May 14, 2013 · 9:50 AM ET ROBERT KRULWICH/NPR iStockphoto.com
…And not just your basic six-sided hexagon. They like “perfect” hexagons, meaning all six sides are of equal length. They go for the jewelers’ version — precise, just so. Why?
Well, this is a very old question. More than 2,000 years ago, in 36 B.C., a Roman soldier/scholar/writer, Marcus Terentius Varro, proposed an answer, which ever since has been called “The Honeybee Conjecture.” Varro thought there might be a deep reason for this bee behavior. Maybe a honeycomb built of hexagons can hold more honey. Maybe hexagons require less building wax. Maybe there’s a hidden logic here.
I like this idea — that below the flux, the chaos of everyday life there might be elegant reasons for what we see. “The Honeybee Conjecture” is an example of mathematics unlocking a mystery of nature, so here, with help from physicist/writer Alan Lightman, (who recently wrote about this in Orion Magazine) is Varro’s hunch.
The Essential Honeycomb
Honeycombs, we all know, store honey. Honey is obviously valuable to bees. It feeds their young. It sustains the hive. It makes the wax that holds the honeycomb together. It takes thousands and thousands of bee hours, tens of thousands of flights across the meadow, to gather nectar from flower after flower after flower, so it’s reasonable to suppose that back at the hive, bees want a tight, secure storage structure that is as simple to build as possible.
So how to build it? Well, suppose you start your honeycomb with a cell like this … a totally random shape, no equal sides, just a squiggle …
If you start this way, what will your next cell look like? Well, you don’t want big gaps between cells. You want the structure tight. So the next cell will have to be customized to cling to the first, like this …
And the third cell, once again, will have to be designed to fit the first two. Each cell would be a little different, and that means, says Alan Lightman …
… this method of constructing a honeycomb would require that the worker bees work sequentially, one at a time, first making once cell, then fitting the next cell to that, and so on.
But that’s not the bee way. Look at any YouTube version of bees building a honeycomb, says Alan, and you won’t see a lot of bees lounging about, waiting for their turn to build a cell. Instead, everybody’s working. They do this collectively, simultaneously and constantly.
So a “squiggle cell plan” creates idle bees. It wastes time. For bees to assemble a honeycomb the way bees actually do it, it’s simpler for each cell to be exactly the same. If the sides are all equal — “perfectly” hexagonal — every cell fits tight with every other cell. Everybody can pitch in. That way, a honeycomb is basically an easy jigsaw puzzle. All the parts fit.
OK, that explains why honeycomb cells are same-sized. But back to our first question: Why the preference for hexagons? Is there something special about a six-sided shape?
Some shapes you know right away aren’t good. A honeycomb built from spheres would have little spaces between each unit …
… creating gaps that would need extra wax for patching. So you can see why a honeycomb built from spheres wouldn’t be ideal. Pentagons, octagons also produce gaps. What’s better?
“It is a mathematical truth,” Lightman writes, “that there are only three geometrical figures with equal sides that can fit together on a flat surface without leaving gaps: equilateral triangles, squares and hexagons.”
So which to choose? The triangle? The square? Or the hexagon? Which one is best? Here’s where our Roman, Marcus Terentius Varro made his great contribution. His “conjecture” — and that’s what it was, a mathematical guess — proposed that a structure built from hexagons is probably a wee bit more compact than a structure built from squares or triangles. A hexagonal honeycomb, he thought, would have “the smallest total perimeter.” He couldn’t prove it mathematically, but that’s what he thought.
Compactness matters. The more compact your structure, the less wax you need to construct the honeycomb. Wax is expensive. A bee must consume about eight ounces of honey to produce a single ounce of wax. So if you are watching your wax bill, you want the most compact building plan you can find.
And guess what? “[The honeycomb is] absolutely perfect in economizing labor and wax.” Charles Darwin]
Two thousand thirty-five years after Marcus Terentius Varro proposed his conjecture, a mathematician at the University of Michigan, Thomas Hales, solved the riddle. It turns out, Varro was right. A hexagonal structure is indeed more compact. In 1999, Hales produced a mathematical proof that said so.
As the ancient Greeks suspected, as Varro claimed, as bee lovers have always thought, as Charles Darwin himself once wrote, the honeycomb is a masterpiece of engineering. It is “absolutely perfect in economizing labor and wax.”
The bees, presumably, shrugged. As Alan Lightman says, “They knew it was true all along.”
…are seen from different perspectives, showing the repeating “cuboct” lattice structure, made from many identical flat cross-shaped pieces (credit: Kenneth Cheung/MIT)
MIT researchers have developed a lightweight structure whose tiny blocks can be snapped together much like the bricks of a child’s construction toy.
