“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.
1.05
0.8
0.55
0.3
0.05
1.05 0.8 0.55 0.3 0.05
–0.2