Why are honeycomb cells hexagonal in shape? - Quora Quora Written Jan 31, 2016 Consider the fact that the most space-efficient way to pack circles is in a hexagonal structure. Consider that in a triangle, the point [math]O[/math] that has the least total distance from the three vertices [math]A,B,C [/math] has angles such that [math]AOB=BOC=COA=120^\circ[/math]. Essentially, a hexagonal structure minimizes the surface area to volume ratio. Wax is very expensive (in terms of energy) for bees to produce, so they would like to try to store as much as they can with the minimum wax possible. Why don't they store it in spheres, then? Well, first of all, each of the compartments need to be open to the rest of the hive, so they can't surround a compartment with other compartments as that would totally defeat the purpose of the hive. Why don't they make larger hexagons instead of such tiny ones? It's the same reason that we store things in individual packages, we need to organize things by type of object, expiration date, etc.

As an aside, if you've seen what bubbles are like when they are clumped together, you will realize that the surfaces all tend to meet at angles of 120 degrees. The reasons for this are similar; one needs to minimize surface area for a given volume. (This was all in a lecture at SMT some years ago.) It is actually smart geometry of nature.Hexagon is the only shape which shares common boundaries with other hexagon and doesn't waste any space.Imagine if it was an all circular small chambers,there will be small gap left between them. ​ ​On the other hand,triangle or square can fit,but have lower surface area than hexagon.Other geometrical shapes like pentagon,heptagon,ocatgon will waste space,as like circle.This arrangement is done in account of use of wax by bees.Tomake the honey comb,bees have strategy that the small chambers should have maximum area in it,with usage of less bee's wax and minimum wastage of space.And hexagon fulfills all this.But the beauty is that bees are no engineers or architectures,still they know how to create perfect geometrical shapes without any tools.

Written Mar 12, 2016 Two possible explanations exist as to why honeycomb is composed of hexagons, rather than any other shape. First, the hexagonal tiling creates a partition with equal-sized cells, while minimizing the totalperimeter of the cells. Known in geometry as the honeycomb conjecture, this was given byJan Brożek and proved much later byThomas Hales. Thus, a hexagonal structure uses the least material to create a lattice of cells within a given volume. A second reason, given by D'Arcy Wentworth Thompson, is that the shape simply results from the process of individual bees putting cells together: somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this, he notes that queen cells, which are constructed singly, are irregular and lumpy with no apparent attempt at efficiency.[2] The closed ends of the honeycomb cells are also an example of geometric efficiency, albeit three-dimensional and little-noticed.

The ends are trihedral (i.e., composed of three planes) sections of rhombic dodecahedra, with the dihedral angles of all adjacent surfaces measuring 120°, the angle that minimizes surface area for a given volume. (The angle formed by the edges at the pyramidal apex, known as the tetrahedral angle, is approximately 109° 28' 16" (= arccos(−1/3)).) ​ The three-dimensional geometry of a honeycomb cell The shape of the cells is such that two opposing honeycomb layers nest into each other, with each facet of the closed ends being shared by opposing cells. ​ Opposing layers of honeycomb cells fit together Individual cells do not show this geometric perfection: in a regular comb, deviations of a few percent from the "perfect" hexagonal shape occur. In transition zones between the larger cells of drone comb and the smaller cells of worker comb, or when the bees encounter obstacles, the shapes are often distorted.

Cells are also angled up about 13° from horizontal to prevent honey from dripping out.[3] ​ Honeycomb section containing transition from worker to drone (larger) cells – here bees make irregular and five-cornered cells (marked with red dots). ​ Western honeybees and honeycomb In 1965, László Fejes Tóth discovered the trihedral pyramidal shape (which is composed of three rhombi) used by the honeybee is not the theoretically optimal three-dimensional geometry. A cell end composed of two hexagons and two smaller rhombuses would actually be .035% (or about one part per 2850) more efficient. This difference is too minute to measure on an actual honeycomb, and irrelevant to the hive economy in terms of efficient use of wax, considering wild comb varies considerably from any mathematical notion of "ideal" geometry. Source: Wikipedia Hexagonal Honeycomb Pattern Hexagonal Honeycomb Pattern ~ the beautiful mathematics of the honeycomb pattern only nature can give. Why is the shape of a honeycomb cell hexagonal, and not square, round, triangle, rectangle, rhombus, or even pentagon? Has any kid asked you this question before?

If you can't get away by telling them because the bees are insects and insects have six legs, and thus they make six-sided homes, read this as it might help somewhat, or perhaps even leave you and all the adults and kids agog. The answer in a nutshell - hexagon is the most efficient shape for the bees' living and storing of food. The bees already have enough awesome abilities to wow us - their pollination of over 90 fruit and vegetable crops worldwide, their phenomenal sense of smell , their incredibly speedy wings that make their famous, distinctive buzz, their highly organized society and intricate cooperation, and their communication with one another by "dancing" . Turning to the structural design of their homes, it also suggests that they excel in economics and mathematics! Studies on the geometry of honeycomb pattern explain that no other shape can create more space with the given material. Circles for instance leave space, and square makes smaller area. In addition, your engineer friends can confirm with you that the hexagon structure provides the maximum strength and that it is used in designing airplane wings and satellites walls. Though the wax cell walls may be only about 0.05mm thick, each cell can support 25 times its own weight.

The bees have chosen the hexagon as a building cell for their combs to achieve the maximum strength to store their food - honey in it: a comb of 100 g weight can hold in it up to 4 kg of honey! Producing wax is a laborious process. For each gram of wax produced, the bee needs to consume 6 to 7 grams of honey. The bees are genetically able to calculate, design, and construct a structure that provides the most living and storage space for their larvae, honey and pollen utilising the least resources and energy! Wow, these insects are born engineers and economists! And modern design today has borrowed the bee's building principle for achieving spaciousness, strength, and sturdiness in creation of countless indoor structures, furniture, and household and office products. Yet another spectacular work of the honeybees and the creator of this universe! It takes a bee to get the honey out. ~ Arthur Guiterman Hexagonal Honeycomb Pattern in Purity Test Many have claimed the amazing formation of hexagonal patterns on a film of honey as a proof of purity. Too curious to dismiss it, I followed the instructions given: 1. Put a blob of honey onto a light coloured round plate. 2.

Pour in some cold water to cover the honey. 3. Swirl the plate several times. 4. If the honey is pure, you would see a honeycomb image/pattern on the honey. And the outcome? Right before my very eyes, with a couple of swirls, I saw a honeycomb pattern consisting of big hexagonal shapes form on the surface of the honey patch, and with several more swirls, the hexagonal cells split into smaller ones, and here's a snapshot of what was observed: My verdict?. I'm still not sure how this points to honey purity . I've also tried the same test with adulterated honey (half very concentrated syrup and half honey mixed) and there no pattern was observed. But that was only one type of syrup, and I suppose amount of syrup versus honey also matters, there are too many variables involved in such a purity test. Also, evidence on the presence of pure honey does not automatically rule out the presence of other substances or ingredients such as starch, maltose, corn syrup, etc. So, is this test a test of 100% pure honey or just the presence of pure honey?

