A weakly universal cellular automaton in the hyperbolic 3D space with three states

In this paper, we follow the track of previous papers by the same author, with various collaborators or alone, see [2, 7, 13, 14, 10, 9], which make use of the same basic model, the railway model, see [16, 5, 9]. In order to be within the space constraint for the paper, we just refer to the above mentioned paper both for what is the railway model and for what is hyperbolic geometrY. For the latter one, we just mention something new in Section 2. A more developped version of the paper can be found on arXiv, see [11]. In the previous papers, the number of states of a weakly universal cellular automaton was reduced from 24 states to 9 ones in the pentagrid and fixed at 6 for the heptagrid. In [10], I succeded to reduce this number to 4 in the heptagrid. The reduction for 6 states to 4 states, using the same model, was obtained by replacing the implementation of the tracks of the railway model. In all previous papers, the track is implemented as a one-dimensional structure where each cell of the track has two other neighbours on the track exactly, considering that the cell also belongs to its neighbourhood. The locomotive follows the track by successively replacing two contiguous cells of the track: the cells occupied by the front and by the rear of the locomotive. The locomotive has its own colours and the track has another one which is also different from the blank, the colour of the quiescent state. In the mentioned paper, this traditional implementation is replaced by a new one. There, the track is no materialized but suggested only. It is delimited by milestones which may not define a continuous structure. At this point, my attention was drawn by a referee of a submission to a journal explaining the 4-state result that it is easy to implement rule 110 in the heptagrid, using three states only. This is true, but this trick produces an automaton which is not really a planar automaton and does not improve our knowledge neither on rule 110 nor on cellular automata in the hyperbolic plane. This implementation with three states can also be easily adapted to the dodecagrid of the hyperbolic 3D space and suffers the same defect of bringing in no new idea.