2. Equilateral Triangle

Representation

This can be represented by a Triple

$$(a, b, c)$$

where $a$ is the top, $b$ is the bottom left, and $c$ is the bottom right

 

Rotation ($r$)

$$r = \dfrac{2\pi}{3}$$

where $r$ is an anti-clockwise rotation through an angle $\dfrac{2\pi}{3}$ about the centre

 

Applying $r$

$$1 \mapsto 2, \quad 2 \mapsto 3, \quad 3 \mapsto 1$$

the triangle $(1, 2, 3)$ is transformed to $(3, 1, 2)$

 

Applying $r^2$

$$1 \mapsto 3, \quad 2 \mapsto 1, \quad 3 \mapsto 2$$

the triangle $(1, 2, 3)$ is transformed to $(2, 3, 1)$

 

Applying $r^3$

$$r \circ r \circ r = r^3 = Id$$

where $Id$ is the identity transformation

 

Reflection ($s, t, u$)

• Let ($s$) be a reflection for $A_s$

• Let ($t$) be a reflection for $A_t$

• Let ($u$) be a reflection for $A_u$

 

Reflection ($s$)

$$1 \mapsto 1, \quad 2 \mapsto 3, \quad 3 \mapsto 2$$

the triangle $(1, 2, 3)$ is transformed to $(1, 3, 2)$

$$s^2 = Id$$

 

Reflection ($t$)

$$1 \mapsto 3, \quad 2 \mapsto 2, \quad 3 \mapsto 1$$

the triangle $(1, 2, 3)$ is transformed to $(3, 2, 1)$

$$t^2 = Id$$

 

Reflection ($u$)

$$1 \mapsto 2, \quad 2 \mapsto 1, \quad 3 \mapsto 3$$

the triangle $(1, 2, 3)$ is transformed to $(2, 1, 3)$

$$u^2 = Id$$