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

Two additional notes on a finite object: a vocabulary of lines on a 12 × 12 square, and the identification of this vocabulary with the level sets of a square plate's normal mode. [A] A vocabulary of axes generated by an inverse doubling process.It is shown that the construction's vocabulary is generated rather than chosen. A single operation suffices—doubling the offset of a line relative to the center—and the origin is fixed: a level established by the doubling process has no external antecedent; thus, it cannot be generated and must be posited, whereas all other levels are accessible. There are exactly two such levels, 0° and 120°, with the second existing precisely because 3 divides 12. Starting from these two levels, the orthogonal vocabulary closes in two steps to form 24 offsets, and the diagonal vocabulary in three; the latter includes the inscribed rhombus, which belongs to the vocabulary but is not depicted. The inverse structure consists of two towers of height two rather than a simple chain. The parity rule transforming a set of axes into a figure is injective regarding the band systems; the integrity check enabled by the construction constitutes a [1152, 13] linear code with a minimum distance of 288 based on the triangle reading; and the period of 12 is itself constrained, with the descent terminating without a cycle precisely when N = 2^a or 3·2^a. A distinct result, verified across the sixty published images, identifies thefifteen families of an existing pattern system with the fifteen non-emptyunions of four first-generation families. [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 corresponds exactly to the reflectionθ ↦ 180° − θ. This function is an exact eigenfunction of a Kirchhoffplate with four guided edges, and the degenerate combinations cos(4πx/a) ±cos(4πy/a) are exact guided modes whose nodal sets are two of thediagonal families—one of which is the 48-cell skeletoncommon to the sixty images in the independent corpus. This same function is also theexact function of a second, trivially realizable system: acoustic pressure ina rectangular cavity with rigid walls, where the governing equation is Helmholtzrather than biharmonic. The two systems share eigenfunctions but notspectra, such that a simple ratio of measured frequencies allows them to be distinguished.A falsifiable interferometric prediction is stated. No physical measurementshave been performed. Contents. The two notes in PDF formats; teaaxis catalog in JSON format; eight scripts that output each cited figure, along with theresults of their last execution; the two figures from [A] in SVG format, generated by one of thesescripts from the same catalog. Licenses. Text and figures: CC BY-NC 4.0. Code: AGPL v3;commercial license upon request.