## § A walkway of lanterns (TODO)

#### § Semidirect products

• $(\alpha \equiv \{ a, b, \dots\}, +, 0)$
• $(\omega \equiv \{ X, Y, \dots\}, \times, 1)$
• $\cdot ~: ~\omega \rightarrow Automorphisms(\alpha)$
• rotations: $\mathbb Z 5$
• reflection: $\mathbb Z 2$
• $D_5 = \mathbb Z5 \rtimes \mathbb Z2$
\begin{aligned} \begin{bmatrix} 1 & 0 \\ a & X \end{bmatrix} \begin{bmatrix} 1 & 0 \\ b & Y \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ a + X \cdot b & XY \end{bmatrix} \end{aligned}
• $(Y \mapsto b) \xrightarrow{act} (X \mapsto a)$
• $XY \mapsto a + X \cdot b$

#### § A walkway of lanterns

• Imagine $\mathbb Z$ as a long walkway. you start at 0. You are but a poor lamp lighter.
• Where are the lamps? At each $i \in \mathbb Z$, you have a lamp that is either on, or off. So you have $\mathbb Z2$.
• $L \equiv \mathbb Z \rightarrow \mathbb Z2$ is our space of lanterns. You can act on this space by either moving using $\mathbb Z$, or toggling a lamp using $\mathbb Z2$. $\mathbb Z2^{\mathbb Z} \rtimes \mathbb Z$
• $g = (lights:\langle-1, 0, 1\rangle, loc:10)$
• $move_3: (lights: \langle \rangle, loc: 3)$
• $move_3 \cdot g = (lights:\langle-1, 0, 1\rangle, loc:13)$
• $togglex = (lights:\langle 0, 2 \rangle, loc: 0)$
• $togglex \cdot g = (lights: \langle -1, 0, 1, 13, 15 \rangle, loc:13)$
• $toggley = (lights: \langle -13, -12 \rangle, loc:0)$
• $toggley\cdot g= (lights:\langle -1 \rangle, loc:13)$