ubergeekism/README.md

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 educational 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,
• Delaunay triangulation,
• Bowyer-Watson algorithm,
• convex hull,
• Chan's algorithm,
• graph (adjacency list, adjacency matrix),
• hash table.

The current code is written in Python.

Penrose graph

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 segments 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.

Penrose tiling

Because the L-system draws the Penrose tiling segments by segments, we need to compute how each segment is related to the diamonds to rebuild the tiling corresponding to all those edges.

Fortunately, computing a Delaunay triangulation of the Penrose vertices brings back the Penrose graph (how cool is that!?) and stores plain shapes (triangles) instead of unordered segments. This is done thanks to the Bowyer-Watson algorithm.

But this triangulation contains edges that link the set of exterior vertices, which are not in the Penrose graph. This is solved by computing the convex hull, with the Chan's algorithm, and removing the triangles that contains those edges from the triangulation.

TODO

More coolness:

• Compute the Voronoï diagram 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,
• Draw the neighborhood with splines across the center of diamonds segments,
• 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.