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
ABOUT

An independent research program

Substrate geometry classifies forms by the operational invariants of how they behave, such as how many ways a body can rest, rather than by symmetry. The work here is by Vincent Wesley Couey and is independent: it is not affiliated with a university or a funder.

The current focus is gombocs, convex bodies with exactly one stable and one unstable equilibrium. Existence is known; the questions here are quantitative: how large is the convex range of an explicit family, and how much room does one explicit smooth body have.

Corrections are welcome. If a number here disagrees with its source file, the source file wins and the page is wrong; tell us and we will fix it at the source.