## Mostly Misaligned Mirrors

Recently my stochastic professor introduced me to a problem he has been pondering for over two decades: on the two-dimensional integer lattice $\mathbb{Z}^2$ one shall flip a three-sided coin for each point and uniformly place one of three mirrors, $\{\diagup,\,\cdot\,,\diagdown\}$, where $\,\cdot\,$ denotes not placing a mirror. After having populated the world, one picks their favorite integer tuple and points a beam of light in one of the four cardinal directions. With what probability does the light fall into a loop, never fully escaping?

## Rain

Inspired by the current weather – my favourite weather, the rain – I made this simple simulation. There are no controls, just look at it quietly and peacefully. # Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech 1st of May, 2015

## Star

I had an idea to make a simulation which generates stars. Using my ‘getCirclePos()’ function, two radii – an inner and an outer one – and a number of points I realized the idea in this program.

#### How it works $\frac{360^\circ}{\text{number of points}} = \text{circle sector}$ $\frac{360^\circ}{\text{number of points} \cdot 2} = \text{shifted circle sector}$

Using those to sectors – and alternating between the inner (shifted) and outer radius – the program connects the points and out comes a star shape.

#### How to use

• ‘d’ toggles debug mode (gray circles)
• ‘f’ toggles points being drawn (blue points at the corners)

# Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech 24th of April, 2015