A strongly universal cellular automaton with four states

TL;DR

This paper demonstrates the existence of a strongly universal cellular automaton in hyperbolic 3D space with four states, achieved by relaxing rotation invariance constraints compared to previous automata.

Contribution

It introduces a four-state strongly universal cellular automaton in hyperbolic space, relaxing rotation invariance constraints from prior five-state models.

Findings

Automaton operates in hyperbolic 3D space (dodecagrid)

Uses only four states due to relaxed symmetry constraints

Maintains strong universality despite reduced states

Abstract

In this paper, we prove that there is a strongly universal cellular automaton in the dodecagrid, the tesselllation {5,3,4} of the hyperbolic 3D space, with four states but, it is not rotation invariant as the automaton of arXiv:2104.01561 is with five states. The present paper is not an improvement of arXiv:2104.01561. The reduction to four states of the automaton of the present paper results from a relaxation of the condition of rotation invariance. However, the relaxation is not complete: the rules are invariant with respect to rotations of the dodecahedron which leave it globally invariant which also leave invariant a couple of opposite faces of the dodecahedron. As there are twenty five such rotations, it can be considered that the reduction to four states is significant although the relaxation it implied.