# 4.4: Symmetries

- Page ID
- 89855

## 4.4.1 Explore Symmetries

A set of points A has symmetry of type T for some transformation T if and only if T(A) = A.

Confirm T(x, y) = (-y, x) is a symmetry of the set {(1, 1), (-1, 1), (-1, -1), (1, -1)}.

Demonstrate T(x, y) = (-y, x) is not a symmetry of the set {(0, 0), (1, 0), (1, 1), (0, 1)}.

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?

List all symmetries of a square by labeling the vertices and giving the type and parameters for the transformations.

List all symmetries of a regular n-sided polygon (n-gon) by labeling the vertices and giving the type and parameters for the transformations.

For one of the regular n-gons check the following.

- What is the composition of two rotational symmetries?
- What is the composition of two reflection symmetries?
- What is the smallest number of symmetries you can use to generate all the symmetries?

Draw some regular n-gon. Color in the n-gon so that the colored figure maintains the rotational symmetries, but not the reflectional symmetries.

Draw some regular n-gon. Color in the n-gon so that the colored figure maintains the reflectional symmetries, but not the rotational symmetries.

Draw a figure that has translational symmetry.

Draw a figure that has translational symmetry and exactly one reflectional symmetry.

Draw a figure that has translational symmetry and rotational symmetry.

Draw a figure that has dilational symmetry.

## 4.4.2 Explore Tesselations

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.

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.

Analyze the tesselation 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.

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
_{1}n_{2}... 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
_{1}n_{2}... 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 × 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 ○ at the front of the signature for each pair of these translations.

See the example signatures in Figures \(\PageIndex{1}\) to \(\PageIndex{4}\).

Find the signatures of two tesselations from the class archive at here. You may not choose two with the same signature.

Find the signature of the tesselation in Figure \(\PageIndex{1}\).

Begin the tesselation project.