Chapter 14 – Symmetry

You may want to draw a line right through the middle of a figure you have created to fold it into two exact halves. However, you cannot make use of a pencil and draw the line physically every time. Instead, you have to imagine the line to divide a particular shape symmetrically. This line that divides an object into two halves is known as a line of symmetry. While you can draw it vertically, you can also draw a horizontal or diagonal line to meet your objective. Remember that you are free to have multiple lines of symmetry for a single figure as well. Here are a few examples to help you understand the concept better:

• • 1 Line – You can divide a figure symmetrically around a single axis. Look at the word “ATOYOTA.” You can draw a line through ‘Y’ to obtain two identical halves.
  • 2 Lines – A figure can be made symmetrical by passing a couple of lines through it. You can do it using a vertical and a horizontal line passing through the letters H and X.
  • Infinite Lines– You may pass as many lines as you want through certain figures to get symmetrical parts. The most obvious example here is a circle. Since no specific lines are forming the figure, you can divide it into multiple symmetrical parts by passing infinite lines through it.

If you think about other geometrical figures apart from a circle, then you will find it interesting to note that the closed figures known as polygons can be divided symmetrically as well. Think of a triangle; it has three sides. You can divide an equilateral triangle into symmetrical halves with the help of a line. A square can be divided similarly by two diagonal lines. You are welcome to use multiple lines of symmetry to divide other polygons such as pentagon, hexagon, heptagon, octagon and so on.

You can get a symmetrical figure when you rotate a figure, move it to another position or flip it completely. Symmetry is of four different types described based on varied situations:

• • Translation– When an object slides about an axis, the movement is known as translation. The object moves down in the same orientation and remains the same in shape and size.
  • Rotational– When you turn an object around a point, it becomes identical to itself even in position. The rotational symmetry line is therefore, a point. The angle of rotational symmetry is quite small for a figure to coincide with itself. Almost all polygons can have rotational symmetry with other objects such as petals displaying properties of symmetry and rotational symmetry too.
  • Reflection– Reflexive symmetry is described as a situation where a half of the object reflects the other completely. It can also be described as a mirror image. The two longitudinal halves of a human body or a butterfly can serve as classic examples of reflection symmetry.
  • Glide- A combination of reflection and translation symmetry can be described as glide symmetry. It has commutative properties that reinforce the theory that even if the objects’ order is altered, the results remain the same. You will learn more of this in higher classes once you get introduced to calculus and theories of probability.

You will have much fun trying to identify symmetry in different shapes. Take a figure depicting half a fish horizontally, for instance. You can make use of reflexive symmetry to complete the figure by obtaining a full fish. Use rotational symmetry to draw a flower of uniform proportions. You also have to remember that not all figures happen to be symmetrical. You will come across a scalene triangle in geometry and other irregular objects in nature that do not conform to the theory of symmetry.

The idea of symmetry may be pretty obvious, but you need to examine the world around you to understand the implications and mathematical reasoning behind it. You will find it easier to organize things once you master the concept and realize how symmetry enables us to create a harmonious effect.