Ideas in Geometry/Symmetry Groups

Learning about Symmetry Groups

Symmetries

There are two different major types of symmetries that can be found in math: rotational and reflective.

Rotational Symmetry describes the different rotations that a shape can make, while still looking exactly the same.

There are four major facts that can be useful with rotational symmetry:

- Rotations can be described in either fractions or degrees

- Rotations can be either clockwise or counterclockwise

- Many symmetrical shapes will have a rotation for every side of their shape

- Every shape has a rotation that brings the shape all the way back around, which can be referred to as a full rotation or null rotation

The pictures below illustrate a few examples of rotational symmetry (arrows illustrate the sides that can be rotated):

Reflective Symmetry describes the different ways that a shape can be reflected / folded across itself, without any of the shape hanging over the edge.

There are some basic facts that can be useful pertaining to reflective symmetry:

- It is possible for a shape to have no reflections

- The lines on the drawings are referred to as lines of reflection

The pictures below illustrate a few examples of rotational symmetry (with lines of symmetry included on the drawings):

Now that we have an understanding of the basics of symmetry, there are a few more topics to cover to fully understand symmetry groups.

MODS

When telling time on a clock, we know that there are only 12 hours, and after 12, we must go back to 1, correct? This is referred to as mod 12.

The basic characteristics of mods can be found in the curriculum for Math 119, which is where the following information comes from:

For n a positive whole number and 0≤a,b≤n - 1, we define the sum of a and b mod n in the same way:

-If a + b < n, then the sum of a and b mod n is a+b.

- if a+b≥n, then the sum of a and b mod n is a+b-n (the n must be subtracted to wrap back around to the beginning again)

When trying this to another example, say mod 5, we follow the same procedures. Lets look at some example problems:

a) 2+4 = 1 (because 2+3 = 5 and since there is 1 remaining, we bring it back around to 1)

b) 3+1 = 4 (since this number does not pass 5 and have to come back around, therefore the final answer is 4)

c) 4+3 = 2 (since it only takes 1 to get to 5, we then have 2 left over, which makes the final answer 2)

Taking It One Step Further

The Following information is borrowed from Math 119's Ideas in Geometry Textbook:

A group is a set with an operation * such that:

-For all x, y in the group, x * y is also in the group.

-There is a special member of the group I called the identity, such that for any x in the group we have x * I = I * x = x

-For any x in the group there is a member if the group x^-1 called x's inverse such that x * x^-1 = x^-1 * x = I

-The operation * is associative. That is, x * y * z = x * (y * z) = (x * y) * z

Thus, in the example of the set {0, 1, 2, 3, 4}, with addition mod 5, it is considered a group.

Conclusion

There are much deeper levels that can be dove into before fully understanding the subject of symmetry groups, but this should give most a basic understanding of the subject matter, which will allow for more extensive thought down the line. The basic conclusion that we can come to is that through the use of the symmetries at the beginning (reflective & rotational), which make up the symmetry group for that particular group, along with the mods, we can gain a general understanding of what Symmetry Groups truly are. It may not be a full understanding, but it is more than we knew before.