Axes and Level Sets of a Jacquard Pattern System — two companion notes

Two companion notes on one finite object: a vocabulary of straight lines on a12 × 12 square, and the identification of that vocabulary with the level sets ofan eigenmode of a square plate. [A] A Vocabulary of Axes Generated by an Inverse Doubling.The construction's vocabulary is shown to be generated rather than chosen. Oneoperation suffices — doubling a line's offset about the centre — and the originis forced: a level fixed by the doubling has no preimage outside itself, so itcannot be generated and must be posited, while every other level is reachable.There are exactly two such levels, 0° and 120°, the second existing preciselybecause 3 divides 12. From those two the orthogonal vocabulary closes in twosteps on 24 offsets and the diagonal one in three, the latter including theinscribed diamond, which belongs to the vocabulary and is not drawn. Thebackward structure is two towers of height two rather than a single chain; theparity rule that turns a set of axes into a figure is injective on band systems;the integrity check the construction affords is a linear code, [1152, 13] ofminimum distance 288 on the triangle reading; and the period 12 is itselfconstrained, the descent terminating without a cycle exactly for N = 2^a or3·2^a. A separate result, checked on all sixty published images, identifies thefifteen families of an existing pattern system with the fifteen non-empty unionsof four families of the first generation. [P] A Generative Construction of the Level-Set Family of aSquare-Plate Eigenmode. Twelve of the sixteen axis families arecomplete level sets of the single function cos(πx/3) at the seven levels cos θ,θ = 0°, 30°, …, 180°; the construction's involution is exactly the reflectionθ ↦ 180° − θ. That function is an exact eigenfunction of the Kirchhoff platewith all four edges guided, and the degenerate combinations cos(4πx/a) ±cos(4πy/a) are exact guided modes whose nodal sets are two of the diagonalfamilies — one of which is the 48-cell skeleton that all sixty images of theindependent corpus carry in common. The same function is also the exacteigenfunction of a second, trivially realisable system, the acoustic pressure ina rigid-walled rectangular cavity, where the governing equation is Helmholtzrather than biharmonic: the two share their eigenfunctions and not theirspectra, so a single ratio of measured frequencies discriminates between them.One falsifiable interferometric prediction is stated. No physical measurementwas performed. Contents. Both notes as PDF and Markdown source; the axiscatalogue as JSON; eight scripts that print every number cited, with the outputof their last run; the two figures of [A] as SVG, generated by one of thosescripts from the same catalogue. Licences. Texts and figures: CC BY-NC 4.0. Code: AGPL v3;commercial licence on request.