Übergeekism

This is an attempt at using as many as possible cool computer science stuff to produce a single image.

Algorithms may not be implemented in the most efficient manner, as the aim is to have elegant and simple code for educationnal purpose.

Until now, the following algorithms/data structure/concepts are used:

• the (logo) turtle,
• Lindenmayer systems,
• Penrose tiling,
• Travelling salesman problem,
• ant colony algorithm,
• A* shortest path,
• graph (adjacency list, adjacency matrix),
• hash table.

The current code is written in Python.

Penrose tiling

The main shape visible on the image is a Penrose tiling (type P3), which is a non-periodic tiling with an absurd level of coolness.

The edges are recursively built with a Lindenmayer system. Yes, it is capable of building a Penrose tiling if you know which grammar to use. Yes, this is insanely cool.

The Lindenamyer system works by drawing edges one after another, we thus use a (LOGO) turtle to draw them.

Because the L-system grammar is not very efficient to build the tiling, we insert edges in a data structure that contains an unordered collection of unique element: a hash table.

Travelling Salesman Problem

The Penrose tiling defines a graph, which connects a set of vertices with a set of edges. We can consider the vertices as cities and edges as roads between them.

Now we want to find the shortest possible route that visits each city exactly once and returns to the origin city. This is the Travelling Salesman Problem. We use an Ant Colony Optimization algorithm to (try) to solve it.

Because each city is not connected to every other cities, we need to find the shortest path between two cities. This is done with the help of the A-star algorithm.

The ant colony algorithm output a path that connect every cities, which is drawn on the image, but it also stores a so-called pheromones matrix, which can be drawn as edges with variable transparency/width.

TODO

More coolness:

• Compute the Delaunay triangulation from the Penrose graph vertices,
• Compute the Voronoï diagram (Bowyer-Watson or Fortune's algorithm?) from the triangulation,
• Remove Voronoï edges that intersects with the Penrose graph,
• quad trees may be useful somewhere to query neighbors points?,
• The center of remaining segments is the center of the Penrose tiles,
• Build back the neighborhood of those tiles from the Voronoï diagram,
• Run a cellular automata on this Penrose tiling,
• Draw a planner on it.

Maybe even more coolness?

• percolation theory?

Improvements:

• Use a triangular matrix for pheromones in ACO.