Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
This problem explores the shapes and symmetries in some national flags.
Published November 2006,December 2006,February 2011.
They all have symmetry.
Symmetry is the basis of all patterns in art, music, bell
ringing, knitting, dancing, crystals, elementary particles and
Something is said to be symmetric if it is not changed by one or
more of these operations (reflection, rotation or translation).
Lots of art is based on symmetry, here is a very old
A square is symmetric. How many symmetries does
A square will look the same under any combination of these
symmetries but if we label the corners of the square and apply
rotations and reflections we end up with "different" squares. Here
are four examples (I have named them 'a', 'b', 'c' and 'd'.
The simplest symmetry we can have is the"do
nothing "symmetry which we shall call 'e'.
So, for example:
Answers are at the end of this article.
We have looked at combining rotations and combining reflections
but what happens if we combine a rotation with a reflection? Let's
rotation ba = ?
how about two reflections? bc = ?
Answers are at the end of the article.
Here are some other combinations you might like to
check for yourself:
By now you might be
Well let me
My name is Chris and
fortunately I have three friends called Andrew, Bryony and Daphne
(that makes A, B, C and D) who all like dancing.
We make ABCD - four corners of a square. You might
already be seeing the connection! If not, here's a hint:
Key Fact: the symmetries
of the square correspond to different dance moves.
This dance move is called an "inner-twiddle " or
This dance move is called an "outer-twiddle " or
And this coresponds to a dance called a
Fou r " or
Now find three friends and try it!
We see the same patterms in bell ringing and in
not see whether you can find other places?
A square has 8 symmetries; 4 rotation symmetries and 4
bb, cc and dd all equal e.
ba = c
ab = d
bc = a