What question did this study set out to answer?

The aim is to explore how Penrose tilings exhibit 5-fold symmetry, which is forbidden in periodic crystals, through the study of quasicrystals.

July 24, 2026Open Access

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Penrose Tilings: Forbidden 5-Fold Symmetry Realized in Quasicrystals — E8 Intelligence Research

ACAndrew Stewart Caldin

Key Points

The aim is to explore how Penrose tilings exhibit 5-fold symmetry, which is forbidden in periodic crystals, through the study of quasicrystals.
Investigated the properties of Penrose tilings and their association with 5-fold symmetry.
Analyzed the mathematical characteristics, including the golden ratio and inflation/deflation factors.
Examined quasicrystal diffraction patterns to identify sharp Bragg peaks with 5-fold symmetry.
Penrose tilings demonstrate 5-fold symmetry, contrary to periodic crystals as shown through quasicrystal diffraction patterns.
The inflation factor was found to be approximately 2.618, and the deflation factor approximately 0.382, based on the golden ratio.
Aperiodicity in the tiles is enforced by matching rules, resulting in unique geometric patterns.

Abstract

FINDING: Penrose tilings are non-periodic, quasiperiodic tilings with 5-fold rotational symmetry, forbidden in periodic crystals, realized physically in quasicrystals. MATH: - 5-fold rotational symmetry (forbidden in periodic lattices by crystallographic restriction theorem). - Golden ratio φ = (1+√5)/2 ≈ 1.618 appears in tile edge ratios, inflation/deflation scaling factor. - Inflation factor = φ² = φ+1 ≈ 2.618; deflation factor = 1/φ² ≈ 0.382. - Rhombus acute angles: 36° (π/5) and 72° (2π/5); obtuse angles: 144° (4π/5) and 108° (3π/5). - Matching rules enforce aperiodicity; substitution matrix eigenvalues are φ and -1/φ. - Quasicrystal diffraction patterns show sharp Bragg peaks with 5-fold symmetry (icosahedral point group). CONNECTION: - φ and its powers (0.382, 0.618, 1.618, 2.618) are the geometric ratios governing tile shapes and inflation. - 5-fold symmetry links to icosahedral and dodecahedral symmetry in 3D quasicrystals. - Base-60 not directly present, bu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Cite This Study

Andrew Stewart Caldin (Wed,) studied this question.

synapsesocial.com/papers/6a630201395161722cd166db https://doi.org/https://doi.org/10.5281/zenodo.21484544

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