Post #6: Symmetries of a dihedral square, known as D_4 or D_8.

 

Consider a square centered at (0,0) on the Cartesian plane, with corners at (1,1), (1,-1), (-1,-1) and (-1,1). If we rotate the square any multiple of 90°, the four corners will line up at the original corners. If the square has no identifying marks on it, it looks exactly the same as the original drawing.

 

Let us assume we can pick the square up and turn it upside down. It will again look the same if we make these flip moves through the lines x=0, y=0, x=y or x=-y, the lines as 0°, 90°, 45° and 135°, respectively. This means there are eight positions that work to make an unmarked square look exactly as it originally did. Because the square is assumed to be two-sided, this is called a dihedral group, and this particular example is either called D4 or D8, depending on what book you read. Four is the number of sides and eight is the number of elements in the group. This author prefers D4.

 

 One way to represent these rotations and flips is through 2x2 matrix multiplication. The list presented here is split into the rotations, which includes the identity, technically a rotation that is some multiple of 360°, and the flips, which are designated M for mirror.

If you took linear algebra at some point, you might recall that AB and BA are not always equal. The special cases here are the 90° rotation, which does not commute with the mirror moves, and the 270° rotation, which also does not commute with the mirror moves. More than that M0° and M90° do not commute with M45° and M135°.

If you need a reminder of how matrix multiplication works, the number in position ij of the matrix AB is the dot product of  row i in the matrix A with column j in matrix B. While it may be a little confusing at first, in these particular matrices, ever entry will turn out to be either, 1, 0 or -1.

 

Tomorrow: The Cayley Table for D4.