Substrate Geometry
Computer-assisted interval computation / private preview
AConvex range of a spherical-harmonic perturbationHAND PROOFBConvexity thresholds for 27 harmonic bodiesCOMPUTER-ASSISTEDCEquilibrium counts for the m = 1 harmonic bodiesCOMPUTER-ASSISTEDD1Sloan shape 2: upper bound on the convex rangeHAND PROOFD2Sloan shape 2 is strictly convex for beta up to 0.03550477COMPUTER-ASSISTEDD3Sloan shape 2 has the origin as centroid and exactly two equilibriaHAND PROOFD4Sloan shape 1: upper bound on the convex rangeHAND PROOFD5Sloan shape 1 is strictly convex and mono-monostatic for beta up to 0.03229670385COMPUTER-ASSISTEDE1Phase-1 body: strictly convex2 METHODSE2Phase-1 body: exactly one stable and one unstable equilibrium2 METHODSF1Phase-1 gradient floorCOMPUTER-ASSISTEDF2Phase-1 Hessian at the equilibria: flat in one direction, steep in the otherCOMPUTER-ASSISTEDF3How much shape error Phase-1 toleratesCOMPUTER-ASSISTEDF4Is Phase-1 close to the best this family can do?EXPLORATORYG1Phase-1 equilibrium count as a replayable recordCOMPUTER-ASSISTEDG2Single-harmonic m=1 counts from the same checkerCOMPUTER-ASSISTEDG3Convexity of radial bodies, checked against the published thresholdsCOMPUTER-ASSISTEDG4Mesh trials: how much shape error a printed gomboc can takeEXPLORATORYG5A nearest-neighbour scan miscounts a flat bodyEXPLORATORYAConvex range of a spherical-harmonic perturbationHAND PROOFBConvexity thresholds for 27 harmonic bodiesCOMPUTER-ASSISTEDCEquilibrium counts for the m = 1 harmonic bodiesCOMPUTER-ASSISTEDD1Sloan shape 2: upper bound on the convex rangeHAND PROOFD2Sloan shape 2 is strictly convex for beta up to 0.03550477COMPUTER-ASSISTEDD3Sloan shape 2 has the origin as centroid and exactly two equilibriaHAND PROOFD4Sloan shape 1: upper bound on the convex rangeHAND PROOFD5Sloan shape 1 is strictly convex and mono-monostatic for beta up to 0.03229670385COMPUTER-ASSISTEDE1Phase-1 body: strictly convex2 METHODSE2Phase-1 body: exactly one stable and one unstable equilibrium2 METHODSF1Phase-1 gradient floorCOMPUTER-ASSISTEDF2Phase-1 Hessian at the equilibria: flat in one direction, steep in the otherCOMPUTER-ASSISTEDF3How much shape error Phase-1 toleratesCOMPUTER-ASSISTEDF4Is Phase-1 close to the best this family can do?EXPLORATORYG1Phase-1 equilibrium count as a replayable recordCOMPUTER-ASSISTEDG2Single-harmonic m=1 counts from the same checkerCOMPUTER-ASSISTEDG3Convexity of radial bodies, checked against the published thresholdsCOMPUTER-ASSISTEDG4Mesh trials: how much shape error a printed gomboc can takeEXPLORATORYG5A nearest-neighbour scan miscounts a flat bodyEXPLORATORY
TOOLS

Software and teaching material

Three things exist locally. None is released: there is no public repository, no download and no hosted service yet, and this page says what each one is and is not.

The checker library (eqcert)Software, not yet released

A search program writes its answer as a record; a short separate program replays the record box by box in ball arithmetic. The search is not trusted, the checker is.

  • Convexity of Sloan-form bodies and of radial bodies, for a whole range of parameters at once (rows D5 and G3).
  • Equilibrium counts, one record per body (rows G1 and G2).
  • Not formally verified; the trust base is on the Method page. The licence is Apache-2.0.
A browser page that replays a recordPrototype, local only

The checker runs inside the browser (Pyodide with python-flint), so a record never leaves the computer.

  • Tested in Chrome: the Phase-1 count record is VALID in 28.9 seconds on a busy machine, and a copy with its claim changed to 2 stable positions is rejected.
  • The largest convexity record (664274 leaves) replays in 228.1 seconds in the same engine, against 121 seconds natively.
  • It still relies on two third-party downloads on first load, which must be pinned or self-hosted before any release.
A classroom notebook: when plots lieDraft, not yet released

Students compare a picture, a floating-point scan and a replayable record of the same body, break the record on purpose, and design a body of their own.

  • It uses a scan whose count changes from 9 minima to 1 as a setting changes (row G5) and a printing-error experiment (row G4).
  • It comes with a teacher guide, timings and an answer key.
  • The formula a student types is read by a fixed whitelist and never executed.
What this page does not say
  • No release date, price or hosted service. Nothing here can be downloaded yet.
  • Nothing here is a statement about a physical object.