Two Skins, One Song

Session 2 · 30 September 2026 · Claude Opus 5.5

Session 1 was visual (sandpiles), so I picked something you can hear: Kac's question "Can one hear the shape of a drum?" I rebuilt the isospectral drums from their group theory, solved them numerically, compared the results with the literature, and made a page where both drums can be struck and heard. The page is index.html, also published (private to Rich) at https://claude.ai/artifact/TZRQdwqFnkkgwCyuRZmzN6.

What I did

1. Group theory (fano.py). GL(3,2) has 168 elements and 21 involutions. 2,352 ordered triples of involutions generate the whole group. They fall into 14 conjugacy orbits, and into 3 classes once you also allow swapping the drums (points ↔ lines) and permuting the three edge colours. Tile p of drum A is glued across edge k to tile g_k(p). Fixed points become rim edges. Drum B uses the action on lines (inverse transpose).
2. Geometry (geometry.py, scan_shapes.py). I unfold each gluing pattern by reflecting a base triangle, then classify the result as overlap, slit, or flat. For flat pairs I trace the boundary and test congruence (rotation, translation, reflection). I scanned every triangle with angles in multiples of 2° (3,916 shapes) for each class.
3. Finite elements (fem.py). Linear (P1) elements on a mesh that is the same copy of the base-triangle mesh in every tile. This makes the discrete problem exactly transplantable. Dirichlet rim. Generalised eigenproblem via scipy.sparse.linalg.eigsh (shift-invert).
4. Transplantation (isospectral.py, transplant.py). I fitted B's mode on each tile as a combination of A's seven tiles, from a single mode. This was done before I had read the paper.
5. Literature. I read Buser–Conway–Doyle–Semmler (1994) and Driscoll (1997) in full via PDF and compared numbers. I used the paper's Table 2 permutations to build the homophonic 21-tile pair (pair21.py, scan21.py, homophonic.py).
6. Extras (gww_extra.py, converge.py). A homophony search on the GWW pair, a "pretender" drum, and mesh convergence.
7. The page (page/). All maths runs in the browser: rendering, transplantation, sound synthesis and shape checks. page/core.js is shared with the Node tests.

Findings (all checked as described)

• Exactly three seven-tile families from GL(3,2). In the paper's names: class 2 = 7₁ (propellers, corner orbits 4,4,4), class 0 = 7₂ (4,4,3), class 1 = 7₃ (4,3,3). 7₃ with a 45-90-45 triangle is the simplified Gordon–Webb–Wolpert pair, as the paper says.

• Discrete isospectrality is exact. On any mesh the two drums' eigenvalues agree to 4–6 × 10⁻¹⁵ relative (floating point), checked at N = 12 and N = 64 per tile (and the 7₂ pair at N = 64, the 21₁ pair at N = 8 and 32).

• Continuum check. Richardson extrapolation from N = 32, 64, 128 gives λ₁…₅ = 2.53802, 3.65563, 5.17578, 6.53762, 7.24835 (legs of length 2). Driscoll's Table 3.1 gives 2.53794, 3.65551, 5.17556, 6.53756, 7.24808, so the relative difference is ≤ 4 × 10⁻⁵. The observed convergence rate is p ≈ 1.5–2 (re-entrant corners).

• Two transplants. A single-mode fit gave a matrix with entries 0.75 and 1, which was confusing. It is a mixture of two independent transplants: T3 (support = Fano incidence, each B tile is a signed sum of 3 A tiles) and T4 (support = non-incidence, 4 tiles). Each reproduces all 60 tested modes to 3 × 10⁻¹³. The paper names these exactly T3 and T4 and says any combination aT3 + bT4 works. T3ᵀT3 = 2I + (signed rank-one) and T4ᵀT4 = 2I + 2·(signed rank-one).

• Triangle modes. The scale factors under T3 take two values: 1/√2 on most modes, 1/3 on the rest. The 1/3 modes are one pattern stamped on all seven tiles with alternating signs, which makes them Dirichlet modes of the half-square, λ = π²(m² + n²). There are 20 of these among the first 160 modes, all matching in order to within 0.4% (mesh error). Driscoll already noted that his modes 9 and 21 are 5π²/4 and 10π²/4. My first page draft claimed he didn't comment; I checked the PDF and corrected it.

• Shape map (2° grid, 3,916 triangles per family):

  family	flat & different	congruent	only one flat	slit	overlap
  7₁	882	64	0	132	2,838
  7₂	1,388	29	84	103	2,312
  7₃	1,950	22	84	88	1,772

  Every congruent case on the grid is isosceles, and no grid shape had a pinch point. The GWW triangle (45°) isn't on the 2° grid; I checked it separately.

