The Metallic Roof: tan² Rung Law, Only Gold Dwells Finitely, and the Tower-in-the-Roof Theorem (INDYNA Research Note 26)

Abstract

Rung law. In the $2m$-gon ring with the conjugate shell pair $(\pm 2\cos(\pi/m))$, the family crossing with the alternating family sits exactly at $g^\ast=\tan^2(\pi/2m)$: silver $(1{+}\sqrt2)^{-2}$ for $m{=}4$, twelvefold $(2{+}\sqrt3)^{-2}$ for $m{=}6$ — the metallic units are the tangents of the half-corner. Only gold dwells. The crystallographic restriction grants finite non-crystallographic reflection groups only to fivefoldness ($H_2,H_3,H_4$): silver and the twelvefold possess the ladder arithmetic (Pisot) but no finite house; on Ammann–Beenker approximants the near-rung of the $7/5$ cell dissolves at $17/12$ (1417-vertex torus, no kink in the covering window) — the silver ground ladder does not exist even asymptotically. Tower in the roof. Odd powers of every metal are themselves metallic means ($n$ = odd Lucas $1,4,11,29,\dots$ giving $\varphi,\varphi^3,\varphi^5,\dots$; Pell–Lucas $2,14,82$ giving $(1{+}\sqrt2)^{1,3,5}$), proof: $N(\mu)=-1\Rightarrow \mu^{\rm odd}-\mu^{-\rm odd}\in\mathbb{Z}$. Even powers form the sister family $x^2=nx-1$ ($n$ = even Lucas; $2{+}\sqrt3=((1{+}\sqrt3)/\sqrt2)^2$ included). In particular the fourth mean — mercury — equals $\varphi^3$ exactly: mercury lives in gold's house (the chemistry of amalgam, echoed by arithmetic) and conducts exactly gold's frequencies in steps of three. Licence CC-BY-4.0. Continues: Note 25. Supplement to: INDYNA Law Register (L149–L154, L180). Part of the INDYNA Notes series (Notes 16–34). All numbers reproduce from the INDYNA law register (master data record, doi:10.5281/zenodo.21891088). Licence CC-BY-4.0.