## Symmetry Group of the Square V8 Fall 2020 (Updated)

### Description

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

- Verify that these elements are, in fact, closed under composition, have an identity element, and have inverses. Hence, they form a finite group.
- Discover the multiplication table of this group.

**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 $D_{4}$.

### Instructions

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.

*Details*

- The
**Reset**button will restore the square to its original orientation. - The animation rate can be changed by using the
**Animation Rate**menu. - The axes of rotations can be displayed by using the
**Show/Hide Axes of Rotation**button. - The square along with its markers can be rotated in space by simultaneously holding down the
**Control Key**and**Clicking**and**Dragging**the mouse.