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Christmas maths activity: Stars and rotational symmetry
One of the most used Christmas motives are stars. As ornaments they appear in various designs and with various numbers of spikes, but one thing is common to most Christmas star designs: they posess rotational symmetry.
by Franka Miriam Brueckler
Rotational symmetry is one of the possible types of symmetry. An object is said to posess it if one can turn the object for an angle to a position indistinguishable from the original one. The order of the rotational symmetry is the number of such matches in one full turn.
An easy and cheap hands-on activity to experience this property is making of stars from paper, foil or some other easily cuttable and foldable material (since our activity involves folding, cardboard would not be a good choice; you can glue your “thin” star onto cardboard later if you want to use it as Christmas tree decoration). If you want to make a star with rotational order of your choice (we suggest numbers from 4 to 10 so that your folded model does not get too thick), start with a square sheet of, say, paper. Mark the midpoints of the sides and the midpoint of the paper with folds as shown in the picture below. If your chosen order of symmetry is even, you can make the “horizontal” fold go full from left to right.
Now, for any chosen order n of symmetry you have to construct the angle equal to 1/n of full angle with vertex in the center of the paper. Say, if you want to make a star with symmetry order 10, you need a central angle of 360°/10 = 36°. You can of course do that using a protractor and tracing the corresponding lines with a thin pencil. For some angles it is worthwile to try to construct them in a mathematically precise way by folding. We present the constructions for the two simple cases (rotational orders 4 and 8, i.e. central angles of 90° and 45°), and one not that simple case (rotational order 6, i.e. central angle of 60°). Although there exist exact constructions for some other orders (e.g. for the mentioned order 10), they are relatively complicated and we leave the description of these folding sequences for some other article.
Now, if your chosen order is 4, all you have to do is fold your square twice in halves, to obtain foldlines as in the picture below.
If your chosen order is 8, you also need to fold the diagonals, to obtain the following situation:
The instructions for folding a central angle of 60° are as follows, and illustrated by the pictures below. Starting from the initial horizontal fold in half, fold the upper half once more into halves, to obtain an additional fold as in the first of the pictures below. Now comes the tricky part: bring the upper midpoint onto the fold obtained in the previous step (second picture below) in such a way that the obtained fold passes through the centre of the paper (third picture below, the colour red represents the back side of your sheet of paper). It sounds complicated, but is not that hard – just try it! When you open your paper you will see the constructed angle of 60° (fourth picture below). Now repeat the second and third step mirrorwise (to the left) and you will obtain the six needed angles of 60° (fifth picture below).
Whatever your chosen order n of symmetry was, and whichever method of obtaining n lines under angles 360°/n through the centre (protractor and pencil, or folded constructions from above), you should fold your paper repeatedly along these lines until you obtain a triangle (in the case of order 4 you will obtain a square) with n layers. Cut through these layers along a line of your choice positioned like the green line in the picture below. The remaining part of paper is the basic triangle whose rotation around the centre about the angle 360°/n once, twice, … , “n-ce” results in your star – unfold your paper to see your star. Now, compare your star with the stars others have made and check out which order of rotational symmetry was the most popular in your class, family, group of friends …