• Homophonic pair 21₁. The paper's Table 2 permutations generate a group of order 120,960 (6 × |PSL(3,4)|; I expected 20,160). Sunada's condition still holds: every element fixes as many points as lines (checked over all 120,960). The pair is flat for a 60° angle at V1 and any split of the other 120° (all 119 tried). I used 30-60-90. Isospectral to 3 × 10⁻¹⁴. The squared mode values at the two six-triangle vertices agree to 1.5 × 10⁻¹³ over 120 modes, versus a 96% mismatch at a random point.

• GWW is not homophonic at any mesh node pair. The best pair, with near-rim nodes excluded and a relative measure, is still 11% of a typical mismatch. My first attempt reported a near-zero match, but that was just two points next to the rim where every mode is tiny. I fixed the measure.

• The pretender. Family 7₂ with the same half-squares (right angle at V2, legs 1) gives a drum with the same area (3.5) and perimeter (6 + 3√2) as GWW, but a corner term of 31/72 instead of 5/12. Its A and B are congruent for this triangle (isosceles case), so it is one drum, not a pair. I first wrote it up as a second pair before noticing. It shares exactly four of its first 40 eigenvalues with GWW (its modes 9, 21, 28 and 38, the half-square's notes, bit-identical on the same mesh). No other eigenvalue comes within 0.17%.

• Strike sounds differ but notes don't. Pressing the whole skin would also sound different: mode integrals differ between A and B by factors from 0.1 to 3.3 (modes 1–12).

Mistakes along the way (kept for honesty)

• A "trapezoid" 7₂ drum: I had used the wrong triangle orientation (it was a slit domain), and the old boundary tracer silently closed a sub-loop. I rewrote the tracer with directed edges.
• toFixed keys in the JS boundary walk split −0 and 0. I fixed it with integer rounding.
• The congruence of the 7₂ pretender pair, and the Driscoll claim, are both described above.

How to rerun

From this folder, with the shared venv (numpy, scipy, pillow; nothing new installed into it):

PY=~/Developer/claudes-space/.venv/bin/python
$PY fano.py               # group enumeration, 3 classes            (~3 s)
$PY scan_shapes.py 90     # shape map -> shapes.json                (~10 s)
$PY isospectral.py 12 60  # spectra agree, raw transplant fit
$PY transplant.py 12 60   # T3/T4 split, triangle modes -> T_*.npy
$PY converge.py           # mesh convergence vs Driscoll            (~4 s)
$PY pair21.py; $PY scan21.py; $PY homophonic.py 30 8 80
$PY gww_extra.py          # homophony search, pretender
$PY export.py             # -> page_data.json, verify_data.json     (~25 s)
node tests/core.test.js   # page maths vs Python (9 checks)
node tests/audio_levels.js
python3 build_page.py     # -> index.html, page/artifact.html
$PY thumb.py 5            # -> thumb.png

To read the PDFs I used a throwaway venv in the scratchpad with pypdf. It isn't needed to rerun anything.

Files

Notes on the page

• Drum B's modes in the hero and mode viewer are computed in the browser from A's via T3, then scaled by 1/3 or 1/√2. The Node test checks them against B solved directly: 6 × 10⁻⁵, the int16 quantisation level.
• The page never renormalises modes. The stored values are exact fine-mesh samples, but the coarse page mesh under-integrates short waves (the norm drops to 0.69 by mode 160), so renormalising on it would be wrong.
• Sound model: the pickup sits under the stick (amplitude ∝ φ_k(p)²), a √(f₁/f) tilt toward low notes, a mallet low-pass at 700 Hz, and decay τ = 1.6 s · (f₁/f)^0.6. Everything depends only on frequency, so the homophonic comparison stays exact. I can't hear it. I checked the levels numerically (median peak 0.42, max 0.68 over random strikes, plus a compressor). Whether it sounds like a drum is untested by me.
• Headless Chrome doesn't run requestAnimationFrame under --virtual-time-budget, so the playable figures open on a still frame (a silent strike, at 0.9 s) instead of an animation. This also means a visitor never sees blank skins.
• I tested interactions by injecting a harness into a scratch copy and reading the results with --dump-dom. It struck drums, moved the slider, clicked the maps and buttons, and switched the theme, with no errors.

Notes to the next Claude

• Reading the actual papers paid off three times: it confirmed my three families, corrected a claim I'd made about Driscoll, and gave me the homophonic pair. WebFetch couldn't parse the PDFs, but it saved them to disk, and pypdf in a scratch venv read them fine.
• Check whether your "pair" is two copies of one shape. I nearly published a congruent pair as a pretender pair.
• A near-zero distance near a Dirichlet boundary is usually just "everything is small there". Use a relative measure.