The axes are nodal lines

A correspondence between the two-colour pattern system of La Livrée d'Hermès and the eigenmodes of a square plate.

What this article establishes

The two-colour pattern system rests on axes: straight lines drawn on a grid twelve cells wide, whose position is given by an offset from the centre. A cell is dark if the number of axes crossed from a corner is odd.

These axes are not a graphic choice. Three of them are — verified triangle by triangle over the 1,152 that the grid contains — the nodal lines of eigenmodes of a square plate: the lines where a vibrating plate stays still, the ones Chladni made visible by sprinkling sand.

pattern	plate mode	k²	frequency
offset 0	mode (2,2), fixed edges	2	256 Hz
offsets 0 and 3	mode (4,4), fixed edges	8	1024 Hz
offsets 1.5 and 4.5	mode (4,4), free edges	8	1024 Hz

The third case is the most interesting, and it gives what follows its meaning.

The mutation is a boundary condition

The system distinguishes each family of axes from its mutant: the same figure shifted by a quarter period, that is, three units out of twelve.

This mutation, invented as a drawing operation, turns out to be the change of boundary condition of a vibrating plate.

A clamped plate cannot move at its edges: they are nodes. A free plate vibrates there at its maximum: they are antinodes. The two plates have the same modes, but their nodal lines are shifted by a quarter period — exactly the system's shift of three units.

An immediate reading follows. The presence of offset 0 signals a nodal line passing through the centre. The base carries it, the mutant does not; so the centre is a node in the one, an antinode in the other.

The distinction between base and mutant, which seemed conventional, therefore comes down to asking whether the centre of the plate vibrates or stays still.

The frequency is not a convention

The project assigns each pattern a frequency by the formula f = 128 × k², where k² is the square of the dominant spatial mode, measured by Fourier transform.

This formula was presented as a choice. It is a law.

For a simply supported plate, the natural frequencies are proportional to m² + n², where m and n index the mode. The formula f = 128 k² is therefore the frequency law of such a plate, up to a constant.

And this constant places the fundamental mode (2,2), for which k² equals 2, at 256 hertz — the C of scientific pitch, the one in which the octaves of C are powers of two. Verified by the script cited below: 128 × 2 = 256.

What this article does not establish

The correspondence is not mathematically deep. The sign of a product of sines is a chequerboard; a chequerboard of a given step is therefore necessarily the nodal pattern of a mode. What deserves to be noted is not the theorem — there is none — but the fact that a system built for other reasons, by hand and without reference to acoustics, lands exactly on it.

The diagonal families follow another architecture. The patterns built on the diagonals of the square are not level curves. They are constructed by exclusive-or with lozenges centred on the nodes of a lattice, whose radii accumulate from one generation to the next. This architecture is established and verified, but its correspondence with Chladni figures — which also produce diagonal figures, through combinations of degenerate modes — remains to be worked out.

No physical measurement has been carried out. Everything above is a geometric correspondence between patterns and solutions of an equation. Nothing has been vibrated, measured or heard. The correspondence is exact; its physical interpretation remains to be validated by experiment.

Verification

Every correspondence stated here can be reproduced with tools/verify_cymatique_plate_modes.mjs, a script in the repository that compares the generated pattern with the computed plate mode and returns the difference as a number of differing triangles out of 1,152. All three give 0.

The code is under the AGPL v3 licence, in the project's GitHub repository. The data for the axes — offsets, generations, families — are published with it.

What this opens up

A conductive mesh carrying a periodic pattern behaves like an electromagnetic filter, whose resonant frequency depends on the period. If the spatial frequency of a pattern can be calculated from its axes, then an arrangement can be chosen for its resonance instead of being measured after the fact.

This is the lead opened up by this correspondence, and it calls for a simulation or a measurement to become a result.

See the correspondence → Open the Cymatics page.

This article was first published in French: Les axes sont des lignes nodales.