For a closed triangulated surface both methods are exact, not approximate, and they are exact by different routes. The divergence theorem gives the volume as a sum of signed tetrahedra over faces,
with the interior contributions cancelling. Cavalieri gives it as an integral of cross-sectional areas along an axis, and for a polyhedron each of those areas is itself exact: slice the mesh with a plane, orient each intersection segment by the triangle’s outward normal, and accumulate Green’s theorem over the segments. No polygon chaining is needed, which matters here, because a section through a folded object arrives in several disconnected pieces and a chaining step is exactly where that goes wrong.
On one displaced icosphere of 5 120 faces, with the areas integrated by midpoint rule at N heights:
| N | Cavalieri | divergence theorem | relative difference |
|---|---|---|---|
| 500 | 4.220294675 | 4.220292391 | 5.41 × 10−7 |
| 2 000 | 4.220292393 | 4.220292391 | 4.81 × 10−10 |
| 8 000 | 4.220292388 | 4.220292391 | 6.14 × 10−10 |
Blocks [1] and [2] of the script check each half against the analytic sphere first, so the agreement in the table is not two errors cancelling: the inscribed polyhedron’s volume gap falls by a factor of 3.98 per subdivision, and the slicer tracks π(1 − z²) to 4.7 × 10−3, which is the polyhedron’s error and not the slicer’s.
A 1635 result about indivisibles and an 1820s result about flux, applied to the same polyhedron, return the same number to ten significant figures. Neither is an approximation here. That is the platform everything below is measured against, and it is the reason a null result in section 4 can be trusted as a statement about the estimator rather than about the apparatus.
A unit icosphere is displaced radially by a sum of 160 cosine modes, each a plane wave in the direction of a random unit vector, with frequencies drawn log-uniformly from 2 to 80 and amplitudes following a power law A(ω) = ω−β. Small β leaves the high frequencies alive and gives a rough surface; large β damps them and gives a smooth one. The displacement is normalised and scaled to 22 % of the radius, so the body stays star-shaped and the comparison is about spectrum rather than amplitude.
Roughness is reported as the isoperimetric ratio S / V2/3, which is scale-free, minimal for the sphere at 4.8360, and is the same quantity a folding index measures.
For each surface: sections at spacing T = H/n with the first offset drawn uniformly in [0, T) — systematic sampling with a random start, which is what makes the Cavalieri estimator unbiased. The estimate is V̂ = T Σ A(zk). Thirty offsets per surface, two surfaces per spectrum, and the realized coefficient of variation of V̂ taken against the exact volume of that same mesh. Then the exponent γ fitted to CV ∝ n−γ.
| β | S / V2/3 | n = 8 | 16 | 32 | 64 | 128 | γ |
|---|---|---|---|---|---|---|---|
| 0.3 | 6.226 | 3.90e−3 | 2.00e−3 | 3.40e−4 | 8.45e−5 | 3.42e−5 | 1.823 |
| 0.8 | 5.213 | 6.45e−3 | 1.17e−3 | 1.45e−4 | 8.61e−5 | 2.30e−5 | 2.002 |
| 1.5 | 4.924 | 6.79e−3 | 1.61e−3 | 2.76e−4 | 4.74e−5 | 2.04e−5 | 2.185 |
| 2.5 | 4.872 | 6.76e−3 | 1.43e−3 | 3.87e−4 | 1.08e−4 | 2.59e−5 | 1.979 |
Mesh of 20 480 faces; sphere floor 4.8360 for reference. Eight sections at the roughest setting already puts the estimator within four parts in a thousand of the true volume, and 128 sections within three parts in a hundred thousand.
γ is about 2 in every row. It does not fall as the surface gets rougher; it does not rise either; and the ordering is not monotone — the roughest surface gives the lowest exponent, 1.82, but the second-smoothest gives the highest, 2.19, and the smoothest sits between them at 1.98. With thirty offsets a coefficient of variation carries roughly 13 % relative standard error, which leaves any difference in γ below about 0.3 undetectable by this design. The spread in the table is inside that.
Over the range built here, the decay exponent of the Cavalieri error does not depend detectably on the roughness of the object. The honest reading is that this design cannot see an effect smaller than 0.3 in γ, and found none larger.
The roughest surface this construction produced has an isoperimetric ratio of 6.23. The sphere is 4.84. Taking round numbers for a human brain — a cortical surface on the order of 1 600 cm² against a volume on the order of 1 100 cm³, figures chosen to set a scale and not offered as measurements — the same ratio is about 15.
So the experiment stops at rather less than half the folding it was built to probe, measured on the one scale-free index available. A 22 % radial displacement on a sphere is a corrugation, not a convolution. Whatever the grooves do to the error law, this paper did not reach the regime where they would do it.
Whether the decay exponent of the Cavalieri error degrades on a genuinely convoluted surface remains untested here. Closing it needs three things this construction did not have: a surface whose isoperimetric ratio reaches the cortical range, which means either much larger displacement amplitudes on a mesh fine enough to resolve them or a real segmented cortical surface used as the phantom; enough random offsets to resolve differences in γ well below 0.3; and section counts that straddle the folding wavelength, since at n = 128 on a band that stops at ω = 80 the sections are still resolving the roughness rather than aliasing it. The last of those is the likeliest reason the effect did not appear: an estimator that resolves the corrugation has no reason to care that it is there.
The published variance predictors for Cavalieri sampling are parameterised by a fractional smoothness constant, conventionally taken in [0, 1], where the upper end describes “fairly regular, quasi-ellipsoidal objects” with a smooth area function and the lower end objects whose area function has discontinuous derivatives. None of those predictors is implemented or tested in this paper. They are named here only to say what was not checked: whether the constant a practitioner defaults to misprices the error on a folded object. That is a different question from the one measured above, and it is the more useful one.
The same caution applies to fractality. Reported fractal dimensions for the human cortical surface span roughly 2.1 to 2.8 depending on method, and the reviews are explicit that using fractal concepts to index complexity is not the same as establishing genuine scale-invariance. This paper builds surfaces with a controlled power-law spectrum and reports an isoperimetric ratio; it does not estimate a fractal dimension and no claim here depends on one.
2026-09-13. The diagnosis in section 5 is superseded. WP-112 builds area functions whose Hölder exponent is set by construction and finds the asymptotic decay exponent to be 2.003 at every roughness from H = 0.1 to 0.9, moving only when the order at which the area function vanishes at the ends of its support is changed. A closed surface vanishes linearly at its poles, so the null reported here could not have been broken by corrugating the interior at any amplitude. The measurements on this page stand; the reason given for them does not. What interior roughness does control is the number of sections needed before the law holds.