Could this comb pattern be related to any concentrated form of sugar syrup (and not just honey) since we know similar pattern can occur with rock sugar and candy bars? (See Honeycomb Sweets below.) Whatever the answers may be, the observation described above is enough to wow me. Incredible, beautiful phenomenon! Honeycomb Sweets Sweets with a "Honeycomb" label or manufactured to appear like real honeycombs seem to be able to present themeselves as healthier options. See in here some of these sweets that many of us have got ourselves hooked on: World of Honeycombs Apiological: mathematical speculations about bees (Part 1: Honeycomb geometry) | The Aperiodical Apiological: mathematical speculations about bees (Part 1: Honeycomb geometry) in Features , Irregulars Bees have encouraged mathematical speculation for two millennia, since classical scholars tried to explain the geometrically appealing shape of honeycombs. How do bees tackle complex problems that humans would express mathematically? In this series we’ll explore three situations where understanding the maths could help explain the uncanny instincts of bees. Honeycomb geometry A curvy wild honeycomb.

Honeybees collect nectar from flowers and use it to produce honey, which they then store in honeycombs made of beeswax (in turn derived from honey). A question that has puzzled many inquiring minds across the ages is: why are honeycombs made of hexagonal cells? The Roman scholar Varro, in his 1st century BC book-long poem De Agri Cultura (“On Agriculture”), briefly states “Does not the chamber in the comb have six angles, the same number as the bee has feet? The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space 1 .” This quote is the earliest known source suggesting a link between the hexagonal shape of the honeycomb and a mathematical property of the hexagon, made more explicit a few centuries later by Pappus of Alexandria (sometimes considered to be the last Ancient Greek mathematician). Writing after the Roman Empire’s glory days, Pappus points out that there are three regular polygons that tile the plane without gaps—triangles, squares and hexagons—and bees, in their wisdom, choose the design that holds the most honey given a set amount of building material 2 .

The idea that bees economically choose the regular tiling polygon with the greatest area for a given perimeter satisfied the ancients. But why should all the honeycomb cells be identical—could bees do any better than using regular polygons? The rhomboidal base of a wax foundation is behind the flatness of domestic honeycombs. Image from Creative Commons The amazingly regular honeycombs we are used to seeing are built by domestic honeybees, using perfectly regular wax foundations provided by their beekeepers as a guide. Wild honeybees, however, don’t have the luxury of nestboxes fitted with wax foundations. Their honeycomb cells, though still very regular, aren’t always hexagonal—sometimes pentagons and heptagons creep in. After choosing an irregularly-shaped tree cavity, many worker bees will start independently building the hive’s hexagons at once, meeting in the middle along seams. It is along these seams where the five and seven-sided irregularities and other defects tend to appear 3 . More intriguingly, what happens if you provide honeybees with the wrong size of foundation as a guide? H. Randall Hepburn tried this with variously-sized hexagonal foundations 4 (with flat bases unlike the standard ridged commercial ones).

The largest caused the bees to to build gorgeous rosettes—they filled each large hexagonal foundation with five or six cells (again, typically five or seven-sided) surrounding a central cell. The next size down caused the bees to build in an irregular pattern, while the size smaller than that led to the bees building hexagonal cells on each vertex of the foundational hexagons, and leaving a hexagonal void or ‘false cell’ in the centre. Cell construction by African honeybees on foundations of various sizes. ©Entomological Society of Southern Africa – reproduced with permission of the Society Same perimeters, but different areas as in (P). So bees don’t necessarily need to use regular polygons at all. Allowing for the possibility of a mixture of shapes leads to two subtly different mathematical problems: (A) Which mixture of shapes that tile the plane, and each have unit area, have the minimum average perimeter? (P) Which mixture of shapes that tile the plane, and each have unit perimeter, have the maximum average area?

Fejes Tóth, a Hungarian mathematician, had proved by 1963 that the regular hexagonal tiling was the solution to the isoperimetric problem, (P), but only proved that it solved the iso-areal problem, (A), under the assumption that all the shapes were convex. This convexity condition is more restrictive than it may at first seem: a bulging edge on one shape leads to a concave indentation on a neighbouring shape, so the only convex shapes that can tile the plane are polygons. It wasn’t until 1999 that Thomas Hales proved that hexagons also divide a plane into shapes of equal area with the least perimeter. A key step in the proof (though not Hales’ breakthrough) is showing that, on average, such shapes would have six sides. Take a planar graph (ie. with no edges crossing) on a sphere. Using the Euler characteristic of a planar graph (which also holds on a sphere), we know that the number of the vertices, edges and faces of the graph satisfies $V-E+F=2$. Nudging the edges slightly, it turns out to be sufficient to consider only vertices of degree 3.

It’s quite easy to convince oneself with a diagram (like the one above) that this is plausible. Moving edges by a tiny amount, you can make sure no more than three lines meet at one vertex. You might change the numbers of edges and vertices, but without changing the perimeter or area by much. If the $i$th face has $e_i$ edges (and vertices), then adding these up for each face, we would count each edge twice and each vertex three times 5 . So, by the Euler characteristic of the whole graph, overall we must have $$V-E+F=\sum_i \left ( \frac{e_i}{3}-\frac{e_i}{2} + 1 \right )=\sum_i \left ( 1 – \frac{e_i}{6} \right )=2.$$As the number of faces increases, $1 – \frac{e_i}{6}$ must become very small, and so the average of $e_i$ over all faces $i$ tends to $6$, which is the number of sides we were hoping for. 6 This step can be used to show a finite version of the theorem on the plane, which is a good start for the infinite version.

Open end (above) and closed rhombic end (below) of a rhombic dodecahedral honeycomb cell. So out of all the prisms arranged side-by-side, hexagonal prisms use the least beeswax to build a unit bee-sized volume. But honeycombs are not made up of hexagonal prisms: the hidden end of the honeycomb cell is not flat. There are two layers of cells, back-to-back and offset, and the end-cap between them is a pyramid made of three rhombuses, so the whole shape could also be described as ‘half of an elongated rhombic dodecahedron’. The ridged wax honeycomb foundation pictured near the start mimics these rhombuses. This shape is more efficient than a hexagonal prism with a flat base. What bees haven’t realised, is that there’s a slightly more efficient way to cap off their hexagonal cells. Keeping the hexagonal footprint of their hive, they could make repeated use of a truncated octahedron, formed by lopping off the vertices of an octahedron. This would lead to a saving of almost 2% in the idealised version with walls of negligible thickness, which isn’t much.

