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Symmetry: In and Out of Mathematics

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What is it?

In order to explain what symmetry is we need an axis, an imaginary straight line. Symmetry only exists with respect to an axis:


Now that we have an axis we want to know whether or not two images or figures are symmetrical with respect to this axis. To find out there is a very simple trick: we just have to imagine that it is a piece of paper folded in half. If the figures coincide with each other when we fold the paper, then they are symmetrical, and if they don’t coincide, then they aren’t.

We can try an example on real paper. We fold it in half to make the axis for symmetry and with a large marker that bleeds through the paper, we draw the figure we would like – don’t forget to protect the table that you are working on. For example:


Leave the paper to dry and then unfold the paper. The picture has also transferred to the other side and has created two symmetrical figures with respect to the axis. With the paper folded, they coincide exactly.


Another way to understand it is like a reflection in a mirror. A mirror is like an axis of symmetry, and if one image is reflected in a mirror then they are symmetrical and if it is not, then they aren’t.

Symmetry Exercises on Smartick

In the post ”Smartick’s New Contents,” we have added examples of symmetry activities to the sequence. The difficulty varies through the shape of the figures and their orientation on the axis of symmetry. This difficulty increases little by little, facilitating learning and comprehension about this concept. These exercises favor the development of spatial awareness and geometric reasoning.

We have exercises where they need to analyze if two figures are symmetrical:


In others they need to draw symmetrical figures:


Or place a series of dots in a symmetrical way with respect to an oblique axis:


Typical Errors in Symmetry Exercises

Are two figures symmetrical with respect to an axis? There are two mistakes that students often make when faced with this question.

  • Thinking that if two figures are identical then they are symmetrical:


To correct this mistake it is useful to think of the grid like a piece of paper that you can fold for an axis of symmetry. If the figures do not coincide when folded, then they are not symmetrical. Another way is to think of the axis as a mirror and if that figure is not reflected, then they are not symmetrical.

  • Another common thing that confuses students is thinking that if a figure has a specular reflection then they are symmetrical, regardless of their position with respect to the axis:


We can use the same strategies as before to resolve this mistake. If we fold a piece a paper for an axis of symmetry, and the figures do not coincide and one figure is not a reflection of the other in the mirror of the axis – then these two figures are not symmetrical with respect to the axis.

Symmetry Outside of Mathematics

Symmetry surrounds us and is everywhere:

  • In a mirror or a reflection in water: the image reflected is symmetrical to the real image.
  • In ourselves: we have a right hand and a left hand, a right ear and a left ear, and each pair is symmetrical. Our body is divided into two symmetrical parts, right and left, with respect to a vertical axis that runs down the center, from our head to our feet.
  • The majority of houses and buildings have facades that are symmetrical with respect to a vertical axis.
  • Cars, toasters, mobile phones, a glass, a plate, a bottle, the television, the sofa… the majority of everyday objects have one or more axes of symmetry.
  • We can also find symmetry in art. Artists use it in painting, sculpture, music and countless other disciplines.
  • As well as nature. Most animals and plants have some type of symmetry: bilateral, radial…

If you want to learn more about geometry and mathematics, go to Smartick and try it for free.

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