*Eric Istre*Show bio

Eric has taught high school mathematics for more than 20 years and has a master's degree in educational administration.

Lesson Transcript

Instructor:
*Eric Istre*
Show bio

Eric has taught high school mathematics for more than 20 years and has a master's degree in educational administration.

Translational symmetry exists in patterns that we see in nature and in man-made objects. This lesson will define it and provide a few examples with accompanying explanations and images.

Have you ever seen a wallpaper border with a pattern on it? Did you notice that a particular object or group of objects is repeated over and over again? Patterns like this occur in many man-made objects like quilts and rugs, as well as bricks on a house or the tiles of a patio. Patterns of this type also occur in nature in things like a honeycomb or the markings on the skin of a snake. There are plenty of other examples as well. If you have seen any of these things, then you have likely seen something with translational symmetry.

**Translational symmetry** is when something has undergone a movement, a shift or a slide, in a specified direction through a specified distance without any rotation or reflection. The distances between points within the figure will not change. The angles within the figure will not change. The size and shape of the figure will not change. The only thing that changes is its location. It may be moved right or left. It may be moved up or down. It may be moved through a combination of these two, but these are the only possibilities.

Take a look at this image of a house with smoke coming out of the chimney.

The house on the left has been translated up and right to create another house. It has been translated down and right to make another house. It has also been translated further to the right to make another house. The pattern is clear. If we performed the same translation on each of the new houses, the pattern would continue. This pattern could be performed an infinite number of times, and the pattern would be predictable. Translational symmetry is only a characteristic of infinite patterns. However, the concept can be applied to finite patterns if we use our imagination and pretend that the pattern we see actually continues forever. Since the pattern is treated as if it continues indefinitely in the horizontal and vertical directions, then the entire visible portion of the pattern could be translated. It might look like this after such a translation:

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As long as a portion of the figure can be translated in such a way as to make up exactly or fit perfectly onto another portion, regardless of which portion is chosen, then it has translational symmetry.

What if you are looking at a pattern, and you are trying to determine if it has translational symmetry? How can you do this? If the pattern can be divided by straight lines into a sequence of images that are identical, then there is translational symmetry. The straight lines can be vertical, horizontal, or diagonal. In this image, a vertical line has been drawn that has divided the pattern into identical parts.

Here a horizontal line has been drawn to separate the pattern into identical parts.

Finally, here a diagonal line has been drawn to subdivide the figure into identical parts.

Remember that the identical parts could be moved to lie perfectly on top of each other.

**Translational symmetry** is common in many of the patterns we see. It technically only exists in infinite patterns, but we can apply the concept to finite patterns with a bit of imagination. It occurs when a piece of a pattern has been moved a specific distance and direction so that it fits perfectly onto itself. We can determine if it exists by using a straight vertical, horizontal, or diagonal line to divide the figure into identical parts. If we are able to do that, then the pattern has translational symmetry.

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