Using a curvy non-polyhedral variant of the truncated octahedron, Lord Kelvin, the 19th century physicist, held the record for ‘least surface area for a fixed volume’ in the three-dimensional version of the honeycomb conjecture for a full century 7 . This was assumed to be the best possible until 1994, when two physicists, Denis Weaire and Robert Phelan, armed with computers, devised an ever-so-slightly more efficient way of dividing space into cells of equal volume, using two different curvy solids (2.2% more efficient than the rhombic dodecahedra). It was after this discovery that the long-standing two-dimensional honeycomb conjecture seemed less obviously true: a suggestion from Weaire is what nudged Hales to attempt his proof 8 . Despite the fact that honeybees do not achieve a theoretical minimum surface area for a set volume, honeycombs are still a sensible way to reduce the need for building materials. However, we should question the assumption that evolution is purely trying to economise on the bees’ behalf, and ask whether this actually is the main reason why honeybees build hexagonal honeycombs.

Building a typical wild nest takes approximately a kilogram of wax, produced by consuming about eight times that amount of honey. Since this is also a third of the amount of honey the bees require to survive the winter, there is a strong evolutionary incentive to use as little wax as possible. There is evidence that bees do economise in another sense when building hives: they have some preference for nesting sites where honeycombs are already present 9 , so they don’t need to build new ones. An alternative explanation to the economic argument is an engineering one: that hexagonal honeycombs are strong structures. It seems pretty hard to design experiments to determine how much each factor contributes to the bees’ motivations 10 , so for the time being the explanations may well remain as Just-So stories. One useful way of looking at the causes of a particular animal behaviour is to ask four questions, known as Tinbergen’s questions . The idea is to explain the current and historical reasons for a behaviour’s existence by asking questions about the individual and the species. (Current & species: Function) Why have honeybees evolved to build combs in this shape? (Historical & species: Phylogeny) How has honeycomb building evolved over time?

(Current & individual: Mechanism) How do the bees construct the honeycomb? (Historical & individual: Ontogeny) How does the behaviour develop in an individual bee? We’ve been looking at one possible answer to the first question, however this still leaves three unanswered. The ontogeny, or biological and neurological development of bees’ building behaviours is well beyond the scope of this article, which is convenient because I know absolutely nothing about it. As for the other two questions, Charles Darwin (among others) had a good crack at them. Among the biological investigations recorded in Darwin’s “ On the Origin of Species ” were experiments and observations relating to honeycomb formation. He proposed that a rough evolutionary progression (the phylogeny) could be inferred from various types of modern bee nests. Bumblebee nests are rough conglomerations of almost spherical cells found underground. The nest of the Mexican stingless bee , Melipona domestica, lies somewhere between that of the bumblebee and honeybee, with a hexagonal arrangement but more rounded cells and cylindrical walls. Looking at these other bee nests, Darwin suggested that honeybees once had nests similar to bumblebees, then more like those of the Mexican stingless bee, before their current honeycombs.

Through conversation and correspondence, he came to the same conclusion as some others before him, that if bees built regular arrangements of cylindrical or spherical cells closer and closer together over the generations, then at some point they would crowd together into the famous honeycomb arrangement. The Voronoi cells of five points. In more formal modern mathematical terms, this is the idea of Voronoi cells . Take some points in space: the region closer to one point than any other is known as that point’s Voronoi cell. If you take points in two slightly offset layers, all spaced equally to their closest neighbours (ie. joining the points would form equilateral triangles and hexagons), then the three-dimensional Voronoi cells form the honeybees’ hexagonal cells with their rhombic bases (partial rhombic dodecahedra). Darwin wanted to know whether this theory matched up with the way bees actually build their hives. He watched very closely as his own honeybees constructed their nest and his observations support the Voronoi cell idea of how the nests evolved and how they are built. Darwin conducted some brilliantly simple experiments that involved introducing either a thick wax block or thin coloured wax sheet into a hive.

With a wax block introduced, each bee dug a small hemisphere into the block, until it touched or got extremely close to an adjacent bee’s excavation, at which point the bees built the hexagonal walls up, leaving a smooth basin. This suggests that in the usual situation, the rhomboid basins are caused by pressures from bees working on cells on the other side, whereas in this case there were none. With a thin sheet introduced, the bees started excavating but stopped before they broke through to the opposite side, leaving very shallow and flat basins. This is all supports Darwin’s suggested mechanism for the cell construction. In the classic 1917 book “ On Growth and Form ” D’Arcy Thompson points out that the beeswax is warm and slightly liquid when being used for building, which may cause tension effects to come into play. The same effects cause bubbles to naturally seek shapes that minimise their surface area. This is attractive as an alternative to the Voronoi cells idea, and so gets frequently repeated, but to me it doesn’t yet seem to be supported by the evidence. The surface tension explanation is most vocally put forward in the paper Honeybee combs: construction through a liquid equilibrium process?

(££) 11 . The authors suggest that—in analogy with experiments where wax is melted around close-packed rubber bungs—the bees first construct cylindrical cells, and then heat them until the wax melts enough for surface tension to take over and form the hexagons. 12 A rebuttal, Hexagonal comb cells of honeybees are not produced via a liquid equilibrium process (££) 13 , points out some weaknesses in the paper above. The bees are seen to actively move their mandibles across the cell walls ( video , 22mb AVI), mechanically shaping them, as well as feeling them with their antennas. Social wasps build their hexagonal nests with cellulose, a rather different material to wax. Cellulose doesn’t have the same fluid properties under heating either, so it is merely softened by the wasps using saliva. This means that wasps achieve similar results to the honeybees by actively kneading cells into shape without surface tension effects. If in doubt, we should prefer arguments that have more explicative power, and not resort to special cases for bees and wasps. Central to the argument is the question of whether bees construct intermediate cylinders.

This observation only seems to appear once in the literature, in a paper published in 2013 (after the above rebuttal) 14 . This paper shows a close-up photo of the few circular cells found after the honeybees were smoked out during the construction of their hive. The circular cells were near the edge of the growing comb, a bit shorter than the usual 10mm, and the walls nearest the base of the circular cells were hexagonal. After the bees returned and completed their work, these cells were modified to their normal hexagonal specifications. Unfortunately, this observation doesn’t clinch the argument either way. While it is necessary for surface tension to play a major role, intermediate cylinders are consistent with both explanations. The fact that bees rework their cells as the comb expands was noticed by Darwin, who observed that some vermillion-coloured wax introduced at the edge would become dispersed throughout much of the comb. It’s going to take a very well-designed experiment to convince me that bees rely largely on surface tension effects, as opposed to moulding the exquisite hexagons themselves. After all, the worker bees are rarely depicted as lazy.

Further reading: Charles Darwin’s “On the origin of species”: is freely available on Archive.org and Project Gutenberg , for instance, and cheaply available in book form. Chapter VIII: Instinct, Cell-making instinct of the Hive-Bee, is the section referred to above. After reading about Darwin’s experiments in his own words, I strongly recommend reading an account of Darwin’s honeycomb correspondence which includes some modern recreations of Darwin’s original experiments. “ On Growth and Form ” by D’Arcy Thompson not only has a detailed history of the study of honeycomb geometry, but has many interesting facts about biological and geological shapes and their growth, as well as speculation about their root causes. And on the pure mathematical geometry side, Fejes Tóth’s “ What the bees know and what they do not know ” gives the best overview of minimisation problems that bees’ honeycombs do solve, and then many more they do not solve. What Is It About Bees And Hexagons? : Krulwich Wonders... : NPR What Is It About Bees And Hexagons? May 14, 20139:50 AM ET Robert Krulwich Solved!

