Symmetries of Polyhedra

Section 1.2 Symmetries of Polyhedra

Now that we understand the terminology and basic structure of polyhedra, we can consider their symmetries. Symmetry is an additional description of the structure of polyhedra which rather than noting which faces are adjacent, describes which ones are the same in special ways. We will describe three types of symmetries: rotational, reflectional (mirror over a plane), and mirrors through a point.

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Subsection 1.2.1 Illustrate Symmetries

Rotational symmetry means that if we rotate the object, the object looks as if it had not been rotated. This rotation must be less than 360° (all the way around) to be interesting. Consider the example of rotational symmetry below.

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Figure 1.2.1. 3 and 2 fold Rotational Symmetries

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When we rotate around the vertical axis (the blue arrow pointing up) so that corner 1 moves to corner 2, the shape (triangular prism) looks unchanged. After three rotations of this type, every corner is back in the original location. This is why we refer to it as order 3. When we rotate around the horizontal axis (the blue arrow pointing forward) so that the top moves to the bottom and vice versa, the shape looks unchanged. The order is 2 because we only rotate twice.

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Reflectional symmetry means one half of the object is the mirror image of the other. Consider the example of reflectional symmetry below which shows how corners of one half of the object match those on the other half.

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Figure 1.2.2. Reflectional Symmetries

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The top and bottom halves of this shape are the same. Also, the left and right sides of this shape are the same. Thus this shape has two, distinct reflectional symmetries. Do you see any others?

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In 3D there is another type of reflection. Instead of reflecting across a plane (mirror), it reflects through a point. This means that every point can be paired to another point so that the line segment connecting them goes through this one point. Consider the example of reflection through a point below.

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Figure 1.2.3. Reflection through a Point Symmetries

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For reflection over a plane, points on either side of the plane swap. For reflection through a point we see more swapping. Top and bottom swap. Left and right swap. Front and back swap. Reflection through a point is the same as reflecting through three, mutually perpendicular planes. Play with the figure below to see how each vertex changes location.

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Figure 1.2.4. Reflection Through a Point

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Subsection 1.2.2 Identify Symmetries

To identify symmetries of a shape, we will look for matching faces and the figure out how the shape has to move (rotate or reflect) for one face to move to the location of the matching one. The following illustrates this process.

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Figure 1.2.5. Identifying Symmetries

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Checkpoint 1.2.6.

Start by rotating the image below so you have seen all the faces.

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(a)

Pick a face and find a second face that is the same shape.

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(b)

Identify what symmetry would move the first face onto the second. Specify the location of a mirror or rotation axis.

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(c)

Test if that symmetry maps every other face to a matching face. If so it is a symmetry.

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(d)

Repeat these steps until you have checked every pair of matching faces.

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Recall that more than one symmetry may exist to map a face to another. You are trying only to find a way to map each face to another not necessarily every possible symmetry.

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Figure 1.2.7. Identify Symmetries of this Polyhedon

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The next example illustrates identifying symmetries when we can not see all the faces at once.

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Figure 1.2.8. Multiple Symmetries

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