Chapter Questions
Show that an isometry of Rn is one-to-one
Exhibit a plane figure whose plane symmetry group is Z5.
Show that the group of rotations in R3 of a 3-prism (that is, a prismwith equilateral ends, as in the following figure) is isomorphic to D3.
What is the order of the (entire) symmetry group in R3 of a 3-prism?
What is the order of the symmetry group in R3 of an n-prism?
Show that the symmetry group in R3 of a box of dimensions 20 330 3 40 is isomorphic to Z2 % Z2 % Z2.
Describe the symmetry group of a line segment viewed asa. a subset of R1.b. a subset of R2.c. a subset of R3. (This exercise is referred to in this chapter.)
(From the “Ask Marilyn” column in Parade Magazine, December 11,1994.)* The letters of the alphabet can be sorted into the followingcategories:1. FGJLNPQRSZ2. BCDEK3. AMTUVWY4. HIOX What defines the categories?
Exactly how many elements of order 4 does the group in Example 1 have?
Why is inversion [that is, $\phi(x, y)=(-x,-y)]$ not listed as one of the four kinds of isometries in $R^{2}$ ?
Explain why inversion through a point in $\mathbf{R}^{3}$ cannot be realized by ? rotation in $\mathbf{R}^{3}$.
Reflection across a line $L$ in $\mathbf{R}^{3}$ is the isometry that takes each point $Q$ to the point $Q^{\prime}$ with the property that $L$ is a perpendicular bisector of the line segment joining $Q$ and $Q^{\prime} .$ Describe a rotation that has this same effect.
In $\mathbf{R}^{2}$, a rotation fixes a point; in $\mathbf{R}^{3}$, a rotation fixes a line. In $\mathbf{R}^{4}$, what does a rotation fix? Generalize these observations to $\mathbf{R}^{n}$.
\text { Show that an isometry of a plane preserves angles. }
Show that an isometry of a plane is completely determined by the image of three noncollinear points.
Suppose that an isometry of a plane leaves three noncollinear points fixed. Which isometry is it?
Suppose that an isometry of a plane fixes exactly one point. What type of isometry must it be?
Suppose that $A$ and $B$ are rotations of $180^{\circ}$ about the points $a$ and $b$, respectively. What is $A$ followed by $B ?$ How is the composite motion related to the points $a$ and $b ?$