A bee-buzzing, honey-licking 2,000-year-old mystery that begins here, with this beehive. Look at the honeycomb in the photo and ask yourself: (I know you've been wondering this all your life, but have been too shy to ask out loud ... ) Why is every cell in this honeycomb a hexagon? iStockphoto.com Bees, after all, could build honeycombs from rectangles or squares or triangles ... Robert Krulwich/NPR But for some reason, bees choose hexagons. Always hexagons. 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? Robert Krulwich/NPR 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. Article continues after sponsorship 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 ...

Robert Krulwich/NPR 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 ... Robert Krulwich/NPR 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. Robert Krulwich/NPR 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 ... Robert Krulwich/NPR ... 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." Robert Krulwich/NPR 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." Alan Lightman's essay " The Symmetrical Universe ," originally published in Orion Magazine, will be included in his new book The Accidental Universe: The World You Thought You Knew to be published early next year. I also recommend Ivars Peterson's essay in Science News, The Honeycomb Conjecture . How do bees produce the hexagonal-shaped cells of a honeycomb? - Quora Science How do bees produce the hexagonal-shaped cells of a honeycomb? How do hundreds of bees, working together, produce such intricately patterned habitats? Is there a "bee architect" in charge of the overall design of each hive and honeycomb? Written Feb 8, 2015 Jeff is right, it is instinct. Each individual bee knows from birth how to make a hexagon comb... if that's what they use.

Not all bees make the perfect, hexagonal combs like the honeybee, Apis mellifera, does. Some bees make hexagonal combs with rounded edges that don't quite fit perfectly. Others make round combs that are slightly six sided, and others make round combs. Evolution predicts the round-cell bees will be more basal and the hexagon-cell bees more derived, with the other bees as "links" in between... which is exactly the case, as morphology and DNA confirmed. Score yet another one for science! Charles Darwin himself actually did an elegant test of this, by sticking sheets of wax into beehives and watching what shapes were made. Even if he made round or hexagonal dents in the wax, the bees would make whatever combs they were used to. I believe he describes this in "Origin of Species," but I don't have a copy of it at this time. Written Mar 1, 2015 A worker bee will come across a piece of unfinished comb and, if she has a supply of wax, she will be stimulated to add to the comb. She secretes scales of wax from the front of her thorax (their chest), which she then chews to a malleable consistency.

She builds each cell from the bottom up. A single cell may be built by multiple bees. Each bee takes its cue about what to do from the outermost part of the cell. If it's short, she will add to its height. If it's open and full of honey, she will make a lid. If it's closed and the hive is hungry, she will eat the lid to access the honey - and so on. The thickness of the cell walls is tested by the antennae across the top of the cell. Thin walls are added to; thick walls are shaved away. As mentioned in another answer, the cells start off round. It's thought that wax heated by the bodies of the bees melts and flows into the gaps, thus forming the hexagons. However, a 2009 study by Kadmon et al. suggests that this last process is assisted by ultrasonic buzzes, which may be why different species of bee achieve different levels of accuracy with their hexagons.

The hexagonal shape of the honeycomb cells depends on the construction behavior of bees : Scientific Reports The hexagonal shape of the honeycomb cells depends on the construction behavior of bees Scientific Reports 6, Article number: 28341 (2016) doi :10.1038/srep28341 20 June 2016 Abstract The hexagonal shape of the honey bee cells has attracted the attention of humans for centuries. It is now accepted that bees build cylindrical cells that later transform into hexagonal prisms through a process that it is still debated. The early explanations involving the geometers’ skills of bees have been abandoned in favor of new hypotheses involving the action of physical forces, but recent data suggest that mechanical shaping by bees plays a role. However, the observed geometry can arise only if isodiametric cells are previously arranged in a way that each one is surrounded by six other similar cells; here I suggest that this is a consequence of the building program adopted by bees and propose a possible behavioral rule ultimately accounting for the hexagonal shape of bee cells. Introduction The hexagonal shape of the cells of the bees’ honeycomb has intrigued laypeople and scientists for ages.

Why do bees build this kind of cells? How can they achieve such a spectacular structure? Early natural philosophers, like Marcus Terentius Varro 1 , based on the observation that hexagons possess the highest surface/perimeter ratio, compared to other polygons that can be used for tiling the plane, suggested that honey bees build their hexagonal cells in order to achieve the best economy of material. Eighteen centuries later, Darwin surpassed the early teleological explanations noting that the swarm wasting least honey in the secretion of wax would have succeeded best 2 . Precise data about food conversion rates in the production of beeswax 3 support the intuition of the great naturalist. However, for a complete understanding of a given phenomenon, both the proximate and ultimate sets of causes have to be explained and interpreted 4 ; in other words, both the why and the how questions need to be addressed. Varro suggested the correspondence between the bees’ legs number and the angles of the comb cell might hint of a possible construction method 1 ; in any case, the geometers’ skills of bees were not questioned for a long time.

However, cells are not hexagonal when they are built but rather circular, as already suspected by ancient observers and easily demonstrated by studying wax combs in the early stages of construction 5 , 6 . Rasmus Bartholin suggested that hexagons would result automatically from the pressure of each bee trying to enlarge as much as possible each cell 7 , while Thompson argued that it is surface tension that molds the soft wax into a hexagonal pattern of cells 8 , as a result of the formation of three 120° angles between the wax walls converging in a triple junction. More recently, Pirk and collaborators provided support to this latter hypothesis by studying the wax condensing around an array of rubber bungs 5 , while Karihaloo and coworkers reached a similar conclusion by modelling the effect of surface tension at the triple junction between adjacent cells 6 . This latter evidence, supporting the interpretation of the honeycomb architecture as the result of blind physical forces rather than biological engineering, lead Ball to state that there does not seem to be much room left for the honey bees’ engineering prowess 9 . However, a recent study questioned the possibility that surface tension alone can explain the process.

In particular, Bauer and Bienefeld 10 argued that bees do not heat up wax to the temperatures needed for the wax to reach the liquid equilibrium and proposed an alternative explanation based on the mechanical shaping of cells. In any case, available data about the thermal properties of beeswaxes 11 indicate that the onset of the wax melting temperature is very close to the range observed by Bauer and Bienefeld 10 . In conclusion, the formation of the hexagons is still a matter of debate and a definitive conclusion has not yet been reached. Anyway, regardless of the mechanism leading to the hexagonal shape of the primitively circular cells, an array of hexagonal prisms can be obtained with any of such mechanisms only if isodiametric cylinders are previously arranged in a regular tightly packed array, with each cylinder surrounded by six other same sized cylinders. In sum, despite the considerable attention it recently attracted, the mechanism through which hexagons are formed appears to be of secondary importance; instead, the central point appears to be the formation of the hexagonal pattern of isodiametric cylinders. Under this perspective, the questions to be addressed are: Is a hexagonal pattern of cylinders an essential prerequisite for the formation of hexagonal cells?

