Groups (4)
How do the reflections of a square fit together?
If you worked out the joint effects of the three pairs of
reflections which I suggested, you will have seen straight away
that P, Q, R and S are not going to combine together in the same
sort of self-contained way that E, U, V and W do.
Not only that, but the order that we do the transformations in
matters. This is what we get.
P followed by P gives E.
P followed by R gives U.
R followed by P gives W.
Here is what the table showing all the
interactions of P, Q, R and S looks like.
Combining reflections of a square
The reflection labelled in the vertical column
is done first.
| |
P |
Q |
R |
S |
| P |
E |
V |
U |
W |
| Q |
V |
E |
W |
U |
| R |
W |
U |
E |
V |
| S |
U |
W |
V |
E |
Amazingly, we find that the whole interior of this table is made up of
the transformations E, U, V and W. We also see that it was essential
to say which reflection we were doing first since the final result
sometimes depends on this.
Finally, what will happen if we make a grand table including all 8 of the
symmetry transformations for the square?
We have already worked out two quarters of the answer.
Do you have any idea what the other two quarters will look like?
You might possibly like to try working out this table for yourself.
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