Definition 4.4.1. Symmetry.
A set of points \(A\) has symmetry of type \(T\) for some transformation \(T\) if and only if \(T(A)=A.\)
Section 4.4 Symmetries
Subsection 4.4.1 Explore Symmetries
Checkpoint 4.4.2.
Confirm \(T(x,y)=(-y,x)\) is a symmetry of the set \(\{ (1,1), (-1,1), (-1,-1), (1,-1) \}.\)
Solution.
To determine if \(T(x,y)\) is truly a symmetry of the set, we must input the values of the set into the formula. If the result is belongs in the set, we move on and check the other values. All values should yield others in the set in order for \(T(x,y)\) to be symmetric.
\begin{align*}
T(1,1) = & (-1,1)\\
T(-1,1) = & (-1,-1)\\
T(-1,-1) = & (1,-1)\\
T(1,-1) = & (1,1)
\end{align*}
Thus we find that \(T(x,y)\) is symmetric for the set.
Checkpoint 4.4.3.
Demonstrate \(T(x,y)=(-y,x)\) is not a symmetry of the set \(\{ (0,0), (1,0), (1,1), (0,1) \}.\)
Checkpoint 4.4.4.
What points do you have to add to \(\{ (1,1), (-1,1), (-1,-1), (1,-1), (1,0) \}\) so \(T(x,y)=(-y,x)\) is a symmetry of this set?
Solution.
Find which points must be added to the set by finding the transformation for each point currently in the set. Use \(T(x, y) = (-y, x).\)
- \(\displaystyle (1, 1) \rightarrow (-1, 1)\)
- \(\displaystyle (-1, 1) \rightarrow (-1, -1)\)
- \(\displaystyle (-1, -1) \rightarrow (1, -1)\)
- \(\displaystyle (1, -1) \rightarrow (1, 1)\)
- \(\displaystyle (1, 0) \rightarrow (0, 1)\)
Note that all of the points on the right of the arrows except one are already in the set. So we must add the missing point, \((0, 1).\)
Also note that the transformation of \((0, 1)\) is \((-1, 0),\) which is not included in the set. Repeat to find that the transformation of \((-1, 0)\) is \((0, -1)\) which is again not included. Repeat the process once more to find that the transformation of \((0, -1)\) is already included, so we are done.
The points we added are \((0, 1), (-1, 0),\) and \((0, -1).\)
Checkpoint 4.4.5.
List all symmetries of a square by labeling the vertices and giving the type and parameters for the transformations.
Solution.
There are 8 symmetries of a square: four rotations and four reflections. The square can be rotated by 90°, 180°, 270°, and 360° counterclockwise. Note that the rotation by 360° can be considered the identity transformation.
The square can be reflected on four axes: vertical, horizontal, and two diagonals.
Checkpoint 4.4.6.
List all symmetries of a regular \(n\)-sided polygon (\(n\)-gon) by labeling the vertices and giving the type and parameters for the transformations.
Solution.
A regular \(n\)-sided polygon has \(2n\) symmetries: \(n\) reflections and \(n\) rotations. Each rotation is of an integer multiple of \(2\pi/n\) radians around the center of the polygon. The lines of reflection depend on whether \(n\) is even or odd. If \(n\) is even, then the lines of reflection are found by joining opposing vertices or midpoints of opposing sides. If \(n\) is even, each line of reflection is found by joining a vertex to the midpoint of the opposing side.
Checkpoint 4.4.7.
For one of the regular \(n\)-gons check the following.
- What is the composition of two rotational symmetries?
- What is the composition of two reflectional symmetries?
- What is the smallest number of symmetries you can use to generate all the symmetries?
Checkpoint 4.4.8.
Draw some regular \(n\)-gon. Color in the \(n\)-gon so that the colored figure maintains the rotational symmetries, but not the reflectional symmetries.
Checkpoint 4.4.9.
Draw some regular \(n\)-gon. Color in the \(n\)-gon so that the colored figure maintains the reflectional symmetries, but not the rotational symmetries.
Solution.
Note we found that the composition of two reflectional symmetries is a rotation for a regular polygon. If we draw a pattern (or coloring) on a regular polygon, to maintain reflectional symmetries, we may reflect it across all \(n\) symmetry lines until it's satisfactory. Note we can take the pattern, reflect it across some line, then reflect the pattern in the new position across another line; the pattern found in the final, third position can also be found by rotation. Without loss of generality, we can do this with all combinations of reflections, to show any rotation will result in the same shape as before. Therefore, for a regular polygon, if the figure maintains reflectional symmetry, it must maintain rotational symmetry. So, the purpose of this checkpoint can not be fulfilled.
Checkpoint 4.4.10.
Draw a figure that has translational symmetry.
Checkpoint 4.4.11.
Draw a figure that has translational symmetry and exactly one reflectional symmetry.
Checkpoint 4.4.12.
Draw a figure that has translational symmetry and rotational symmetry.
Checkpoint 4.4.13.
Draw a figure that has dilational symmetry.
Subsection 4.4.2 Explore Tesselations
Definition 4.4.14. Tesselation.
A covering of the plane is a tesselation if and only if it consists of a single shape infinitely reproduced using a finite set of transformations.
Definition 4.4.15. Tiling.
A covering of the plane is a tiling if and only if it consists of a finite set of shapes infinitely reproduced using a finite set of transformations.
Checkpoint 4.4.16.
Analyze the tesselation in Figure 4.4.21 as follows.
- Identify the generating shape.
- Identify the smallest set of transformations that can generate the tesselation.
- List all symmetries of the tesselation.
- Identify the smallest set of symmetries of the tesselation that can generate all the symmetries of the tesselation.
Definition 4.4.17. Conway Label.
The following labeling of tesselations derives from the book The Symmetries of Things by John Conway, Heidi Burgiel, and Chaim Goodman-Strauss. Follow these steps in order to identify and label the type of symmetry group of a tesselation. The resulting notation is called the signature
- Identify all lines of reflection.
- If two or more lines of reflection intersect in a point, write \(*n_1n_2 \ldots\) where \(n_1,\) \(n_2,\) are the number of lines intersecting at each unique point of intersection.
- If any line of reflection does not intersect other lines of reflection, just write one \(*\) for each of these.
- Identify any rotations that are not the composition of reflections already listed.
- Write \(n_1n_2 \ldots \) in front of any \(*\) for each rotation where \(n_1,\) \(n_2\) are the order of the rotations.
- Identify any glide reflections that are not the composition of reflections or rotations already listed.
- Write \(\times\) at the end of the signature for each of these glide reflections.
- Identify any translations that are not the composition of other symmetries already listed.
- Write \(\circ\) at the front of the signature for each pair of these translations.
See the example signatures in Figure 4.4.22 to Figure 4.4.23.
Checkpoint 4.4.18.
Find the signatures of two tesselations from the class archive at
http://www.math.uaa.alaska.edu/~afmaf/classes/math305/notes/tessel_gallery/gallery.html. You may not choose two with the same signature.Checkpoint 4.4.19.
Find the signature of the tesselation in Figure 4.4.21.
Checkpoint 4.4.20.
Begin the tesselation project.
Subsection 4.4.3 Tesselation Images
