Lecture Notes: Symmetries & Symmetry Groups

For more information about symmetry, see chapter 11 of your text. In particular, see section 11.7.

The symmetry of an object or shape is a motion that moves the object back onto itself. In other words the object looks the same after applying the motion as it did before.

Examples:

1. Symmetries of a Square: As discussed in class, the symmetries of the square are rotations by 90°, 180°, 270° and 360°, as well reflections about the horizontal (l₄), the vertical (l₃), and the two diagonal axis (l₁ and l₂).

°	l₁	l₂	l₃	l₄	R₉₀	R₁₈₀	R₂₇₀	e = R₃₆₀
l₁	R₃₆₀	R₁₈₀	R₉₀	R₂₇₀	l₃	l₂	l₄	l₁
l₂	R₁₈₀	R₃₆₀	R₂₇₀	R₉₀	l₄	l₁	l₃	l₂
l₃	R₂₇₀	R₉₀	R₃₆₀	R₁₈₀	l₂	l₄	l₁	l₃
l₄	R₉₀	R₂₇₀	R₁₈₀	R₃₆₀	l₁	l₃	l₂	l₄
R₉₀	l₃	l₄	l₁	l₂	R₁₈₀	R₂₇₀	R₃₆₀	R₉₀
R₁₈₀	l₂	l₁	l₄	l₃	R₂₇₀	R₃₆₀	R₉₀	R₁₈₀
R₂₇₀	l₃	l₄	l₂	l₁	R₃₆₀	R₉₀	R₁₈₀	R₂₇₀
e = R₃₆₀	l₁	l₂	l₃	l₄	R₉₀	R₁₈₀	R₂₇₀	R₃₆₀

Filling in the above table shows that, the set of symmetries of the square along with the operation composition, , forms a group known as the Dihedral group D₄. The square is said to have D₄ symmetry type. Note: The element in the R₉₀ row and the l₃ column corresponds to l₃ ° R₉₀ , i.e. perform a rotation counter-clockwise 90 degrees and then flip around the l₃ axis and you'll get the same thing as if you flip around the l₁ axis.

2. Symmetries of an equilateral triangle: The symmetries of an equilateral triangle are rotations by 120°, 240°, and 360°, as well as reflections about the three symmetry axes, l₁, l₂, and l₃.

3. Symmetries of a propeller: The symmetries of a propeller are rotations by 90°, 180°, 270°, and 360°. The set of symmetries of a propeller with the composition operation form an Albelian group, as shown in the following table. Clearly, rotations or 360° act as the identity element under composition. All rotations have an inverse under composition. Example, the composition of a 270° rotation and a 90° rotation is the same as a 360° rotation, or no rotation at all. The group formed by the symmetries of a propeller and the composition operation is known as Z₄. The propeller is said to have Z₄ symmetry.

°	R₉₀	R₁₈₀	R₂₇₀	e = R₃₆₀
R₉₀	R₁₈₀	R₂₇₀	e	R₉₀
R₁₈₀	R₂₇₀	e	R₉₀	R₁₈₀
R₂₇₀	e	R₉₀	R₁₈₀	R₂₇₀
e = R₃₆₀	R₉₀	R₁₈₀	R₂₇₀	R₃₆₀

4. Z₂ symmetry: We’ve already seen examples of shapes with D₄ and Z₂ symmetry. There are also shapes with what is known as Z₂ symmetry. These are shapes whose symmetry is just the set of two elements, Rotations of 180° and rotations of 360°. The face cards of most decks have Z₂ symmetry, as do the letters Z, and S. The set {R₃₆₀, R₁₈₀} along with composition forms an Albelian group known as Z₂. The following shapes all have Z₂ symmetry.

°	R₁₈₀	e = R₃₆₀
R₁₈₀	e	R₁₈₀
e = R₃₆₀	R₁₈₀	R₃₆₀

5. D₁ symmetry: The set {R₃₆₀, l₁}, i.e. rotation of 360° and reflection about one axis, also form an albelian group with composition. This group is called D₁. Examples of D₁ symmetry are shown below.

°	l₁	e = R₃₆₀
l₁	e	l₁
e = R₃₆₀	l₁	e

6. Z₁ symmetry: The R₃₆₀ forms a rather trivial one element group known as Z₁. Elements whose only symmetry is rotation by 360° are said to have Z₁ symmetry. Examples of Z₁ symmetry are shown below.

°	e = R₃₆₀
e = R₃₆₀	e