How do bees build cylinders of exactly the same size? How do bees arrange the cylinders in a hexagonal pattern? Several methods have been exploited so far for the study of the building behavior of bees, and Darwin himself dedicated to the subject some interesting pages of the “Origin of the species” 2 . However, many bees are involved in this behavior normally occurring, under natural conditions, within a dark cavity. This makes the observation somehow more difficult than in the case of other hymenopterans that build nests made of hexagonal cells as well (see, for example, the studies carried out by West-Eberhard on paper wasps 12 ). For this reason, some clever expedients were adopted, ranging from the use of painted wax mentioned by Darwin 2 to the thermographic observation of building bees while repairing their nest altered by the careful intervention of the experimenter 10 . In this study I try to address at best the above questions by using only methods that allow the study of such a complex behavior while minimizing the possible perturbation of the process under observation.

Results and Discussion To see if the arrangement of cylindrical cells in a hexagonal pattern, with each cell surrounded by six other cells, is an essential prerequisite for the formation of the hexagonal cells, I carried out a systematic observation of honeycombs, searching for possible imperfections that could reveal the consequences of building errors involving the reciprocal arrangement of cells. This allowed to record a limited number of building errors; in these few cases, it was noted that when cells are not surrounded by six other cells, their final shape is not hexagonal but rather matches that of a polygon with as many sides as the number of surrounding cells ( Fig. 1a and Fig. S1 for more examples). This indirectly confirms the influence of the pre-existing pattern on the final shape of cells. Figure 1: Geometry and construction of the honeycomb. (a) The pattern of cells influences their final shape: cell 1, surrounded by six other cells, has a hexagonal shape, with 120° angles between sides, whereas cell 2 has a different shape, with 90° angles between some consecutive sides.

(b) The margin of a comb with stubs of cells at different stages of construction suggests a possible construction scheme (1: the construction of the cell base is started in the groove between two pre-existing cells; 2: when the cell base is as large as the cell diameter, the walls are started; 3: the walls encircle the first stub of the cell). Full size image Conversely, if another behavioral rule is used, whereby the addition of cell floor can start anywhere, without being confined in the groove between adjacent cells ( Fig. S4 ), an array of aligned cells is obtained ( Fig. 2a ; Supplementary excel file 1 ). In order to see if the proposed building rule is actually exploited in the construction of combs by bees, sequential pictures of honeycombs under construction were taken at two hours intervals. The study of the tip of such combs revealed a growth pattern similar to that postulated here ( Fig. 2b and Fig. S5 for further examples). Honeycombs are made of two layers of contiguous cells, each one sharing its closed end with three cells of the opposite side ( Fig.

S6a ); the perfect matching of the two sides of the comb is another feature that attracted remarkable attention for its geometrical implications and it is still waiting for an explanation. Despite the study of this further feature was outside the scope of this work, it is worth noting that the rule described above seems to be consistent with the observed geometry, provided that the two sides of the honeycomb grow in synchrony and the beginning of the construction of the cell base, in the groove between two adjacent cells on one side of the comb, coincides with the construction of the lateral walls of a cell on the opposite side ( Fig. S6b ). In conclusion, regardless of the mechanism transforming the close packed cylinders into hexagonal prisms, here I draw the attention to the causes responsible for the geometrical pattern of cylinders on which that mechanism can then act. I suggest that a set of behavioral rules including the initiation of a new cell from the groove between two pre-existing cells and the erection of the cell’s walls when the cells base reaches a certain size, are likely involved.

The computer simulation shows that a small difference in the set of behavioral rules followed by bees would account for a big difference in the arrangements of cells and consequently on the shape of cells with big consequences in terms of the wax needed to build the comb. This proposed rule appears to be remarkably similar to that reported by Downing and Jeanne for the wasp Polistes fuscatus 18 , despite the different physical properties of the used material and the distant taxonomic relationship, highlighting an interesting case of convergent evolution of the construction behavior pattern. These reflections may help to bring some clarity in the longstanding debate about the hexagonal shape of the honeycomb cells and provide inputs for more experimental work on the subject as well as stimulating some comparative work on a feature that is shared by other hymenopterans. Methods Assessing the influence of the pattern of cells on their final shape Although regularity represents one of the most striking features of the honeycomb, errors can sometimes occur 3 .

In order to observe such errors, to see if they can reveal the possible consequences of mismatches involving the reciprocal arrangement of cells, the combs built by an Apis mellifera ligustica x carnica colony maintained in the experimental apiary of the Dipartimento di Scienze AgroAlimentari, Ambientali e Animali of the University of Udine, were observed and photographed in the field ( Fig. 1a and Fig. S1 ). Estimation of the coefficient of variation of the cell diameter in a honeycomb To test the hypothesis that the “ruler” of bees (i.e. the body parts used for estimating the distances by bees) is adequate to realize regular cylinders of a given diameter, I measured the coefficient of variation of the cells of a natural comb and compared this figure to that of the bee’s body parts. To do so, five close-up pictures of different portions of a comb were taken and the diameter of cells was measured using ImageJ (Rasband, W.S., ImageJ, U. S. National Institutes of Health, Bethesda, Maryland, USA, , 1997–2015).

To minimize possible parallax errors, only the diameter of the six cells surrounding the central one was considered and the coefficient of variation calculated dividing the standard deviation by the average diameter of the six cells. The overall coefficient of variation, reported in the text, is the average of five estimates, whereas the standard deviation describes the variability of such estimates around the reported mean ( Supplementary data file 1 ). Since there is abundant observational evidence about the use of antennae for measuring distances in the honey bee, I measured the length of the flagellum of the antennae of 5 batches of 5 bees, using the “segmented line” tool of ImageJ, as illustrated in Supplementary data file 2 , and calculated the coefficient of variation of this morphological feature as described above ( Supplementary data file 2 ). Computer simulation of the process of comb construction According to Downing and Jeanne 18 , the unifying features regarding the regulation of construction behavior are: 1. construction results from a linear series of if-then decisions; 2. cues regulating decision points come from previous construction; 3. the construction behavior pattern is inherited. In this case, the use of computer simulations can help to study the possible ordered patterns emerging from such processes.

Therefore, to study the geometry of a comb built according to a certain set of behavioral rules, a spreadsheet was created according to the above-mentioned logic (see Fig. S4 for the flowchart of the program and Supplementary excel file 1 ). This kind of tool was exploited to make it possible for the interested reader to interpret the behavioral rules that are modeled by looking at the formulas that have been used for designing the spreadsheet. The spreadsheet simulates a two dimensional view of a simplified honeycomb made of quadrangular cells with a fixed dimension (3 × 3 squares); note that, in this way, the progressive enlargement of the virtual comb is constrained in two dimensions. In the spreadsheet, the condition of each square, at each time step is based on the condition of the surrounding squares at the preceding step as modified according to a set of fixed rules, applied in an ordered sequence ( Fig. S4 ). Three conditions are possible for a square: empty or covered with floor (representing the pavement of the bee cells) or wall (representing the walls separating adjacent bee cells). Already filled squares are left untouched (i.e. no rearrangement of square condition is allowed).

