## Volume 32, Number 1 (2023)

**Cultivating the Garden of Eden
**

Randall D. Beer

Garden of Eden (GOE) states in cellular automata are grid configurations that have no precursors; that is, they can only occur as initial conditions. Finding individual configurations that minimize or maximize some criterion of interest (e.g., grid size, density, etc.) has been a popular sport in recreational mathematics, but systematic studies of the set of GOEs for a cellular automaton have been rare. This paper presents the current results of an ongoing computational study of GOE configurations in Conway’s Game of Life (GoL) cellular automaton. Specifically, we describe the current status of a map of the layout of GOEs and non-GOEs in 1-density/size space, characterize how the density-dependent structure of the number of precursors varies with increasing grid size as we approach the point where GOEs begin to occur, provide a catalog of all known GOE configurations up to a grid size of 11×11, and initiate a study of the structure of the network of constraints that characterize GOE versus non-GOE configurations.

Cite this publication as:

R. D. Beer, “Cultivating the Garden of Eden,” Complex Systems, 32(1), 2023 pp. 1–17.

https://doi.org/10.25088/ComplexSystems.32.1.1

**The Domino Problem of the Hyperbolic Plane Is Undecidable: New Proof
**

Maurice Margenstern

The present paper revisits the proof given in a paper of the author published in 2008 proving that the general tiling problem of the hyperbolic plane is algorithmically unsolvable by proving a slightly stronger version using only a regular polygon as the basic shape of the tiles. The problem was raised by a paper of Raphael Robinson in 1971, in his famous simplified proof that the general tiling problem is algorithmically unsolvable for the Euclidean plane, initially proved by Robert Berger in 1966. The present construction improves that of the 2008 paper. It also very strongly reduces the number of prototiles.

Keywords: hyperbolic plane; tilings; tiling problem; algorithmic unsolvability

Cite this publication as:

M. Margenstern, “The Domino Problem of the Hyperbolic Plane Is Undecidable: New Proof,” Complex Systems, 32(1), 2023 pp. 19–56.

https://doi.org/10.25088/ComplexSystems.32.1.19

**A Game of Life Shifted toward a Critical Point
**

Tomoko Sakiyama

The Game of Life (GoL), which produces complex patterns of life, has been employed to describe biological systems through self-organized criticality and scale-free properties. This paper develops two novel GoL models. One model allows each cell to update the time for the state update following interactions with other local cells using parameter tuning. Thus, individual cells replace their behaviors from intermittent state updates with continuous ones. The system evolves unpredictably close to a critical point and occasionally close to extinction for the alive population if an adequate parameter is chosen. This event occurs with a power-law tailed time interval and presents synchronous behaviors, since individual cells modify their state-update intervals and create time continuity. The other model is the same except that the system evolves unpredictably without any parameter tuning. In the second model, actions of individual cells are tuned not by a fixed parameter but by the surrounding situation. We found that the GoL system in the second model behaved in a similar manner in the first model, which suggests that that model shifts toward a critical point autonomously.

*Keywords*: Game of Life; critical point; power-laws; phase transition

Cite this publication as:

T. Sakiyama, “A Game of Life Shifted toward a Critical Point,” Complex Systems, 32(1), 2023 pp. 57–70.

https://doi.org/10.25088/ComplexSystems.32.1.57

**Spatial Scale Effects in COVID-19 Spread Models
**

Marlon N. Gonzaga, Marcelo M. de Oliveira and Allbens A. P. Faria

The COVID-19 pandemic has highlighted epidemiological models as important forecasting methods and planning strategies, with studies conducted using a wide variety of analytical and computational techniques. Knowing that more pandemic episodes may occur, it is essential that epidemiological modeling present increasingly credible results. From this perspective, this paper aims to highlight the influence of spatial distribution on an epidemic dynamic, using agent-based modeling. To calibrate the behavioral profile of the population, data was taken on mobility, population pyramid, individual behavior and government policies of a real population during the pandemic. Two different initial spatial distribution scenarios are tested and the robustness of the infection is analyzed. Totalistic rules were designed to assess the influence of infected individuals in the vicinity of an agent, a factor that must not be ignored in modeling respiratory diseases with viruses capable of spreading by aerosols, such as SARS-CoV-2. It is shown that the scenario with nonuniform distribution of agents is much more robust, generating an epidemic process even when uniform distribution, for the same parameters, did not propagate the infection. Our results also suggest that herd immunity is attained in different levels of recovered individuals, showing higher values in denser regions. In conclusion, it is reinforced that the nonuniform feature of the spatial distribution of individuals plays a key role in the infection dynamics and should receive more attention when building epidemiological models.

*Keywords*: complex systems; epidemiological modeling; spatial scale effects; spatial distribution in power laws; COVID-19

Cite this publication as:

M. N. Gonzaga, M. M. de Oliveira and A. A. P. Faria, “Spatial Scale Effects in COVID-19 Spread Models,” Complex Systems, 32(1), 2023 pp. 71–87.

https://doi.org/10.25088/ComplexSystems.32.1.71

**Use of Recurrence Plots to Find Mutations in Deoxyribonucleic Acid Sequences
**

D. E. Rivas-Cisneros

This paper examines the use of recurrence plots to find changes in the deoxyribonucleic acid (DNA) sequence. The DNA sequence is entered into the recurrence plot algorithm in codons and not consecutively, in order to make it easier to find the nucleotide change. The results show that the codon arrangement makes it easy to identify two types of mutations, insertion and deletion, by means of the recurrence plot. It is also shown that the recurrence plot of a short codon sequence has a homogeneous structure. Similarly, a comparison is made with the digital signal processing methodology. Some limitations are the number of codons that can be entered into the algorithm because computationally it becomes very slow and matrix issues, which are mentioned in the discussion section.

Keywords: recurrence plot; DNA sequence; genetic mutations; digital signal processing

Cite this publication as:

D. E. Rivas-Cisneros, “Use of Recurrence Plots to Find Mutations in Deoxyribonucleic Acid Sequences,” Complex Systems, 32(1), 2023 pp. 89–100.

https://doi.org/10.25088/ComplexSystems.32.1.89

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