Caley table for the symmetry group of the tripod
Like the triskelion, the tripod without feet has rotational symmetry. But it also
has mirror symmetry and therefore has a larger symmetry group. We’ll call U, V
and W the reflections in the three mirror axes.
The larger symmetry group of the tripod, which is of order six, is composed
of the six transformations I, R, S, U, V and W and has the multiplication table
shown.
An interesting transformation is achieved by combining two reflections in
different axes, such as U ° W (where the reflection W is applied first and is
followed by the reflection U). This is actually a rotation of the tripod through 120
degrees, in symbols U ° W = R. Combining the reflections the other way around
W ° U = S, gives a rotation through 240 degrees. In particular U ° W ≠ W ° U.
This is a major difference between a multiplication table for a group and an
ordinary multiplication table with numbers.
Reflections of a tripod
A group in which the order of combining the elements is immaterial is called
an abelian group, named after the Norwegian mathematician Niels Abel. The