Construction starts from the top and proceeds laterally and downwards starting with a piece of floor attached to the upper horizontal line, representing the top bar of the virtual comb. The proposed tool can be regarded as a cellular automaton, similar to those presented by Wolfram 19 ; the results obtained here further support the idea that this kind of computer creations can be useful to achieve a better comprehension of some natural processes. By modifying the settings of the spreadsheet, different rules can be applied with different possible outcomes. To apply the following rules: (a) New wall is added as soon as the adjacent floor reaches a certain size (in either of both directions), (b) New floor can be added either to pre-existing floor or walls, use the following set of criteria for building the floor in Supplementary excel file 1 : floor/wall to the left or right: yes; floor above: yes; wall above: yes; vertical stretch of wall above: no. With these settings, a pattern of aligned cells is obtained ( Fig. 2a , left column of schemes).

Note that if rule b) is modified such that new floor is added only to pre-existing floor, the construction stops as soon as the first cell is encircled by walls; instead, if lateral growth is permitted either in contact with floor or wall but downward growth is allowed only in contact with pre-existing floor, the construction stops after the first line of cells is built ( Supplementary excel file 1 with the following set of criteria for building the floor: floor/wall to the left or right: yes; floor above: yes; wall above: no; vertical stretch of wall above: no). To note the effect that is obtained downward growth is allowed only between pre-existing cells (in this simplified simulation, this is achieved by allowing the deposition of new floor in contact with the vertical wall separating two adjacent closed cells), use the following set of criteria for building the floor in Supplementary excel file 1 : floor/wall to the left or right: yes; floor above: yes; wall above: no; vertical stretch of wall above: yes. In this case, an alternated pattern of cells is obtained ( Fig. 2a , right column of schemes). Of course, the rules regulating the deposition of cell walls can not be changed without completely disrupting the construction process.

To see if a more complete set of rules, but conceptually similar to those presented above, could be exploited to build a more realistic virtual comb made of polygonal cells simulating the circular ones found in nature, another spreadsheet was created. In this case, the rule allowing floor addition in the space between pre-existing cells was implemented with an algorithm such that the distance between the cell walls converging into the groove between existing cells was compared to the cell diameter and new floor was added only in case such distance was smaller ( Fig. S7 ; Supplementary excel file 2 ). Observation of comb construction To study the construction process of natural combs ( Fig. 2b and Fig. S5 ), a frame with combs under construction was photographed, at two hours intervals, from 10.30 to 16.30, during Summer 2015, after removing the building bees with smoke. The frame was placed in the central portion of a hive containing an Apis mellifera ligustica x carnica colony maintained in the experimental apiary of the Dipartimento di Scienze AgroAlimentari, Ambientali e Animali of the University of Udine.

Additional Information How to cite this article: Nazzi, F. The hexagonal shape of the honeycomb cells depends on the construction behavior of bees. Sci. Rep. 6, 28341; doi: 10.1038/srep28341 (2016). References Varro, M. T. Rerum Rusticarum Libri III (37 B.C.). 2. Darwin, C. On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life (John Murray, 1859). 3. Hepburn, H. R. Honeybees and Wax. An experimental natural history (Springer-Verlag, 1986). 4. Cause and effect in biology . Science 134, 1501 (1961). Pirk, C. W. W., Hepburn, H. R., Radloff, S. E. & Tautz, J. Honeybee combs: construction through a liquid equilibrium process? Naturwissenschaften 91, 350–353 (2004).

Karihaloo, B. L., Zhang, K. & Wang, J. Honeybee combs: how the circular cells transform into rounded hexagons . J. R. Soc. Interface. 10.1098/rsif.2013.0299 (2013). 7. Bartholin, E. De Figura Nivis. Dissertatio (Hafniae, 1661). 8. Thompson, D’. A. W. On Growth and Form (Cambridge University Press, 1917). 9. How honeycombs can build themselves . Nature. 10.1038/nature.2013.13398 (2013). Hexagonal comb cells of honeybees are not produced via a liquid equilibrium process . Naturwissenschaften 100, 45–49 (2013). Buchwald, R., Breed, M. D. & Greenberg, A. R. The thermal properties of beeswaxes: unexpected findings . J. Exp. Biol.

211, 121–127 (2008). The social biology of polistine wasps . Misc. Publ. Mus. Zool. Univ. Michigan 140, 1–101 (1969). 13. Ruttner, F. Biogeography and Taxonomy of Honeybees (Springer-Verlag, 1988). 14. Über die fähigkeit der bienenkönigin (Apis mellifica L.) zwischen arbeiten und drohnenzellen zu unterscheiden . Apidologie 1, 115–142 (1970). 15. Grassé, P. P. La reconstruction du nid et les coordinations interindividuelles chez Bellicositermes natalensis et Cubitermes sp. la théorie de la stigmergie: Essai d’interprétation du comportement des termites constructeurs . Insectes Soc. 6, 41–80 (1959). Self-organizing pattern formation on the combs of honey bee colonies . Behav. Ecol. Sociobiol.

28, 61–76 (1991). Downing, H. A. & Jeanne, R. L. The regulation of complex building behaviour in the paper wasp, Polistes fuscatus (Insecta, Hymenoptera, Vespidae) . Anim. Behav. 39, 105–124 (1990). Bees don't do math — hexagonal honeycombs emerge naturally - NBC News advertisement Honeycomb cells start off as circles within the first few seconds of formation (a) and then eventually morph into hexagons (b). The perfect hexagonal shape of honeycomb cells — once thought to be an incredible feat of math-savvy insects — has now been explained by simple mechanics. Scientists have marveled at the angular perfection of honeycomb for centuries, but none have been able to clearly describe how it forms. Engineers in the U.K. and China have taken a step forward by showing that the cells actually start off as circles — molded by the shape of a bee's body — and then flow into a hexagonal pattern seconds later. The researchers reported their findings on July 16 in the Journal of the Royal Society Interface.

"People have always speculated how bees have formed these honeycombs," said Bhushan Karihaloo, an engineer at Cardiff University in the U.K. and co-author of the study, citing Galileo Galilei and Johannes Kepler as two of the luminaries mystified by the problem. "There have been some incredible, esoteric, even bizarre explanations; [some people] believed the bees had an uncanny ability to measure angles. But it's actually much more mundane." [ 10 Amazing Things You Didn't Know About Animals ] Honeycomb up close Using a honeycomb grown at a research facility in Beijing, the researchers were able to carefully ward off the bees and photograph the bare honeycomb seconds after formation, providing the first clear evidence that cells naturally start as circles. They then observed honeybees heating the wax after the initial cell formation — a phenomenon identified in previous studies, but never analyzed in close detail — and found this to be the key step in hexagon-formation. By heating the cells, the bees cause the wax to become molten and flow like lava. Once the wax starts flowing, the cell walls naturally fall flat and take on the shape of a hexagon, like adjoining bubbles in a bath.

