Golden Ratio Inflation Rules Encode 5-Fold Symmetry in Aperiodic Penrose Tilings — E8 Intelligence Research

Q: What does this research mean for the field?

Metallic mean Wang tiles generalize the self-similar aperiodicity of Penrose tilings to any integer n by utilizing substitution rules based on metallic means. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.21503754

July 23, 2026Open Access

Golden Ratio Inflation Rules Encode 5-Fold Symmetry in Aperiodic Penrose Tilings — E8 Intelligence Research

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Key Points

This research explores how Penrose tilings demonstrate 5-fold symmetry using golden ratio inflation rules.
Analyzed Penrose tilings for aperiodicity by applying golden ratio inflation rules.
Calculated inflation and deflation ratios, with eigenvalues of substitution matrices derived from tiling rules.
Generalized findings to metallic mean Wang tiles with substitution rules based on various integers.
Penrose tilings exhibit 5-fold symmetry as a result of golden ratio inflation rules, indicating aperiodicity.
The eigenvalues of the substitution matrix for Penrose tilings are the golden ratio φ and -1/φ.
Metallic mean Wang tiles also display self-similar aperiodicity, showing that these principles extend to other integers.

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Implication

Finding that Penrose tilings show 5-fold symmetry through golden ratio inflation rules suggests new insights into aperiodicity.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a65a61ad3aea3239cd77aa3 https://doi.org/10.5281/zenodo.21503754

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Golden Ratio Inflation Rules Encode 5-Fold Symmetry in Aperiodic Penrose Tilings — E8 Intelligence Research

Key Points

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Implication

Discussion

Cite This Study

Golden Ratio Inflation Rules Encode 5-Fold Symmetry in Aperiodic Penrose Tilings — E8 Intelligence Research

Key Points

No abstract yet

Implication

Discussion

Cite This Study

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