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Symmetries in 3-D Shapes


So far, we've seen how symmetry works with 2D figures. What about 3D shapes though? Can they have symmetry too? 

Symmetry of a Cube

First, let's have a look at a simple 3D shape, the cube. 

Source: User BenFrantzDale, Wikimedia Commons, licensed under CC BY-SA 3.0

Planes of Reflectional Symmetry

Remember that for two dimensional shapes, "reflectional symmetry" refers to a "mirror image" on both sides of a line. On a square, there are four axes of reflectional symmetry:

Source: User Kokcharov, Wikimedia Commons, licensed under CC BY-SA 3.0

Reflectional symmetry involves breaking the two dimensional square into two halves by using a one dimensional line. If you think about breaking a three dimensional square into two halves, you'll see that you need to use a two dimensional plane:

1. A cube has 9 different planes of reflectional symmetry. See if you can find them all! 


Rotational Symmetry of a Cube

A square has rotational symmetry about its central point, as every time you turn it 90 \deg, you come back to the same shape. We call this 4-fold symmetry, since it comes back to the shape 4 times when you spin by degrees(360). 

In the same way, a cube has rotational symmetry. However, instead of rotating about a zero-dimensional point, cubes rotate about one-dimensional lines. The easiest way to think of this is to imagine a kebab with cube shaped pieces, with the skewer as the axis of rotation:

Source: User PDPhotos, Pixabay, Public Domain

As you spin the skewer, the pieces rotate. If they are skewed down the middle, as shown in the diagram below, they will end up being the same shape after spinning by 90 \deg degrees. Just like for squares, we call this 4-fold symmetry. 

1. There are 13 axes of rotational symmetry in total. See how many you can find! It may help to build a model of a cube and skewer so that you can try spinning it yourself. You can make a cube using a net or find out how you can do it using origami.

2. Of those that you found, how many are 2-fold? How many are 3-fold? How many are 4-fold? 


Symmetry of a Tetrahedron

Now that we've had a look at the cube, it's up to you to figure out the case for the tetrahedron. A tetrahedron is a pyramid made up of equilateral triangles:

Source: User Aldoaldoz, Wikimedia Commons, licensed under CC BY-SA 3.0

Just like a cube, the axes of reflectional symmetry will be planes, and the axes of rotational symmetry will be lines. Here are a few to get you started: 

Source: User Cronholm144, Wikimedia Commons, Licensed under CC BY-SA 3.0

1. There are 6 planes of reflectional symmetry for a tetrahedron. How many can you find? 

2. There are 7 axes of rotational symmetry for a tetrahedron. How many can you find? How many are 2-fold? How many are 3-fold? 

Again, you can use a net or origami to make your own tetrahedron, which may help you in visualising the symmetries. 

More Sides, More Symmetries?

Both the cube and the tetrahedron are examples of "regular polyhedra", meaning that they are made up of shapes which have all sides equal. We saw that as we went from triangular faces with three sides to square faces with four sides, the number of symmetries increased. Does this mean that as we look at regular polyhedra with even more sides, the number of symmetries will increase?

Source: Arne Nordmann, Wikimedia Commons, Public Domain

1. See if you can use the internet to research how many planes of reflectional symmetry and axes of rotational symmetry a dodecahedron has. A dodecahedron is a twelve-sided regular polyhedron made from regular pentagons (pictured above). 

2. Of the axes of rotational symmetry, how many are 2-fold, how many are 3-fold, how many are 4-fold and how many are 5-fold? 

If you want to check for yourself, you can once again make it with nets or origami (Good luck!). 

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