Symmetry Group of the Square V8 Fall 2020 (Updated)


Assume a square is at a fixed potion and orientation in space. The symmetry group of such a square is the subgroup of the rigid motions of space that transform the square into itself. To see the effects of these spacial rotations, we need to mark the four vertices of a square with different colors. A rotation that transforms the square into itself will cause a permutation of its vertices which can be observed because of their markings. Consequently, the symmetry group is finite since there are $4! = 24$ permutations of the four vertices. However, most of these permutations are not possible because, we are only considering rotations of space and not more general transformations. Assuming the vertices are marked using the colors red, green , blue, yellow, in counterclockwise order, the elements of the square's symmetry group are

  • I is the identity element; that is, the rigid motion that does nothing!,
  • $R^k, 1 \le k \le 3$, is a counterclockwise rotation around its center through an angle of $\dfrac{k \pi}{2}$,
  • H is the flip around the horizontal line passing through square's center,
  • V is the flip around the vertical line passing through square's center,
  • D is the flip around the diagonal line through the Red and Blue markers,
  • E is the flip around the diagonal line through the Green and Yellow markers,

  • This app can be used to
    Note: In the program below, to aid you in remembering the square's original orientation, we display four fixed spherical markers whose colors match the four vertices colors when the square is in its original orientation. The square's symmetry group is also known as the Dihedral group $D_{4}$.


    The first row of buttons represent the elements of the symmetry group of the square. Clicking a button will cause the corresponding spacial rotation described above to be performed visually on the square it its current orientation transforming it to a new orientation. The label below the square will be updated to show the element of the group that corresponds to its new orientation. Clicking a sequence of buttons corresponds to composing a sequence of spacial rotations.

    Note: Because transformations are evaluated from right to left, the sequence of transformations $f$ followed by $g$ is written formally as $g \circ f$ or more simply as $gf$ when $f$ and $g$ are regarded to be elements of a group.