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# Line Symmetry

Symmetry is something that most of us understand. After all, symmetrical objects are naturally pleasing to the human eye, whereas asymmetrical objects seem lightly unsatisfying for some reason. But do you know the mathematical definition of symmetry? What are the different types of symmetry? And what does symmetry teach us about math? Let's find out:

## The definition of symmetry

Before we get into the different types of symmetry -- including line symmetry -- let's define what this word actually means.

What is symmetry?

Symmetry stems from the ancient Greek term "symmetria," which translates roughly to "agreement in dimensions."

Symmetry essentially has two different definitions: On one hand, it may be used in a very casual manner to refer to beauty, proportionality, and harmony. On the other hand, the mathematical definition of symmetry is much more precise -- and there are specific rules regarding symmetry in geometry.

When mathematicians say that something is symmetrical, they mean that the object is "invariant." This means that it remains even if we transform it in various ways. For example, we might flip an object upside down, reverse it, or rotate it. A truly symmetrical object would remain the same no matter what we do to it.

For example, a perfect circle looks the same whether we rotate it, reverse it, or turn it upside down.

## Line symmetry, explained

Line symmetry is the simplest form of symmetry, and it's easy to see whether objects have line symmetry.

First, divide an object into two congruent (equal) parts. Do these two equal parts look the same? Are they reflections of each other? Note that we can make our lines in any direction (horizontal, vertical, or diagonal) -- as long as the two parts are equal.

Here are a few examples of line symmetry:

Let's look at the square first. This object has a whopping four different lines of symmetry. We can divide it horizontally, vertically, and with two additional diagonal lines. All of these lines create two congruent parts.

Next, let's look at the parallelogram. This object has zero lines of symmetry. No matter how we slice it, we can't create two congruent parts that are perfect reflections of each other.

Finally, let's take a look at the rectangle. This object is clearly symmetrical -- although not quite as symmetrical as the square. As you can see, it can be divided both horizontally and vertically, giving it two different lines of symmetry. It cannot be divided horizontally to create two different congruent parts.

But what about real-world examples of line symmetry?

One obvious example is a human face -- although this isn't really true because everyone's faces are slightly asymmetrical. In fact, perfectly symmetrical faces look somewhat strange. Another obvious example is the reflection of a tree on a lake. On the other hand, this example is also slightly asymmetrical because of the ripples in the water.

## Other types of symmetry

Aside from line symmetry, there are a few other types of symmetry we should cover:

• Rotational Symmetry: An image can be symmetrical even if it fails the line symmetry test. For example, if we rotate this symbol "~" 180 degrees around its center, it is exactly the same. This means that it has an "order" of 2. This symbol "+" has an order of 4, because it appears four times as we rotate it. This symbol "*" has an order of 6!
• Point Symmetry: Point symmetry is a bit more complicated. In order for an object to exhibit point symmetry, every part must have a matching part the same distance from the central point but in the opposite direction. In order to determine whether an object has point symmetry, it helps to graph it on a Cartesian plane. This lets us determine the distance of the object's points from the "origin," which is $\left(0,0\right)$ .

Here are some fun facts about symmetry:

• We are very good at detecting symmetry. In fact, the human brain can detect whether objects are symmetrical within just 150 milliseconds.
• Non-geometrical concepts can be symmetrical or asymmetrical. For example, your relationship with another student is more symmetrical compared to your relationship with your teacher. Why? Because you are equal to other students, while your teacher has the power to flunk you!
• Some examples of complex symmetrical patterns include Celtic knots and Persian rugs.

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