- Why are there only six?
Before: A After : A or B or C
Suppose I know where A goes:
What about B? If A -> A Before: B After: B or C
If A ->B Before:B After: A or C
If A ->C Before: B After A or B
By an analysis of a "tree" we count there are exactly and only 6 possibilities for where the vertices can be transformed.
- What about combining transformations to give new symmetries:
Think of a symmetry as a transformation, for example, F will mean reflection across the line that is the vertical altitude of the equilateral triangle.
Then let's consider a second symmetry, called R, which will rotate the equilateral triangle counterclockwise about its center O by 120 degrees.
We now can think of first performing F to the figure and then performing R to the figure.
We will denote this F*R... meaning F followed by R.
[Note that order can make a difference here, and there is an alternative convention for this notation that would reverse the order and say that R*F means F followed byR.]
Does the resulting transformation also leave the equilateral covering the same position in which it started?
F*R120 = V*R120 = R2
If so it is also a symmetry.... which of the six is it?
What about S*R? This gives a "product" for symmetries.
If S and R are any symmetries of a figure then S*R is also a symmetry of the figure.
A "multiplication" table for Symmetries:
Do Activity. This shows that R120*F = R1
Called this the Group of symmetries of the equilateral triangle.
Given any figure we can talk about the group of its symmetries.
Does a figure always have at least one symmetry? Yes... The Identity symmetry.
Such a symmetry is called the trival symmetry.
So we can compare objects for symmetries....
does the multiplication table for the symmetries look the same in some sense?
What about the symmetries of a Frieze Pattern on a Strip....
Translations and Glide Reflections [180 degree Rotations and Reflections]
This pattern has vertical axes for reflective symmetries between each p and q... and a translation symmetry taking each letter to the next letter of the ssame type... and also twice as far, and three times as far , and more....
The same is true for the following pattern:
The following pattern also has glide reflection symmetry, for example taking p to b and q to d, etc.
It also has a 180 degree rotational symmetry with center midway between the vertical reflection axes and the letter p and d or q and b.
Next time: more on symmetries of frieze patterns ....
What about the symmetries of a tiling?