The new material, the researchers say, could revolutionize the assembly of airplanes, spacecraft, and even larger structures, such as dikes and levees.
The new approach to construction is described in a paper appearing in the journal Science, co-authored by postdoc Kenneth Cheung and Neil Gershenfeld, director of MIT’s Center for Bits and Atoms.
Gershenfeld likens the structure — which is made from tiny, identical, interlocking parts — to chainmail.
The parts, based on a novel geometry that Cheung developed with Gershenfeld, form a structure that is 10 times stiffer for a given weight than existing ultralight materials.
But this new structure can also be disassembled and reassembled easily — such as to repair damage, or to recycle the parts into a different configuration.
In the lab, a sample of the cellular composite material is prepared for testing of its strength properties (credit: Kenneth Cheung/MIT)
Mass-produced large structures without large factories
The individual parts can be mass-produced; Gershenfeld and Cheung are developing a robotic system to assemble them into wings, airplane fuselages, bridges or rockets — among many other possibilities.
The new design combines three fields of research, Gershenfeld says: fiber composites, cellular materials (those made with porous cells) and additive manufacturing (such as 3-D printing, where structures are built by depositing rather than removing material).
With conventional composites — now used in everything from golf clubs and tennis rackets to the components of Boeing’s new 787 airplane — each piece is manufactured as a continuous unit.
Reduced vehicle weight, fuel, and construction costs
So manufacturing large structures, such as airplane wings, requires large factories where fibers and resins can be wound and parts heat-cured as a whole, minimizing the number of separate pieces that must be joined in final assembly. That requirement meant, for example, Boeing’s suppliers have had to build enormous facilities to make parts for the 787.
Pound for pound, the new technique allows for much less material to carry a given load. This could not only reduce the weight of vehicles, for example — which could significantly lower fuel use and operating costs — but also reduce the costs of construction and assembly, while allowing greater design flexibility. The system is useful for “anything you need to move, or put in the air or in space,” says Cheung, who will begin work this fall as an engineer at NASA’s Ames Research Center.
The concept, Gershenfeld says, arose in response to the question, “Can you 3-D print an airplane?” While he and Cheung realized that 3-D printing was an impractical approach at such a large scale, they wondered if it might be possible instead to use the discrete “digital” materials that they were studying.
Assembler robot for large structures
“This satisfies the spirit of the question,” Gershenfeld says, “but it’s assembled rather than printed.” The team is now developing an assembler robot that can crawl, insectlike, over the surface of a growing structure, adding pieces one by one to the existing structure.
In traditional composite manufacturing, the joints between large components tend to be where cracks and structural failures start. While these new structures are made by linking many small composite fiber loops, Cheung and Gershenfeld show that they behave like an elastic solid, with a stiffness, or modulus, equal to that of much heavier traditional structures — because forces are conveyed through the structures inside the pieces and distributed across the lattice structure.
What’s more, when conventional composite materials are stressed to the breaking point, they tend to fail abruptly and at large scale. But the new modular system tends to fail only incrementally, meaning it is more reliable and can more easily be repaired, the researchers say. “It’s a massively redundant system,” Gershenfeld says.
Cheung produced flat, cross-shaped composite pieces that were clipped into a cubic lattice of octahedral cells, a structure called a “cuboct” — which is similar to the crystal structure of the mineral perovskite, a major component of Earth’s crust. While the individual components can be disassembled for repairs or recycling, there’s no risk of them falling apart on their own, the researchers explain. Like the buckle on a seat belt, they are designed to be strong in the directions of forces that might be applied in normal use, and require pressure in an entirely different direction in order to be released.
The possibility of linking multiple types of parts introduces a new degree of design freedom into composite manufacturing. The researchers show that by combining different part types, they can make morphing structures with identical geometry but that bend in different ways in response to loads: Instead of moving only at fixed joints, the entire arm of a robot or wing of an airplane could change shape.
Alain Fontaine, who directs the innovation program for aircraft manufacturer Airbus, says this new approach to building structures “is really disruptive. It opens interesting opportunities in the way to design and manufacture aerostructures.” These technologies, he says, “can open the door to other opportunities” and have significant potential to lower manufacturing costs.
In addition to Gershenfeld and Cheung, the project included MIT undergraduate Joseph Kim and alumna Sarah Hovsepian (now at NASA’s Ames Research Center). The work was supported by the Defense Advanced Research Projects Agency and the sponsors of the Center for Bits and Atoms, with Spirit Aerosystems collaborating on the composite development.
Some of these design concepts appear to be related to R. Buckminster Fuller’s radical tensegrity structures. — Editor
REFERENCES: Kenneth C. Cheung, Neil Gershenfeld, Reversibly Assembled Cellular Composite Materials, Science, 2013, DOI: 10.1126/science.1240889