3d rotor-router algorithm

This page is dedicated to the exploration of the rotor-router algorithm when run in 3 dimensions. One of the main features that can be observed when the algorithm is run in this way is a sphere with internal "spheres" and a surface pattern made of circles at specific locations. As the algorithm runs, the sphere appears to grow and the circles on the surface expand progressively.

The 3D rotor-router algorithm provides a unique and interesting way to visualize and investigate patterns in three dimensions. The sphere and the internal and surface patterns it exhibits are a result of the algorithm's rules and processes, and offer a rich area for study and exploration.

These two images feature an animated cut of a sphere and the surface of a sphere produced by the rotor-router algorithm in 3 dimensions. It is fascinating to observe the nucleus that is formed inside the sphere, and to explore the patterns and structures that emerge as the algorithm runs.

One question that arises when viewing these patterns is whether the 3D rotor-router algorithm has any connections to particle physics. While further study is needed to fully answer this question, it is possible that the algorithm could be used to simulate or model certain phenomena in particle physics, or that it could provide insights into the behavior of particles.

Overall, the 3D rotor-router algorithm is a powerful and intriguing tool that offers possibilities for exploration and investigation.

1- This picture on the right presents the surface seen from "Front" z-axis of the rotor router algorithm in 3dimension.

The rotor used is UP,DOWN,LEFT,RIGHT,FRONT,BACK.

The picture on the left is a cut in the middle of the sphere in x,y plan.The rotor used is UP,DOWN,LEFT,RIGHT,FRONT,BACK.

3- same work with the UP,RIGHT,DOWN,LEFT,FRONT,RIGHT version of the router. left picture show the cut in the center of the sphere and the second one the surface of the sphere.

4- rotor router surface of the sphere but this time with init of an empty cell to value 2 (left) and to 4 (right) . we still use the UP,RIGHT,DOWN,LEFT,FRONT,RIGHT version of the router. We observe the waves on the surface of the sphere.

4- rotor router surface of the sphere but this time filtered to show only 2 directions (2 colors of the rotor). Other directions/colors are transparent.

It is conjectured that the growth rate of the number of single chip moves in the rotor-router algorithm in 3 dimensions (z3) follows the formula: numberOfChipMoves = 4/5Piradius^5. This formula can be derived by taking the integral dV on the volume of a sphere of radius r^2.

This result is also equal to the moment of inertia of a sphere with a density of 3/2. It is important to note that on the x-axis of the graph, the radius is represented as (4/3Piradius^3) for clarity.

Overall, this formula provides a way to understand and predict the growth rate of the rotor-router algorithm in 3 dimensions, and may have implications for the study of other phenomena or systems that exhibit similar growth patterns.

6. In the upper picture is highlighted that the ripple centers are at coordinates which are continuous fractions. eg x=y/n with n integer.

7. approximation of the RR 3D sphere surface can be obtain by setting the pixel color with the following rule.

We take the (modulo 6) +1 of X+Y+Z coordinate of the sphere.

for(zzz=-radius; zzz<=radius; zzz++)

{

for(yyy=-radius; yyy<=radius; yyy++)

{

for(xxx=-radius; xxx<=radius; xxx++)

{

if(xxx*xxx+yyy*yyy+zzz*zzz <= radius*radius)

{

grid[250+xxx][250+yyy][250+zzz]=abs((xxx+yyy+zzz)%6)+1;

}

}

}

}

All images presented in this website are courtesy of Cedric Vandenweghe at https://sites.google.com/site/projectsced/home

Please feel free to leave a comment at rotorrouterprojectsced (at) gmail.com