This is physically the simplest and most stable way for cylinders to merge, Karihaloo said. The team still does not know exactly how the bees go about heating each cell, and explored the mechanics of two plausible scenarios: One in which the bees focus their heat only at points where neighboring cells touch (a total of six points per cell), and another in which the bees heat the entire cell all at once. "My own feeling is that nature tries to minimize energy spent, and from that point of view, I would think that scenario one is most probable," Karihaloo told LiveScience. "But, on the other hand, from the perspective of the bees, they might just want to warm the whole thing and get away with it. That remains to be shown." Building like bees The team calculated the amount of time each scenario should take, and found that circular cells should morph into hexagons within six seconds if they are warmed entirely, and within 36 seconds if only partially warmed. In their future work, the researchers hope these time constraints will help them assess which mechanism the bees use.

Juergen Tautz, a bee biologist in Germany who was not involved in the study, does not believe that bees can direct their heat to specific points in a cell, but still finds this study valuable. "This paper is very important in my perspective because it not only gives us a deep insight into the mechanisms that honeybees manage to build very precise cells, but technology can also [take] from it," Tautz told LiveScience. The team hopes their findings will eventually allow them to create an artificial honeycomb as strong as the natural material, which would be useful in strengthening a variety of building and structural materials. Follow Laura Poppick on Twitter . Follow LiveScience on Twitter , Facebook and Google+ . Original article onLiveScience. Secrets of the beehive | Science | The Guardian Secrets of the beehive How have honeybees beaten the best mathematicians for 2,000 years? Keith Devlin reports Wednesday 25 August 1999 22.47 EDT Share on Messenger Close After nearly two thousand years, mathematicians have now finally proved that honeybees are among the world's most efficient builders.

The 4th century geometer Pappus was one of several ancient Greek mathematicians who suspected that the elegant shape of the honeycomb was a result not of an innate bee-sense of geometric beauty but of nature's efficiency. The repeating pattern of six-sided figures you see in a cross-section of a honeycomb, Pappus guessed, used the least amount of wax to build the walls. His guess, in an essay on "the sagacity of bees", became known as the Honeycomb Conjecture. It resisted all attempts to prove it until a few weeks ago, when mathematician Thomas Hales of the University of Michigan announced that he had cracked the puzzle. Not until the advent of close-up film techniques did scientists know for certain how bees build their honey stores. It is a remarkable feat of high precision engineering. Young worker bees excrete slivers of warm wax, each about the size of a pinhead. Other workers take the freshly produced slivers and carefully position them to form vertical, six-sided, cylindrical chambers (or cells). Each wax partition is less than 0.1mm thick, accurate to a tolerance of 0.002mm.

Each of the six walls is exactly the same width, and the walls meet at an angle of precisely 120 degrees, producing one of the "perfect figures" of geometry, a regular hexagon. Why don't bees make each cell triangular, or square, or some other shape? Why have straight sides in the first place? After all, the warm wax could just as well be formed into curved walls. Although a honeycomb is a three-dimensional object, because the individual cells are all cylindrical, the total area of the wax walls depends solely on the shape of the cross-section of the cells. Thus, the mathematical problem is one of two-dimensional geometry - the kind we learn in school. What it boils down to is finding the two-dimensional shape that can be repeated endlessly to cover a large flat area, for which the total length of all the cell perimeters is the least (so that the total area of the honeycomb walls is as small as possible). Some facts are easy to establish. For instance, only three kinds of regular polygons (straight-edged figures whose edges are all the same length and whose angles are all the same) can be fitted together side-by-side to cover a plane: equilateral triangles, squares, and regular hexagons.

Others will leave gaps. Of the three space-filling figures, squares give a smaller total perimeter than triangles and hexagons do even better than squares. It is also easy to show that, if you restrict yourself to hexagons, the regular ones (ie the ones with equal sides and angles all 120 degrees) give a smaller perimeter than non regular ones. But if you allow combinations of polygons of all kinds, or edges that are not straight lines, then things become a lot more complicated. Relatively little was known until 1943, when Hungarian mathematician L Fejes Toth used an ingenious argument to prove that the regular hexagon pattern does give the smallest total perimeter for all patterns made up of any combination of straight-edged polygons. What happens if the edges can be curved? Toth thought the regular hexagon pattern would still be more efficient than anything else, but he could not prove it. Thought of in terms of a single honeycomb cell, if a wall bulges out, you can store more honey in that cell for the same wall area than if the wall were straight. But when all the cells are packed together, a wall that bulges out for one cell bulges in for the adjacent cell.

Could there be an entire honeycomb of cells with bulging walls where the net increase in efficiency of the outward bulges outweighs the net decrease caused by the inward bulges? If there were such a pattern, the honeycomb conjecture would be false. Intuitively, outward bulges would exactly balance out inward bulges, which is why Toth thought the hexagonal pattern would be the best. But as mathematicians observed, things are not quite as simple as they might seem. Nevertheless, that is exactly what Hales has proved: the bulges do cancel out. He has posted his 19 page proof at: ~hales and experts who have seen it say it seems correct. Last year, Hales proved that the most efficient way to pack equal-sized spheres together in a large crate is to arrange them in a regular pattern of staggered layers, the way greengrocers the world over stack oranges, so that the oranges in each higher layer sit in the hollows made by the four oranges beneath them.

That problem, which was first raised by the astronomer Johannes Kepler in 1611, resisted many attempts at a solution until Hales took 250 pages and a three gigabyte computer package (program plus data) to prove what supermarket employees know instinctively. But the instincts of small creatures go way beyond human capabilities. Take the Tunisian desert ant. This tiny creature can wander across the desert sands for up to 50 metres until it stumbles across the remains of a dead insect, whereupon it bites off a piece and takes it directly back to its nest - a hole no more than one millimetre in diameter. How does it find its way back? By the process the Apollo astronauts used to plot their course to the Moon: dead reckoning (strictly "ded" reckoning from "deductive reckoning). The idea of dead reckoning is to calculate your position relative to your starting point from a knowledge of your speed and your direction of travel. Some ants lay down a scent trail to keep track of their path but not the Tunisian desert ant.

If it is moved after it finds the food, it will head off in exactly the direction it should have to find its nest if it had not been moved, and will stop and start a bewildered search for its nest when it has covered the precise distance that should have brought it home. Many birds that migrate over featureless seas navigate by the stars. Humans need optical equipment, star charts, and trigonometric calculations. The birds seem to learn the pattern of the stars - which has to be learned, since it changes faster than evolutionary adaptation could adjust for - by gazing up at the sky as fledglings in the nest. Those who struggle with mathematics sometimes marvel at the way birds or insects can perform "mathematical" feats. But these creatures do not use mathematics as we do. Natural selection has produced a rich diversity of creatures that can eachperform a particular range of activities to survive. Human beings have extended their own range of instinctive behaviours so that they can mimic some of the activities of their fellow creatures. Using mathematics, science, and technology, we too can navigate around the globe and build elaborate places to live.

