Groups (3)
The symmetry transformations for a square
There are 8 different symmetry transformations for the square.
I've drawn them below, showing the differences between them by the new
positions of the letters, and also described them in words.
In the same order as the drawings, they are
- E .. This is the 'stay-put'
transformation. E is called the
identity.
- U .. This rotates the square through
a quarter turn.
- V .. This rotates the square through
a half turn.
- W .. This rotates the square through
a three-quarters turn.
Each rotation is about the centre of the square and I've drawn them
anti-clockwise since mathematicians count this direction as positive.
There are also 4 reflections.
- P .. This reflects about a vertical
mirror line through the centre.
- Q .. This reflects about a
horizontal mirror line through the centre.
- R .. This reflects about a
diagonal mirror line through AC.
- S .. This reflects about a
diagonal mirror line through BD.
If you choose any pair of rotations from E, U, V and W, and do one followed
by the other, then the result will also be one of E, U, V or W.
We can show all the possible interactions in this neat little
self-contained table. (I've used two colours to distinguish its inside from
its outside.)
Combining rotations of a square
| |
E |
U |
V |
W |
| E |
E |
U |
V |
W |
| U |
U |
V |
W |
E |
| V |
V |
W |
E |
U |
| W |
W |
E |
U |
V |
I am doing the rotation shown in red in the vertical column first and then
following this by the rotation shown in red in the horizontal column.
In fact, this particular table would look exactly the same if I
worked the other way round.
You can also see from this table that each rotation has a 'partner' so that
together they give the stay-put transformation of E.
U and W have each other, V has itself and E is E anyway.
The partner of any transformation is called its
inverse.
Is it possible to make a similar little self-contained table showing the
interactions of the four reflections?
Try working out what happens if you carry out the following pairs of
reflections:
- P followed by P
- P followed by R
- R followed by P
What will the table for P, Q, R and S look like? The
answer is quite surprising!
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