1 to get p'. Label this as R1(p). Then reflect p' in m2 to get p''; label it R2R1(P). Then reflect p''' in m1 to get p'''', etc. In other words, we wish to construct R1(p), R2R1(p), R1R2R1(p)., etc for all products made of R1 and R2. There will be a finite number and then they will start to repeat.
IMPORTANT: All the reflections should be in m1 or m2, other lines should not be used.
The END: When the shapes start to repeat, you can stop. You should have a pattern with 5 lines of symmetry and 5 rotations that are symmetries.
What to turn in.
Turn in the figure with these labels and the answers to the
questions. They can be in a Sketchpad
document or on paper.
- Label all the images as R2R1(P), etc.
- Indicate by color or by label which images are in fact rotations of p and tell the angle of rotation (be sure to include p itself).
- Indicate which images are line reflections of p and draw in the mirror lines for these line reflections. These lines should be lines of symmetry for the whole pattern.
- Use your example to answer these two questions.
Questions A:
- The transformation (R2R1)2 is a rotation T. What is the angle of rotation? From you labels, tell what isometry is the product R1T (not just what kind, exactly how is it related to m1 and to T).
- From this and other examples, if T is a rotation by angle a and center O, explain why the transformation R1T is a line reflection and tell where the mirror line will be located relative to m1.
- How does your answer help explain why there are exactly 5 line reflections and 5 rotations that are symmetries of your figure.
Experiment B:
Figure 1
Construct a right triangle so that the two lines OA and
You can produce this figure with Sketchpad, with a cut-our cardboard triangle, or you can do it with paper folding and scissors.
- Your final figure should be an equilateral triangle made up of 6 copies of triangle OAB.
- There should also be 6 symmetries of the final figure, 3 rotations and 3 line reflections. Label these transformations and also label the images of OAB by the transformations so that you can see the relationship between the number of symmetries and the number of copies of OAB.
- TURN IN the labeled figure that shows these relationships.
Figure 2
Construct an isosceles triangle OAB so that the two lines OA
and
You should get 4 copies of the triangle OAB in your figure.
But you should find 8 symmetries of the figure, not 4.
- Label the figure clearly to show the lines of symmetry and the rotations that are symmetries.
- Explain why the figure has more symmetries than the number of triangles.
- If p is a regular n-gon inscribe in a circle with center O, then how many lines are there through O and a vertex (careful!
TURN IN both figures and the answers to the questions.
Experiment C: Regular n-gons
This part can be done precisely with Sketchpad or on paper with hand-drawn sketches, etc. With Sketchpad you can create n-gons by rotating one point around a center!
TURN IN. The answers to these questions, with some kind of illustrative figure for each (it can be a rough sketch, it does not have to be precise).
- Draw (several regular polygons and tell how many line symmetries a regular n-gon has and how many rotations that are symmetries.
- Do all the mirror lines of the line symmetries pass through the vertices of the n-gon? If not, what are the others?
- What are the (different) angles of rotation for the rotations that are symmetries of a regular n-gon?
- List the symmetries of a rectangle. .
- List the symmetries of a parallelogram.
- List the symmetries of a rhombus.
Experiment D: Rosette Groups
- Draw an example of a figure that has exactly 5 rotations that are symmetries, and no line symmetries. Such a group of rotational symmetries is called a rosette group.
- Suppose a figure has a symmetry T that is O144, the rotation by 144 degrees with center O. By taking powers of T, list the other rotations that MUST be symmetries because of the group property of symmetries.
- Show in a sketch how one point P is rotated by T to P' then to P'' = T(T(P), etc Connect P to P' with a segment and the P' to P'', etc until the point P is reached again.
- Now suppose that another figure has a symmetry T that is O50, the rotation by 50 degrees with center O. List the powers of T that must also be symmetries of the figure. How many such symmetries are there (be sure to include the identity)? If the figure also has a line symmetry M, if you multiple each of these rotations by M, to get MT, etc, you will get a set of line symmetries equal to the number of rotations.
- Putting these together, what regular n-gon has this set of symmetries, exactly?