Symmetry Group of the Hexagon V5 Fall 2017

Description

Assume a regular hexagon is at a fixed potion and orientation in space. The symmetry group of such a hexagon is the subgroup of the rigid motions of space that transform the hexagon into itself. These turn out to be rotations of space that rotate the hexagon into itself. To see the effects of these spacial rotations, we need to mark the six vertices of a hexagon with different colors. A rotation that transforms the hexagon into itself will cause a permutation of its vertices which can be observed because of their markings. Consequently, the symmetry group is finite since their are at most $6! = 720$ permutations of the six 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, black, magenta in counterclockwise order, the elements of the hexagon's symmetry group are

• I is the identity element; that is, the rigid motion that does nothing!,
• $R^k, 1 \le k \le 5$, is a counterclockwise rotation around its center through an angle of $\dfrac{k \pi}{3}$,
• G is the flip around the line through the Green and Black markers,
• B is the flip around the line through the Blue and Magenta markers,
• Y is the flip around the line through the Yellow and Red markers,
• S is the flip around the bisector of the Red-Green and Black-Yellow edges,
• T is the flip around the bisector of the Green-Blue and Black-Magenta edges,
• U is the flip around the bisector of the Blue-Yellow and Red-Magenta edges

• 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 hexagon's original orientation, we display six fixed spherical markers whose colors match the hexagon vertices colors when the hexagon is in its original orientation. The hexagon's symmetry group is also known as the Dihedral group $D_{6}$.

Instructions

The first row of buttons represent the elements of the symmetry group of the hexagon. Clicking a button will cause the corresponding spacial rotation described above to be performed visually on the hexagon it its current orientation transforming it to a new orientation. The label below the hexagon 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 Undo and Redo buttons can be used to Undo and Redo previous transformations of the hexagon.
• The Reset button will restore the hexagon to its original orientation.
• The animation rate can be changed by using the Animation Rate menu.
• The hexagon along with its markers can be rotated in space by simultaneously holding down the Control Key and Clicking and Dragging the mouse.