Communications - October 2012

“base” facet centered at the origin; specifically, an edge from (− s, s) to (s, − s) for some parameter s. This already gives all vertices, by periodic extension. The hexagonal bubble is the convex hull of the two base points, their translates within [0, 1) 2, and their translates by ( 1, 1). One chooses s to minimize the resulting surface area (perimeter).

We similarly construct a tiling shape B in three dimensions. We form a “base” facet centered at the origin, which is a regular hexagon, with vertices ±(0, −t, t), ±(−t, 0, t), and ±(−t, t, 0), for some t ∈ (0, 1/3). Again, this gives all vertices, by periodic extension. We take B to be the convex hull of the 6 base points, along with their 6 translates within [0, 1) 3 and their 6 translates within (0, 1] 3. The polytope B has 14 facets: two opposing base regular hexagons, six larger “isosceles” hexagons, and six rectangles. An illustration is in Figure 4.

One may calculate that B has a surface area ( 8)

Applying Parseval to ∫ g2 = 1 yields that , subject to which Equation ( 8) is minimized when ĝ ( 1, 1,…, 1) = 1. Thus, we are led to the solution for f stated in Theorem 3 and obtain the bound from Equations ( 6) and ( 8). It remains to verify that ∫|⟨∇f, u⟩| ≤ 2p also holds for each vector u. Using the Cauchy–Schwartz inequality, we obtain for our choice of g, and hence f. This completes the proof of Theorem 3.

4. A thREE-DimEnSionAL CuBiCAL FoAm

Although we have asymptotically solved the Cubical Foam Problem up to a small constant factor, in the physically natural case of d = 3, our construction does not improve on the Choe Prism, or even the cube. Here, we present an improved three-dimensional cubical foam, constructed via an ad hoc method.

The two-dimensional minimizer given by Choe in
Figure 2 (left) can be described as follows: Start with a
which is minimized when t ≈ 0.1880, having minimal value
about 5.6121. This already beats the surface area of the
Choe prism.

Figure 4. our new three-dimensional cubical foam. (top left) the unrelaxed tile. (top right) the tile after it has relaxed according to Plateau’s Rules. (Bottom left) the unrelaxed tile forming a foam according to the integer lattice. (Bottom right) illustration of the relaxed foam as soap bubbles.