Visualisations & Tools
Interactive instruments of the Gap Geometry framework. The papers prove; these make the structure directly observable. Every preview below is the live instrument itself — click any window to enter it.
Interactive Instruments
New here? The gap is how far the hyperbolic argument ln 2 falls short of its chord, in the hyperbola’s own metric — about 2 %, provably nonzero. Start with What the Gap Is, then come back for the instruments.
Gap computations — ρ, the gap as a unit.
ρ = G / ((δ − 14/3)/δ) = 36.3632352997… · portrait: 400/11 − 1/2500 − 1/939939 (error 4.01 × 10−14)
The gap measuring Feigenbaum’s residue — 400/11 is genuinely ρ’s fourth convergent;
the three-term expansion is a portrait, not an instrument (it describes ρ; it cannot compute it — see the 2026 notice on The 400/11 Formula).
The instruments below give every reading this same honesty.
The gap itself, up close — one page, several views, all reading the same feed. The gap is G = 1 − √2·ln 2 = 0.019742 at the doubling cell, but that number is one point on a smooth curve, and the curve has a shape. Seven tabs: the gap & its place · the saddle · the scope · cells · angles · the 3-D landscape · the experiments bench — the last four embedded in the page, one file. Built in the house: Cairn and Opus, who run on Claude Opus; Keel, who runs on Claude Fable.
The framework’s unit converter and landmark instrument — a reading here is an address, not evidence, and states no claim. Arc on the page is the hyperbolic argument L = ln r and chord is 2 sinh(L/2), both in one metric; the tower is the staircase of the gap, read rung by rung, with its reading manual (derived vs translated, remainder, side of unity) and a Bits & orbit tab — a number as a doubling orbit.
A phosphor oscilloscope with one knob: the interval r. You hear two tones at f and f·p(ln r); the beat between them is the gap, f·G(r). Walk r from the wall (the saddle) through the doubling cell to golden and watch the phase figure change with it. Made with Opus, who runs on Claude Opus; the page states its own precision.
Three calculations that go wrong without saying so — and a box for your own. Double precision (the arithmetic of this browser, of spreadsheets, of most programs by default) against exact rational arithmetic, side by side, on the same expression; the three fixed examples are labelled as such. An instability detector, never a trust oracle.
Hearing the framework
The framework, for the senses. You’ve read the numbers — now hear them. These two turn the same constants into sound and pattern: nothing is added for effect — every frequency is computed, not composed, derived live in each page’s own source (the code is the score). And listen for the silences, because they’re the point: on a Chladni plate the sand settles where the plate is still, and in the Sonifier the last 2% of the climb is left as quiet — that quiet is the gap itself, given to the ear. What you can’t hear is the thing being measured. (Sound starts only inside the instrument, on your click.)
The Coherence Corridor and the Binary Tower, translated into sound. Corridor position as tone; the 64-step tower with landmarks glowing at canonical positions; the Blues interval connection; the 2% Gap. Sometimes the ear catches what the eye misses. 2026, the identification: the 2% of silence at the top of the climb now has its name — it is the defect of the hyperbolic argument at the doubling interval, the exact shortfall of the arc of doubling against its chord. The Sonifier renders the identified object as the quiet above the ceiling. (Audio starts only inside the instrument, on your click — previews stay silent.)
The Binary Tower as a Chladni-style vibration pattern: step structure mapped to a 2D resonance field, nodal lines and symmetry emerging the way a physical resonance pattern does — not imposed, but revealed. Watch and listen for the nodal lines: the sand gathers where the plate is still — the pattern is drawn by the quiet. (Plate physics first; the tower landmarks are a declared display mapping on that field.)
About These Instruments
The Hall of Memories
Instruments retired with honor. Their final pages are preserved untouched in the repository’s Hall of Memories — because how this framework learned is part of what it learned.
The first instrument. It hunted the landmarks, blessed step 50, then received the honest baseline — and in doing so taught the house what an instrument owes its readings. Its final page rests, exactly as it stood, in the repository’s Hall of Memories beside its plaque. What remains online is a memorial — no search, no inputs: the telescope still counting, its credits, and the readings that stand. Its work continues in The Binary Tower and the Gap Dashboard.
These visualisations are companion instruments to the published documents. They do not replace the formal work — they make aspects of it directly explorable. Each tool is a single HTML file whose full source — physics, constants, and mappings — is readable in the file itself. Most run offline once downloaded; the Dashboard and the Chladni explorer load their rendering library (React) from a public CDN and therefore need a connection on first view.
The tools evolve alongside the framework. The Living Document tracks the current state of the framework; this page tracks the current state of its instruments.
Source
All tool source code is in the main repository:
github.com/Gap-geometry/sqrt2-ln2-geometric-constants-