We consider Langton's ant on the 2D plane, but we replace the square lattice by a Voronoi tessellation obtained from a finite set of points (it could be another tessellation, however directions such as "left" and "right" must be well-defined).
Definition of the directions
Each cell can have a more than four edges, which means more possible directions. We define the following ones: backward (B), starboard (S), right (R), left (L), and port (P) (notations inspired from https://brtmr.de/2015/10/05/hexadecimal-langtons-ant-2.html). For example, when the cell is an octagon, we obtain:
[A little more formal definition of those directions: Given previous cell and current cell, we obtain the direction of the ant (red arrow) and of the last edge crossed (bold edge). Relative to this direction, we number edges from 1 to E (where E is the number of edges of the current cell). B is edge 1; S is edge 2; R is the median edge minus one; L is the median edge plus one; P is the last edge].
Question 1
Given rule SP, which tessellation and initial conditions should we select to get a periodic trajectory with the minimal period?
My best try is a period of 48 with a tessellation containing 20 vertices, 30 edges and 12 faces. (Are there any name for this particular dodecahedron?)
The following figure represents the 50 first steps. At step 1, the ant came from the top-right square.
(Colors: White: The ant has never reached this cell yet; Red: The ant crossed this cell an odd number of times; Blue: The ant crossed this cell an positive and even number of times)
Note that the graph has been deduced from a Voronoi tessellation.
Question 2
Given a rule, can we always build a tessellation and some initial conditions such that the trajectory of the ant is bounded?
