Substrate geometry / gomboc margins
How close to degenerate is a gomboc?
A gomboc is a convex body with exactly one stable and one unstable resting position. We fix one explicit smooth body, enclose its centroid and its equilibria by interval arithmetic, and measure how much room it has.
Every number on this site comes from a script and a log, carries an evidence tier, and lists what it does not cover.
Recently added
The latest results, software and teaching material, newest first. Each item links to the row or page it comes from and says whether it is released.
Equilibrium counts for the harmonic bodies, from the same checker
A second family now goes through the same cover and replay checker. At one small amplitude each, degrees 2 to 5 give (l, l, 2l-2) stable, unstable and saddle positions, the class of result C.
Software, not yet released OpenThe Phase-1 count as a record anyone can replay
The count of one stable and one unstable position is now stored as a record of 16053 boxes that a separate short program re-evaluates in about 5 seconds.
Software, not yet released OpenA convexity check for any radial body
The convexity matrix is evaluated on cube-face charts and changes sign at the published thresholds for degrees 2, 3 and 4. Convexity is shown for every amplitude up to 95 percent of the threshold for degree 2.
Software, not yet released OpenA browser page that replays a record
A prototype page loads the checker into the browser and replays a record from a file. It says VALID for the Phase-1 count in 28.9 seconds and rejects a copy with a changed claim.
Prototype, local only OpenA classroom notebook: when plots lie
A notebook for students on how a picture, a scan and a replayable record can disagree about the same body, with a teacher guide and an answer key.
Draft, not yet released OpenWays in
Results
19 results, each with its tier, its numbers, the second method that checked it, and the part it leaves open.
Open the ledger COMPUTEDBodies
The Phase-1 body at its true parameters, a drawn picture of the flat and steep directions at its equilibria, and a harmonic explorer with a live count.
Open the bodies TOOLSSoftware and teaching
The checker library, a browser page that replays a record, and a classroom notebook. All still local, none released.
See the tools METHODHow it is checked
Ball arithmetic, two independent methods, a cold code review that tried to produce a false certificate, and the trust base we cannot remove.
Read the methodA claim carries the tier it has earned
Four tiers, used the same way in the papers and here. A page cannot label a row above its tier: the build refuses.
A proof written out on paper. A computer may have checked the numbers, but the argument does not depend on one.
An interval-arithmetic computation, repeated by a second method that shares no chart, symmetry reduction or polar cap with the first, over the whole statement.
A statement about ball-arithmetic enclosures, backed by a script and log. Where a second method covers part of it, the row says exactly how much.
A float scan. It shows where to look. It is not a certificate and is never cited as one.
What this does not claim
- No priority for the existence of a smooth gomboc. That was settled by others.
- No formally verified proof. The main results are computer-assisted interval computations that trust ball arithmetic and our derivative code (see Method).
- No margins for any body other than Phase-1, and no claim about all gombocs.
- No physical tolerance. The perturbation bounds are sufficient conditions on a mathematical body.