Methods, provenance and free software
This is an interactive companion to the project’s six lectures and lecture notes. It exposes 66 physical selectors, the 21 requested classical centers, and three archived ETC location neighbors for every physical entry.
Direct classical geometry
Classical and neighbor points are evaluated from cyclic barycentric center functions in the supplied ETC snapshot. X3, X4, X5 and X20 use equivalent geometric constructions to avoid removable formula singularities. Genuine exceptional centers can remain undefined. No dynamic code evaluation or Wolfram installation is required.
Physical center locations
E4=X2, S3=S4=X10 and M3=X360 are exact limiting identities under their stated uniform-thickness or equal-core conventions. M1 is found by a direct finite-wire-field minimization in the browser; Illum uses its equal edge-angle/area stationarity equations. Their solvers are floating-point computations, not interval proofs.
Other PITCs use piecewise linear interpolation of barycentric coordinates in the ordered-angle fundamental region θ1≤θ2≤θ3. The mesh combines the saved 100 uniform shapes, 40 holdouts, special families, thin samples and reference shapes. Equal-angle vertices are symmetrized to preserve reflection and relabeling covariance. No extrapolation is permitted. The display is not a new PDE or conductor solve, and interpolation error is not bounded. Refinement changes, where supplied, concern the underlying saved solver, not the interpolant.
At an equilateral triangle, Q1, Q2 and Q4 represent degenerate individual eigenstates and have no invariant unique value or universal Taylor expansion. Missing locations remain unavailable. Interpolation near spectral crossings should be treated as a visual guide, not an eigenbranch certificate.
Fingerprints
The seven graphs show isosceles vertical position and derivative; the two right-triangle coordinates and their derivatives; and the longitudinal coordinate in a triangle of finite small height ε. The first six horizontal axes use θ=arctan(h) in degrees, from 0.5° to 89.5° by default. The response-unit control selects dc/dh or dc/dθ per degree, with dc/dθ=(π/180)sec²(πθ/180)dc/dh. All families and axes are explicitly normalized in the interface. Saved right-family coordinates and derivatives are exchanged to use the common convention A=(0,0), B=(1,0), C=(0,h).
Solid PITC curves and dots show the saved one-dimensional samples. Dashed ETC curves come from direct formulas on a uniform angular grid spanning the chosen range. PITC curves stop at the limits of the saved data. Hover reports one shared abscissa and color-matched ordinates interpolated on the displayed lines. Each selected PITC can independently include its three ETC neighbors; shared neighbors are deduplicated. Gaps are preserved. Derivatives are saved finite differences, not derivatives of the display interpolant. Finite ε is a probe of the collinear limit and must not be interpreted as a certificate of that limit.
The near-equilateral table gives the first available Taylor coefficients and names the numerical method. Quantum linear responses use perturbation/refined FEM records; coarser second-order terms are explicitly identified. Unavailable terms stay blank. New illumination thin samples and local coefficients were computed for this explorer with stationarity solves and solid-angle maximization.
ETC similarity
The three neighbors are the lowest archived location RMS errors on the 100 common shapes, normalized by triangle diameter (longest side). The snapshot has 72,807 entries, of which 70,853 are finite on the survey grid and 1,954 are excluded. Forty independent holdouts and derivative/response comparisons are reported where available. The shortlist is not a joint fingerprint/attractivity ranking and does not claim an identity.
Attractivity
Only one vertex moves: δv·δc≥0. Proven attractive selectors are distinguished from certificates of nonattractivity, refined negative numerical evidence and positive sampled screens. The interpolant’s response is never used as proof of a continuum center’s attractivity. M6 has strong positive numerical evidence; its global theorem remains open.
Central conics and axes
The Steiner inellipse has center X2 and shape matrix 2Σarea, and is tangent to the three side midpoints. Its axes are the principal directions of the uniform lamina. The wire also has a covariance ellipse; it is drawn separately with a dashed stroke about X10 and generally is not tangent at the side midpoints. Equal axial vertex currents have an unordered pair of field zeros at the Steiner foci; neither focus is arbitrarily promoted to a single center. At isotropic covariance, an axis direction is undefined.
Software and reproducibility
The application uses HTML, CSS, JavaScript, SVG and the MIT-licensed KaTeX renderer, bundled with its fonts. It has no subscriptions, paid API, account dependency, tracking or application backend. Serve the dist folder with any static web server. Opening it directly through file:// can block the JSON fetch.
Download the complete free application and browser atlas · Download the browser atlas JSON · Verification report
Attribution
ETC identifiers and formulas are drawn from Clark Kimberling’s Encyclopedia of Triangle Centers, using the project’s local X1–X72807 snapshot. Only the 153 expressions needed by this interface are included, not the full vendor database. Research data and mathematical evidence come from the Physically Inspired Triangle Centers project, snapshot 1 October 2026. The app’s software is released under MIT; that software license does not relicense external ETC source material.