# Geometric State Sum Models from Quasicrystals

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometric Realism

- Ising models

- Lattice gauge theory (LGT):

- Spin foam

## 3. Kinematics: The 3D Quasicrystal, Empire and Hits

## 4. Dynamics: Geometric State Sum Model and the PEL

#### 4.1. A New Kind of Game of Life in Quasicrystals

#### 4.2. GSS Observables and Emergence

## 5. Discussions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SSH | Self-simulation hypothesis |

PEL | Principle of Efficient Language |

3DPT | 3-Dimensional Penrose Tiling quasicrystal |

PEL | Geometrical State Sum (GSS) |

GR | General Relativity |

LQG | Loop Quantum Gravity |

3D | 3-dimensional |

LGT | Lattice Gauge Theory |

VT | Vertex Type |

PS | Possibility Space |

PRW | Possibility Random Walk |

GoL | Game of Life |

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**Figure 2.**A typical hit map for the 3DPT quasicrystal. We consider a list with 1000 points of a 3DPT tiling and compute vertex window polytope overlap between them, Equation (6).

**Figure 3.**One 3DPT GoL oscillation pattern. The pattern is a oscillator period 2 and the two frames are shown.

**Figure 4.**A recurrent pattern propagation for a 3DPT GoL made mainly from the VTs from Figure 3. (

**1**–

**8**) show eight frames of the dynamics where the pattern highlighted moves from left to right.

**Figure 6.**PRW hit potential from two patterns evolution starting at different positions with one at origin.

**Figure 8.**Evolution considering local information entropy conservation, initial preferred direction and the central hit potential.

**Figure 12.**Information entropy order parameter for different VTn, where the integer n number the 3DPT VTs and we show only 5 of the 24. $hs$ grows different with distance for the different VTs.

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**MDPI and ACS Style**

Amaral, M.; Fang, F.; Hammock, D.; Irwin, K.
Geometric State Sum Models from Quasicrystals. *Foundations* **2021**, *1*, 155-168.
https://doi.org/10.3390/foundations1020011

**AMA Style**

Amaral M, Fang F, Hammock D, Irwin K.
Geometric State Sum Models from Quasicrystals. *Foundations*. 2021; 1(2):155-168.
https://doi.org/10.3390/foundations1020011

**Chicago/Turabian Style**

Amaral, Marcelo, Fang Fang, Dugan Hammock, and Klee Irwin.
2021. "Geometric State Sum Models from Quasicrystals" *Foundations* 1, no. 2: 155-168.
https://doi.org/10.3390/foundations1020011