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What is Symmetry?


2023-09-08 |    0

 

In mathematics, many concepts find their origins in everyday life, such as lines, circles, and probability. When mathematicians work with these concepts, they express their fundamental properties symbolically. For example, in geometry, a circle is not considered a physical object but a set of all points equidistant from a central point.

According to philosophers like Plato and Kant, we can never truly describe something as it is. However, when you observe an object with a particular property and share it with a friend, they will likely agree. This is how mathematical discussions begin, and mathematics encompasses many topics, including symmetry.

Though you may not have heard mathematicians use the term, symmetry is closely related to the concept of automorphism, which is essentially another way of referring to symmetry. Every object in mathematics generates a set of automorphisms, indicating that symmetry plays a significant role in math.

Consider two graphs, L and R: 

graph 

When asked which graph is more symmetric, you may have chosen R, and mathematicians would concur. 

You have probably chosen R. That’s great! But a mathematician can say more here. Namely, all mathematicians will agree that R is 720 times more symmetric than L. Here, we note that math is a kind of extension of our language that allows us to see something new in common things. But let us return to symmetry and how to measure it in math. We’ll need two things. The first one is the set of edges E of our graphs. I’ll write E = {12, 23, 25, 34, 56, 67} for L and E = {12, 13, 14, 15, 16, 17} for R. Thus, E shows which graph points are connected by lines. A graph's points and lines are called vertices and edges, respectively.

The second thing we need is a vertex permutation. Both L and R graphs have 7! = 1×2×3×4×5×6×7 = 5040 vertex permutations. This is just the number of ways we can reorder the names of our vertices. Let us consider two vertex permutations of R:

 graph

In the first permutation, we swapped 2 and 3; in the second, we swapped 1 and 2. You likely observe that, in R, vertices 2, 3, 4, 5, 6, and 7 exhibit a form of symmetry, whereas vertex 1 differs from all of these six vertices. Notably, the second permutation (which involves swapping non-symmetric vertices) must introduce a change that characterizes the graph's symmetry. This change is simply the change of set E.

Indeed, for our first permutation in R, E remains {12, 13, 14, 15, 16, 17}, indicating that the same pairs of vertices remain connected. However, E becomes {12, 23, 24, 25, 26, 27} for the second permutation. Consequently, the symmetry of a graph refers to permutations that preserve the set of edges E. In the case of R, 6! or 720 out of 5040 permutations maintain this property, establishing that R possesses 720 symmetries. Conversely, L has only one symmetry, specifically when the vertex names remain unchanged. 

This concept of symmetry and preserving structures extends to other mathematical objects like groups, rings, and fields. The set of symmetries always forms a group. Therefore, understanding groups is crucial in studying symmetry. Interestingly, groups and graph theory are closely intertwined. Learning graph and set theory with Russian Math Tutors provides insights into all possible groups, making it a fundamental aspect of mathematical exploration.

Exercise:

 

  1. Draw some graph with vertices 1, ..., 7.
  2. Write down the set of edges E of your graph.
  3. Draw some vertex permutation of your graph and the set E for it.
  4. Did E change? Is your permutation symmetric?
  5. Try to draw all symmetric permutations of your graph.

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