Chapter 14.8, Problem 10E

a.

To determine

To describe the $S^3$ mapping and find the symmetry of it.

Answer to Problem 10E

The four lines symmetries of the square do not form a subgroup, as there is no identity.

Explanation of Solution

Given information:

The figure of a square.

A pattern in which there is a congruent copy of a figure completely fill the plane without overlapping is called Tessellation.

A tessellation of the figure. This pattern has point symmetry and translational symmetry.

The figure of tessellation has the four rotational symmetry. Each symmetry has center O and rotates the figure onto itself. Note that $90°$ rotational symmetry is another name for point symmetry.

  $\begin{array}{l}\left(1\right)\text{ 9}0°{\text{ rotational symmetry: R}}_{0,90}\\ \left(1\right)180°{\text{ rotational symmetry: R}}_{0,180}\\ \left(2\right)270°{\text{ rotational symmetry: R}}_{0,270}\\ \left(3\right)360°\text{ rotational symmetry: I is the identity}\text{.}\end{array}$

The identity mapping always maps a figure onto itself, we usually include the identity when listing the symmetries of a figure.

The group table is given below,

$°$	I	${\text{R}}_j$	$R_k$	$H_0$
I	I	${\text{R}}_j$	$R_k$	$H_0$
${\text{R}}_j$	${\text{R}}_j$	I	$H_0$	I
$R_k$	$R_k$	$H_0$	I	${\text{R}}_j$
$H_0$	$H_0$	$R_k$	${\text{R}}_j$	I

Therefore,

The four lines symmetries of the square do not form a subgroup, as there is no identity.

b.

To determine

To describe $T^{-1}$ and find the symmetry of it.

Answer to Problem 10E

The mapping that maps every point to itself is called identity translation, hence the translation of T followed by $T^{-}$ keeps all point fixed.

The symmetry of $T^{-}$ is T

Explanation of Solution

The mapping that maps every point to itself is called identity translation, hence the translation of T followed by $T^{-}$ keeps all point fixed.

Therefore, the symmetry of $T^{-}$ is T

c.

To determine

The find the symmetries of the tessellation.

Answer to Problem 10E

The tessellation has four symmetries.

Explanation of Solution

Given:

The figure of the tessellation of fish.

Concept Used:

The figure of tessellation has the four rotational symmetry. Each symmetry has center O and rotates the figure onto itself. Note that $90°$ rotational symmetry is another name for point symmetry.

  $\begin{array}{l}\left(1\right)\text{ 9}0°{\text{ rotational symmetry: R}}_{0,90}\\ \left(1\right)180°{\text{ rotational symmetry: R}}_{0,180}\\ \left(2\right)270°{\text{ rotational symmetry: R}}_{0,270}\\ \left(3\right)360°\text{ rotational symmetry: I is the identity}\text{.}\end{array}$

The identity mapping always maps a figure onto itself, we usually include the identity when listing the symmetries of a figure.

d.

To determine

To find the symmetries satisfy the four requirements for the group.

Answer to Problem 10E

The symmetries satisfy the four requirements for the group

Explanation of Solution

Given:

The figure of the tessellation of fish.

Concept Used:

The group is commutative if it has four property,

1. The product of two symmetry is another symmetry.
2. The set of symmetries contains the identity.
3. Each symmetry has an inverse that is also a symmetry.
4. Forming of product is an associative group,

  $A°\left(B°C\right)=\left(A°B\right)°C$ for any three group.

Therefore, the group is commutative as it is an abelian group.