fields at large scales. This concept is in line with quantum information and digital physics principles applied to spacetime, and aims to bring new advances in quantum gravity and unification physics.

In the model we develop here the building blocks are very constrained. There is a small set of Lie algebra representations that interact, and the syntax of its fusion rules works as a glue, indicating topological order [@5throot; @ZhenghanWang; @Pachos]. With a tetrahedron building block the problem is translated to an issue of how to tile the space, where we will be mainly concerned with 3-dimensional space, which can be linked with the Fibonacci anyonic fusion Hilbert space $-$ a computational space. The problem of tiling the space brings us to aperiodic order, which generalizes the notion of long range periodic order $-$ from lattices to quasicrystals [@Senechal; @Fang-Irwin]. Quasicrystals also play an important role in aspects of the representation theory of Lie algebras [@ChenMoodyPatera; @Koca1; @Koca2] that are important for the connection with spin foam models.

This paper is organized as follows: in Section \[sec:spinfoam\] we review the usual concept of 3D spin foam quantum gravity, and present its fifth root of unity “quantization” as well as the 