Multiplication Table for the Equilateral Triangle

```Multiplication Table for the Equilateral Triangle
Fall 2010 Edition
E
D
V
This lame picture gives the letters for the three reflections: “D” is the reflection that goes
through the lower left-hand corner, “E” is the reflection that goes through the lower righthand corner, and “V” is the vertical reflection. In addition, we have R, which is the 1/3
rotation, R2, which is the 2/3 rotation, and finally “1”, which is the identity (do nothing)
symmetry.
A multiplication like “VR” means: do a vertical translation first, and then do 1/3 rotation.
If you pay attention to where the corners of the triangle go, then you will see that doing a
vertical reflection followed by a rotation is just the same as doing the reflection that goes
through the lower right corner. In other words, “VR=E”.
We can make a multiplication table out of this. Note that switching order changes the
result, i.e., VR=E, while RV=D. In regular multiplication, order doesn’t matter. In this
multiplication, order matters. In the table, I am assuming that the symmetry on the top
row comes first, then the symmetry on the left-hand side
1
R
R2
V
D
E
1
1
R
R2
V
D
E
R
R
R2
1
E
V
D
R2
R2
1
R
D
E
V
V
V
D
E
1
R
R2
D
D
E
V
R2
1
R
E
E
V
D
R
R2
1
```