> We propose that the benefit of the inclination of the cells is to direct
about 10% of the weight of cell contents onto the midwall, thus increasing
the carrying capacity of the comb.
This may be one end result, but it implies "an intent by the bees" to a
purely physical process, including the inclination of the cells from the
horizontal.
The shape of honeycomb is nothing but physics at work.
A honeycomb cell is a "hexagonal prism capped off on one end by three
rhombi."
The "Kelvin Problem" was a more formal and complete restatement of the
questions posed by Zenodorus (170 BCE), and Pappus of Alexandria (300 CE).
(The "Kepler conjecture" is better-known, as it addresses close-packing
spheres, a great conversation-starter on first dates.)
Anyway, in 1887, Lord Kelvin asked how space could be partitioned into cells
of equal volume with the least area of surface between them. Does in
enclose the most volume in the smallest surface area? (Or, "the most volume
using the least materials"?) This kind of question may seem boring to most,
but there is a certain type of math geek that will remain fascinated on this
for years. It's a gift, maybe an illness. Kelvin's mathematically-derived
structure was a crystalline array of tetradecahedrons (truncated
octahedrons). Close to honeycomb cells, but not exact, and not quite as
efficient.
At the end of the 19th century, Joseph Plateau worked out something that
should have been seen all along - while "foams" (soap films) may seem
random, Plateau realized that the walls of a soap film can only meet two
ways. When they meet at a junction, it is always in threes, and at angles of
120 degrees, and the corners of the individual bubbles of a soap film can
only meet each other in corner-groups of four, with corner angles of about
109 degrees. So, we now had some rules for what was a STABLE bubble. Big
win.
In 1943, Toth got closer to the shape of an actual honeycomb cell with an
attempt at a mathematical solution to Lord Kelvin's problem, and showed that
his shape was that more efficient than Kelvin. He ended up with a hexagonal
cell capped
off by part of a truncated octahedron rather than the bees' "three rhombi".
But Toth's math just produced a different shape, with no tangible advantage
over honeycomb.
In 1994, D. Weaire and R. Phelan improved on this issue of "the least-area
way to partition space into regions of unit volume". They had a surface
area that was 0.3% less than Kelvin's attempt. They could show the math,
and even had soap bubble models to show the shape arising directly from the
close-packing of soap bubbles with only the pressure of adjacent bubbles
acting upon the bubbles. (You can imagine the lavish parties that were
thrown in mathematical circles to celebrate this astounding news, but bitter
arguments persist over whether "mathematical circles" are really true
circles, so most such parties tend to end in fistfights.)
Strangely, the Weaire-Phelan crystal structure was slightly different from
honeycomb. They had two kinds of cells - a 12-sided one and a 14-sided one.
This bugged the heck out of them, until they thinned their soap/glycerin mix
slightly, and THEN got the familiar shape of honeycomb as built by bees.
The bees do NOT enjoy the absolute "optimal structure". They have the
optimal structure one can have WITH BEESWAX. ("Wet" vs "Dry" foam, if you
read the math journals, and beeswax is a "wet" foam, not a "dry" one.)
But the bees are just trying to build closely-packed simple cylinders, and
physics does the rest. The pressure of the adjacent cylinders deform the
cylinders into the honeycomb crystal shape. Just like the soap bubbles, the
wax deforms to the "perfect" shape in the same way that two connected
bubbles will pick the least-surface-area face between the two, and adding a
3rd bubble results in a different, but still minimal surface, and so on.
This is to be expected from the laws of physics, not from insects with
brains the size of the head of a pin.
There aren't good images of Weaire-Phelan structures forming, but there is a
YouTube video of at least the simple case of hexagons arising from round
bubbles on the surface of a dish, fed bubbles by a precision pump making the
bubbles all the same size.
https://youtu.be/GbdQrR2nTRA
If you are interested in this area, or need a winning science fair project
for your kids, two books in my collection are "Soap-bubbles And The Forces
Which Mould Them" (Charles Vernon Boys, 1923) and "The Science of Soap Films
and Soap Bubbles" (Inseberg, Dover Books, 1992). I can email them both in
pdf, mobi, whatever. I have to admit that there was a period in my 20s when
I spent excessive time with soap bubbles in a glass box, a TI-59
programmable calculator, and a supercomputer of the time, working to do what
Weaire and Phelan eventually did achieve. My friends and family were very
understanding, and convinced me to seek counseling and treatment.
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