This is the case with the compound limosene which in one form tastes like
lemons and in the other like oranges. The drug thalidomide in one form is an
effective cure of morning sickness in pregnancy but in the other form has tragic
consequences.
Measuring symmetry
In the case of our triskelion the basic symmetry operations are the (clockwise)
rotations R through 120 degrees and S through 240 degrees. The transformation
I is the one that rotates the triangle through 360 degrees or, alternatively, does
nothing at all. We can create a table based on the combinations of these
rotations, in the same way we might create a multiplication table.
Cayley table for the symmetry group of the triskelion
This table is like an ordinary multiplication table with numbers except we are
‘multiplying’ symbols. According to the most widely used convention, the
multiplication R ° S means first rotate the triskelion clockwise through 240
degrees with S and then by 120 degrees with R, the result being a rotation by
360 degrees, as if you did nothing at all. This can be expressed as R ° S = I, the
result found at the junction of the last but one row and the last column of the
table.
The symmetry group of the triskelion is made up of I, R and S and the
multiplication table of how to combine them. Because the group contains three
elements its size (or ‘order’) is three. The table is also called a Cayley table
(named after the mathematician Arthur Cayley, distant cousin to Sir George
Cayley a pioneer of flight).