One periodic object, one strict inequality
Eight Cells Against a Century
Kelvin's foam held for more than a century. Rotate the eight-cell structure that beat it, recompute the tiny surface-area saving, then test the exact inequality that disproves Kelvin while leaving the true optimum honestly unknown.
The question: how can space be divided into equal-volume cells with the least interface area? Kelvin offered a long-standing answer in 1887. Weaire and Phelan produced this counterexample in 1993, followed by their article in February 1994.
Drag to turn. Arrow keys also rotate. These are the exact planar certificate cells, not a picture of the curved relaxed foam.
At zero, the representatives sit at their periodic site coordinates. Boundary faces continue into neighbouring copies. Pull them apart to inspect both cell types.
Pull left to pass a plane through the generated polygons. This changes only the view, never the measured cells.
Live reconstruction
Building the periodic cells from half-spaces.
Layer 1: the numerical counterexample
Let the smaller bar finish first
The two costs below are published rounded caps from Surface Evolver runs: Weaire-Phelan under 18.4871, Kelvin under 18.67582. They are inputs, not measurements made by this page.
Layer 2: no numerical relaxation required
Make the cells equal, then run the inequality
The planar family has b = c/2 and a = 2c/3. Only c = cuberoot(2) gives both cell types volume 8 at one common periodic scale.
Moving the slider breaks equal volume. The exact preset is stored separately in JavaScript, never rounded through this stepped control.
Published exact expressions
A = 3 + 3sqrt(6) + (6sqrt(5) - 4sqrt(6) - 3) / cuberoot(16)
mu = A3 / V2
Kelvin-pattern floor = (27/16)(sqrt(3/2) + 1)3
Ready. The exact equal-volume setting is loaded.
The relaxed comparison is the historical blow. It uses curved faces found numerically, and its published result is approximately 0.3% less area. Small enough to hide in a careless chart. Large enough to make Kelvin's proposed structure cease to be the answer.
The second layer removes the obvious dismissal. Kusner and Sullivan gave a planar-faced A15 partition whose cost is below a slicing lower bound for every partition with Kelvin's topology and symmetry. No faith in a Surface Evolver stopping point is needed for that strict comparison.
It still does not solve the Kelvin problem. A counterexample closes one proposed answer. It does not identify the global minimizer, and the global optimum remains unknown.
Why the area convention has a factor of two
A cell's full boundary counts every shared interface from that cell's side. Its neighbour counts the same interface again. The published cost uses A, average interface area per cell, which is half the average full cell-boundary area. The live geometry adds all eight full boundaries and divides by 2 x 8.
Why Surface Evolver is evidence, not the certificate
A numerical relaxation can find a local low-area state inside a chosen pattern. It does not certify a global minimum over all partitions. The approximate 0.3% result is the historical numerical comparison. The planar inequality is the rigorous counterexample to the whole Kelvin pattern.
Equal volume is not equal pressure
The two Weaire-Phelan cell types have equal volume but different pressures. Those conditions must not be exchanged. No pressure equality is assumed anywhere in the calculations here.
The building that borrowed the pattern
The Beijing National Aquatics Centre used a cut and modified Weaire-Phelan geometry as a design basis. Its structural members are straight. The building is not a literal equal-volume curved foam, and no calculation on this page treats it as one.
The check
The browser starts from eight A15 sites in a periodic cube, intersects the weighted half-spaces, orders each face, and computes every polygon area from cross products. Volumes come from signed pyramids to those faces. Nothing below is typed as a result.
| recomputed item | live result | test |
|---|---|---|
| 12-faced cell census | 2 cells, 12 pentagons each | PASS |
| 14-faced cell census | 6 cells, 12 pentagons and 2 hexagons each | PASS |
| periodic interfaces | 54 = 48 pentagons + 6 hexagons | PASS |
| largest volume residual | computing | PASS |
| geometry A versus radical A | computing | PASS |
| analytic inequality | 18.57752 < 18.5816 | PASS |
- Published inputs. The relaxed costs under 18.4871 and 18.67582 are published rounded caps. The original approximately 0.3% saving is also published. This page does not possess the authors' final relaxed meshes.
- Computed outputs. The printed-cap ratio, cell vertices, face areas, volumes, A, both analytic costs, and their strict gap are computed live.
- Uncertainty. The relaxed run is numerical and local. Its cap-ratio reconstruction is 0.338%, consistent with the paper's coarser approximately 0.3%, but it is not a more precise historical measurement.
- Free choices. Camera angle, explosion, section plane, face tint, cell focus, and exploratory c are yours. Only exact c = cuberoot(2) is admitted to the certificate.
- Vacuous settings. Comparing a pattern with itself would give zero by identity, so the page offers no such evidence mode. Independently rescaling each cell would force equal volume by definition, so every cell is tested first at one common periodic scale.
- Open edge. The analytic result excludes the Kelvin topology and symmetry. It does not prove that the relaxed Weaire-Phelan foam is globally optimal.
Independent check: node research/weaire-phelan/verify-weaire-phelan.mjs. It uses the paper's explicit vertex families rather than the browser's half-space intersection, then recovers the faces and volumes from their convex hulls.
- D. Weaire and R. Phelan, “A counter-example to Kelvin's conjecture on minimal surfaces,” Philosophical Magazine Letters 69(2), 107-110, 1994. Source for the original counterexample and approximately 0.3% statement.
- Rob Kusner and John M. Sullivan, “Comparing the Weaire-Phelan Equal-Volume Foam to Kelvin's Foam,” Forma 11(3), 233-242, 1996. Source for the cell census, explicit coordinates, radical, Kelvin-pattern bound, and reported relaxed costs.
- Annalisa Cesaroni and Matteo Novaga, “Minimal periodic foams with fixed inradius,” Mathematika 71(2), e70020, 2025. Source for the open status of the Kelvin problem.
- M. Arkinstall and T. Carfrae, “Structural Design and Optimisation of the Beijing National Olympic Swimming Centre,” Australian Journal of Structural Engineering 6(3), 181-190, 2006. Source for the architectural design note.