- 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
A goes to
Then B goes to
And C must go to
A
B
C
C
B
B
A
C
C
A
C
A
B
B
A
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, V will mean reflection across the line that is the vertical altitude of the equilateral triangle.
Then let's consider a second symmetry, called R=R120, which will rotate the equilateral triangle counterclockwise about its center O by 120 degrees.
We now can think of first performing V to the figure and then performing R to the figure.
We will denote this V*R... meaning V 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*V means V followed byR.]
Does the resulting transformation also leave the equilateral covering the same position in which it started?
V*R = R2 ?
If so it is also a symmetry.... which of the six is it?
What about other products? 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 R240*V = R1
*
Id
R120
R240
V
G=R1
H=R2
Id
I
R120
R240
R1
R240
I
R2
V
R2
R1
I
R120
G=R1
I
H=R2
I
This "multiplicative" structure is called 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....
how many?
does the multiplication table for the symmetries look the same in some sense?