Groups (6)

Fitting together the symmetry transformations for some other shapes

The 3-legged symbol has 3 symmetry transformations.

E, the stay-put transformation or identity.
U, the rotation through one third of a full turn about the centre.
V, the rotation through two thirds of a full turn about the centre.

Working out their interactions gives us the following symmetrical table.

Combining the transformations for the 3-legged symbol

	E	U	V
E	E	U	V
U	U	V	E
V	V	E	U

U and V are each inverses of each other, since together they give the stay-put transformation E.

You'll probably have noticed that this group behaves in the same kind of way as the group of rotations for the square. Groups like these which are based on equal turns are called cyclic groups.
The 3-legged symbol gives us the cyclic group of order 3 called C₃.
The 4 rotations of the square give us the cyclic group of order 4 called C₄.

The butterfly again has just two symmetry transformations.

The stay-put transformation of E.
Reflection in the mirror-line through its body, which I'll call P.

It gives the little table I show below.

Combining the transformations for the butterfly

	E	P
E	E	P
P	P	E

How about the shape which I show below?

How many symmetry transformations does it have? Can you make a table showing their interactions? Do you think they form a group?

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