We saw that the
→
→
→
In the same way that a
C3:
{
}
C4:
{
}
We might even choose different generators to end up with the same result...
→
→
→
See, we can use a
C3:
{
}
But beware we must be careful: not all members of our groups are generators.
For example, a
Instead a
{
} = {
}
Another way to see this is with color...
We can transform a C4 shape into a C2 shape by coloring it.
→
Now the only rotations that leave this colored shape unchanged are those of C2.
C2:
{
}
Again we must be careful. Not all colorings of our C4 shapes will transform them into C2 shapes. Some will remove their rotations altogether and leave them with just the 0 turn.
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Challenge: Find all the generators for C4 and C8.
Challenge: Which rotations of C8 generate our C4 group but not C8?
Can you use color to transform the uncolored shapes into C2 shapes?
C4 and C8 shapes
The C9 shape below is made up of pieces that repeat around a circle. Go clockwise around the circle, coloring every other repeated piece in the same way. That is, color a piece, skip a piece, color the next piece the same way as the first, and keep going. Do you end up coloring every piece? Can you use this to prove a 2/9 turn is or isn't a generator for C9?
C9 shape (circular tessellation)
Can you show that a 3/12 turn is not a generator for C12 by coloring every other 3 pieces in the same way? What group does the 3/12 turn generate?
C12 shape (circular tessellation)
For some cyclic groups, any of their transformations can be used as generators. Which groups are these?
shapes with 3, 4, 5, 6, 7, 8 rotations