# Permutations corresponding rigid motions of a rectangle/rhombus

• Nov 8th 2010, 04:44 PM
kathrynmath
Permutations corresponding rigid motions of a rectangle/rhombus
Find the permutations that correspond to the rigid motions of a rectangle that is not a square. Do the same for the rigid motions of a rhombus that is not a square.

I began by drawing the rectangle and rhombus and labeling sides. I know I need to do something with permutations of the vertices. I feel like I need to do something with angles since they are not right angles, but I can not figure out what to do with angles.

Do I deal with angles at all? I know I need to do something with rotations and flips.
• Nov 8th 2010, 04:52 PM
HallsofIvy
You don't need to worry about the angles. It should be clear that "flips" around the side bisectors as well as 180 degree rotation about the center are motions of the rectangle. Since it is NOT a square, flips around the diagonals and 90 degree rotations about the center are NOT motions.

For the rhombus, flips about the diagonals are motions and flips about the side bisectors are not.
• Nov 8th 2010, 05:27 PM
kathrynmath
I don't know it still confuses me on writing the permutations
• Nov 8th 2010, 06:16 PM
HallsofIvy
Quote:

Originally Posted by kathrynmath
I don't know it still confuses me on writing the permutations

Label the four vertices of the rectangle or rhombus "1", "2", "3", "4". Each operation moves one vertex into another. For example, rotation around the perpendicular bisector of sides 12 and 34 rotates 1 to 2 2 to 1, 3 to 4, and 4 to 3. That is the permutation
$\begin{pmatrix}1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3\end{pmatrix}$
or, cycle notation, (12)(34).