Using Symmetry to Describe Tessellations

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  • 0:01 Symmetry of Tessellations
  • 0:51 Line Symmetry
  • 1:34 Translational Symmetry
  • 2:25 Rotational Symmetry
  • 3:10 Glide Symmetry
  • 3:31 Lesson Summary
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Instructor: Kevin Newton

Kevin has edited encyclopedias, taught history, and has an MA in Islamic law/finance. He has since founded his own financial advice firm, Newton Analytical.

Tessellations are a very common way for geometric concepts to be applied in the real world. In this lesson, we'll look at how to use symmetry to help describe different types of tessellations.

Symmetry of Tessellations

Tessellations are patterns that take a shape and repeat it over and over again. We can find tessellations in plenty of places, from floors, to shower curtains, to art. However, how do you think we describe them? Sure, we could say that this tessellation has blue triangles or red squares, but surely there is a way to describe how the tessellation is patterned. Luckily, there is. We can describe a tessellation with respect to its symmetry. Remember that symmetry is when a shape is similar but in a different direction or orientation. In this lesson, we are going to take a look at the four types of symmetry that a tessellation can have, namely line symmetry, translational symmetry, rotational symmetry, and glide symmetry.

Line Symmetry

Let's start with line symmetry because it's the most straightforward. In order to have line symmetry, a shape must be a reflection of itself if a line is drawn down the middle. For example, if you were to draw a diameter line through a circle, the circle will have line symmetry. Now a tessellation doesn't have to be line symmetrical with respect to every line drawn. In fact, more often than not, it may only be line symmetrical if a specific angle of line is drawn. Also, make sure that it is a mirror image. That means that every point and characteristic of the shape is the same distance from the line of symmetry as its corresponding point on the other side of the line.

Translational Symmetry

Not every tessellation that has symmetry is symmetrical across a line. In fact, not even most of them are. Instead, all tessellations have translational symmetry. That's right, all of them. Translational symmetry occurs when you can move, or translate, a shape across the plane. Note that you do not rotate the angle of the shape at all.

Imagine a polka dot shower curtain; if you were to draw a line down the middle of it, chances are that it is not going to have line symmetry. However, it will have translational symmetry, especially if all the dots are the same size and have the same distance between them. Remember, to achieve translational symmetry, you just drag a shape. Since you are repeating a set pattern in a tessellation, that means that any tessellation you create will have translational symmetry.

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