penrose_ratio
plain-language theorem explainer
penrose_ratio defines the thick-to-thin rhombus area ratio in Penrose tilings as the golden ratio φ drawn from the Constants bundle. Quasicrystal chemists cite it when relating tile counts and energy minima in aperiodic structures with five-fold symmetry. The declaration is a direct one-line alias to Constants.phi.
Claim. The thick-to-thin rhombus area ratio in a Penrose tiling equals the golden ratio $φ$.
background
The module Quasicrystal φ-Stability (CM-002) treats quasicrystals as aperiodic tilings that maintain long-range order without translational symmetry. It states that the golden ratio φ appears in Penrose tilings as the ratio of thick to thin rhombus areas, with stability arising because the energy proxy E(r) = (r - 1/φ)² reaches its minimum at the self-similar fixed point φ. Any deviation increases structural strain and is therefore disfavored.
proof idea
This is a one-line definition that directly aliases Constants.phi from the upstream LawOfExistence constants bundle.
why it matters
The definition supplies the area ratio used by the downstream theorem penrose_frequency_ratio, which equates the frequency ratio to φ by reflexivity. It anchors the quasicrystal stability analysis inside the Recognition Science framework, linking the Penrose tiling ratio to the T6 forcing of φ as the self-similar fixed point and to the five-fold symmetries that appear in the eight-tick octave.
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