The fact that it has taken 2,000 years of effort and 19 pages of advanced mathematics to show that the familiar honeycomb is the most efficient pattern for the storage of honey is as much a marvelous testament both to the wonders of nature and to our mathematical ingenuity. Honeycombs' Surprising Secret Revealed Honeycombs' Surprising Secret Revealed By Laura Poppick, Live Science Contributor | July 17, 2013 02:03pm ET MORE Honeycomb cells start off as circles within the first few seconds of formation (a) and then eventually morph into hexagons (b). Credit: B.L. Karihaloo, K. Zhang and J. Wang The perfect hexagonal shape of honeycomb cells — once thought to be an incredible feat of math-savvy insects — has now been explained by simple mechanics. Scientists have marveled at the angular perfection of honeycomb for centuries, but none have been able to clearly describe how it forms. Engineers in the U.K.

and China have taken a step forward by showing that the cells actually start off as circles — molded by the shape of a bee's body — and then flow into a hexagonal pattern seconds later. The researchers reported their findings yesterday (July 16) in the Journal of the Royal Society Interface. "People have always speculated how bees have formed these honeycombs," said Bhushan Karihaloo, an engineer at Cardiff University in the U.K. and co-author of the study, citing Galileo Galilei and Johannes Kepler as two of the luminaries mystified by the problem. "There have been some incredible, esoteric, even bizarre explanations; [some people] believed the bees had an uncanny ability to measure angles. But it's actually much more mundane." [ 10 Amazing Things You Didn't Know About Animals ] Honeycomb up close Using a honeycomb grown at a research facility in Beijing, the researchers were able to carefully ward off the bees and photograph the bare honeycomb seconds after formation, providing the first clear evidence that cells naturally start as circles.

They then observed honeybees heating the wax after the initial cell formation — a phenomenon identified in previous studies, but never analyzed in close detail — and found this to be the key step in hexagon-formation. By heating the cells, the bees cause the wax to become molten and flow like lava. Once the wax starts flowing, the cell walls naturally fall flat and take on the shape of a hexagon, like adjoining bubbles in a bath. This is physically the simplest and most stable way for cylinders to merge, Karihaloo said. The team still does not know exactly how the bees go about heating each cell, and explored the mechanics of two plausible scenarios: One in which the bees focus their heat only at points where neighboring cells touch (a total of six points per cell), and another in which the bees heat the entire cell all at once. "My own feeling is that nature tries to minimize energy spent, and from that point of view, I would think that scenario one is most probable," Karihaloo told LiveScience. "But, on the other hand, from the perspective of the bees, they might just want to warm the whole thing and get away with it.

That remains to be shown." Building like bees The team calculated the amount of time each scenario should take, and found that circular cells should morph into hexagons within six seconds if they are warmed entirely, and within 36 seconds if only partially warmed. In their future work, the researchers hope these time constraints will help them assess which mechanism the bees use. Juergen Tautz, a bee biologist in Germany who was not involved in the study, does not believe that bees can direct their heat to specific points in a cell, but still finds this study valuable. "This paper is very important in my perspective because it not only gives us a deep insight into the mechanisms that honeybees manage to build very precise cells, but technology can also [take] from it," Tautz told LiveScience. The team hopes their findings will eventually allow them to create an artificial honeycomb as strong as the natural material, which would be useful in strengthening a variety of building and structural materials. Editor's note: This story has been updated to correct where the hives were grown. The hives were grown at the Agricultural Research Academy, Beijing, and were not grown specifically for this study. Follow Laura Poppick on Twitter .

Follow LiveScience on Twitter , Facebook and Google+ . Original article on LiveScience . Editor's Recommendations Savannah Bee Company Savannah Bee Company What is Honeycomb? June 9, 2014 What is honeycomb? Honeycomb is a cluster of repeating hexagonal beeswax cells that fill the interior of a honey bee beehive. While adequate, this definition raises more questions than it actually answers. Let’s take a closer look at this miraculous substance and its ingenious design! Why the hexagon? Honeybees are not unique in their selection of the hexagon as a structural building shape. The hexagon has been utilized throughout evolution due to its simplicity, strength, and durability. All vascular plants maintain a water-transporting tissue called xylem. This tissue is designed to move water from the plant roots to the plant leaves where it will be used in the process of photosynthesis to build the glucose molecule. Water from the soil and carbon dioxide from the atmosphere combine to build plant sugars in a process known as carbon fixation. The hexagon is employed by many plant species in the structure of xylem tissue. The hexagon lattice structure is also incorporated into animal bone tissue.

The hexagonal matrix of bone tissue provides strength but also gives bones a lightweight quality. This enhances mobility without creating undue stress on skeletal muscle groups. Honey bees are not the only insect to make use of this brilliant design. Paper wasps (Polistinae) construct their nests from paper using a repeating hexagonal honeycomb pattern. Although these insects do not produce honey, they do use hexagonal cells for developing brood just as the honey bee does. The hexagon is the base shape for a matrix that minimized the used of construction material while at the same time maximizing the strength and reducing the weight of the final product. Honeycomb Minis! What is beeswax? Beeswax is the substance from which honeycomb is made. Beeswax is a lipid-like all other waxes. Honeybees secrete beeswax from eight glands with openings on their lower abdomen. Honeybees must consume honey in order to produce beeswax. The beeswax hardens on the abdomen region and flakes of in clear flat pieces known as scales. These scales are then chewed and softened by worker bees and formed into hexagonal cells within the honeycomb.

Although the scales are initially clear, they become opaque following the chewing and re-forming. Why do honeybees make beeswax comb? Honeycomb is the internal structure of the honeybee hive. The come forms the “rooms” that are the home of the honeybee. A single hexagonal structure is known as a cell. Individual cells serve the colony in different ways. Upon returning from her mating flight, the queen bee lays her eggs in the cells that have been constructed by her female worker bees. These cells are known as brood cells and will serve as incubation chambers for developing honeybees. The brood cells are meticulously maintained by the female worker bees. After all the brood is the future of the colony! Is Honeycomb Nutritious? Honeycomb is basically beeswax and raw honey . The beeswax has very little nutritional value with the exception of trace amounts of vitamin A. However since our system does not break down beeswax, it is a very good source of roughage. Raw honey as we have discussed is very nutritious; packed full of vitamins, enzymes, pollen, and antioxidants. Interestingly, the most nutritious of all raw honey is honey still sealed in the wax cells of the honeycomb.

Honey in its purest form (inside the cells of honeycomb) contains very little water. But honey, when exposed to air, has a tendency to pull water from the atmosphere. So, honey that is removed from the comb and exposed to air holds around 12% more water than honey left sealed in the comb. This dilution creates a slight but measurable difference in nutritional value by weight. Honey that remains sealed in the beeswax honeycomb cells is without question the purest form of raw honey!

Question: What shape are the individual cells in a honeycomb?
Answer: Hexagonal