Honeycomb is the place where honey bees store the honey they obtain from flower pollen and also place their eggs. Its shape is a hexagon, which is the best shape to use the most space with the least amount of material. The hexagons in an entire honeycomb are made by different bees in equal dimensions. Thus, there is no free space in the honeycomb.

Each honeycomb is prepared in collaboration with many individuals and built from the bottom up. The rhombic section is the first base section, followed by two adjacent honeycomb walls, attached to the second rhombic base, and two more honeycombs formed. The third rhombic section and two wall hexagons complete the hexagon. The honeycombs are suspended vertically, the honeycomb on both sides separated by a wall in the middle. Workers knead and soften the wax and place it on walls of various thicknesses, the thickness of which is not more than 0.02 mm thick.

When making hexagonal shapes adjacent to each other for exactly 109 degrees 28 minutes, there is a need for various angle gauges and rulers to ensure smoothness of these shapes at the specified angle. For a human being, there is a great chance of making mistakes in drawing these shapes. You will also need to make various corrections, redraw some hexagons if necessary, and possibly take quite a long time. Surprisingly, as a human being, intelligent and conscious, dealing with all of these things, honey bees perform the same work without any angle meter or ruler.

Honey bees make combs using this flawless angle all over the world. Although there are hundreds of bees around the hive, not a single one can make mistakes. These creatures use two angles exactly 109 degrees 28 minutes and 70 degrees 32 minutes when building their honeycomb. There is no slight deviation. They build the ends of the combs by raising them by 13 degrees. This is important because this slope does not allow the honey to flow out of the honeycomb.

## How Bees Make Honeycombs?

Honeycomb is a system of pores made by worker bees to protect the eggs left by the queen and to store the stored honey, entirely made of beeswax. The construction of honeycombs is the task of worker bees. They affix the flake waxes to the inner walls of the two opposite upper edges of the wooden frame in the sleeve. Thus, they ensure that the boxers are vertical.

The worker bee peels the beeswax from his own body with his hind legs and kneads them into a paste and knees them into the upper inner wall of the frame. Another does the same job and glues the tiny wax ball right next to his friend’s. This is followed by a third so that thousands of worker bees complete the construction of the honeycombs. The most interesting situation in this work of worker bees is that this structure, which everyone brings and leaves one brick in a crowded mass of workers, can fully comply with the geometrical dimensions.

Mathematicians have proved that with a given amount of wax, there is no way to make a more powerful and larger place to contain the maggots that will hatch. Thus, worker bees have shown how to make a structure with a certain amount of equipment and the required size in the most economical way. A French insect scholar named Antoine Ferchault put it as a problem of geometry known as the problem of bees. This problem is:

Let a smooth hexagonal prism be closed with three equal rhombs whose base is the same slope with respect to each other. What should be the angles between the rhombs so that the total surface area of this prism is the smallest? Three prominent mathematicians, one German, one Swiss and one British, worked on the solution to this problem and came to the conclusion: 70 ° 32 ′. Indeed, this was the match between the angles in the honeycomb pores that the worker bees made. Even the most intelligent people have no say on the construction technique of worker bees.

Occasionally, worker bees also begin to produce honeycombs from several points. As the work progresses, the honeycomb construction, which has been started from different points, gradually approaches one another and merges in the middle and fuses. In this case, the fitters on the fusion line are in perfect hexagonal construction like the others and are of equal size. This reveals that the worker bees are not randomly assigned to work, foresee the distances between the starting and ending points and plan the dimensions of the boxers accordingly.

## Detailed Information about Honeycomb Construction

One of the most astonishing features of bees is their smooth hexagonal honeycomb. When a crowded group of bees is watched as they build honeycomb, the first thing that comes to mind is that there will be an uproar as a result of what this group does. It is unlikely that these creatures, who seem to be making independent movements, can form highly ordered structures together. However, contrary to what is seen from the outside, the honeycomb bees work in perfect harmony and in an orderly manner. So much so that although each starts from different places, they can all produce hexagonal cells of the same size. When they combine these hexagons in the middle, the joints are never clear and there is no shift in the angles of the hexagons.

Bees only weave honeycomb when needed in the hive. They build these combs to shelter, stock food and grow their eggs. Honeycombs have a regular structure in every aspect. For example, bee combs are double-faced. Hundreds or even thousands of eyes are found on both sides. These eyes are filled with honey, pollen, and eggs. If a sort is to be made, a honeycomb contains honey from the top to the middle section. There is pollen in the intermediate section and larval chambers at the bottom. The honey tanks also continue on the sides of the hive. However, worker bees store several rows of pollen between the larval chambers and the honey chambers. In this way, honey, larvae, and pollen are not mixed together. Of course, honey and larvae do not interfere with each other in the honeycomb. Otherwise, there would be an insurmountable situation for beekeepers. Beekeepers who wanted to separate a part of the honeycomb would inadvertently damage new members of the bee colony while trying to get honey. In addition, it would be very difficult to eat honey as it would mix with larvae.

Here again, it is a conscious movement that makes it easier. In appearance, there is no difference between cells in the combs (for example, larval cells, pollen, and honey cells). All of them are completely similar. Despite this similarity, however, the queen does not make the mistake of laying eggs in empty honey or pollen cells. Always lays eggs in the right place.

### General Structure of the Honeycombs

If a honeycomb is split in the middle, it is very interesting. The honeycomb has an intermediate wall. This intermediate wall, like the other parts, is made of wax and forms the common ground of the cells arranged on both sides. The floor of the cells is not flat. One is a pit to suit the other. These pits in opposing cells were inserted into each other to save space. The side walls have a structure that allows the cells to be slightly inclined downward relative to the intermediate wall. This slope ensures that honey does not flow from the filled cells.

Furthermore, there is an arrangement in the hive with the cells of the working bees higher and the smaller number of the males below. The queen cells are also built at the bottom again. In addition, honeycomb cells are knitted according to need. For example, when the number of drones in the hive decreases or in wintertime (no male in the hive in winter), it is started to be built from cells that are produced for men and which are larger than the others. Likewise, the queen cell is made only when a new queen is needed for the hive. There are also very important details in the construction of honeycombs. The details such as the production and use of the raw material of the honeycomb and the mathematical calculations to be made when combing the honeycomb are extremely surprising.

### First Stage in Honeycomb Construction: Production of Wax

The main construction material of bee honeycombs is wax. Bees secrete wax from the 4 pairs of secretory glands located under their bellies. There are two small gaps at the junction of these glands. The wax is formed in the form of small fine scales at these intervals. The bees use the hooks on their hind legs, which are made of feathers, to remove these small layers. They pass it to the wax plate and pull it out with their hind legs. Then they push forward to the middle, then to the front legs. (The bees have 6 legs) Finally, they take the plate with the jaw bones and knead them and bring them to a workable consistency. As soon as a candlestick is removed, the latter immediately exits the gap. Temperature is the most important element for the secretion of wax alone. Therefore, when the worker bees start to build the honeycomb, they are interlocked with each other in a chain and become a ball. In this way, the required 35 C degree heat is provided for wax. The kneading process is carried out at this optimum temperature, so that plasticized, construction-friendly wax is readily available.

The color of the wax is white when it is first secreted. As pollen and other substances are mixed, the color turns yellow and brown. The chemical content of the wax is as follows:

- Hydrocarbon 14%
- Monoesters 35%
- Diesters 14%
- Hydroxy Polyesters 8%
- Free acids 12%

Wax production is a process that requires a lot of energy. Therefore, bees consume approximately 22 kg of honey to make 1 kg of wax. The bees remove the wax from the glands at about the size of the head of a pin at a time. It is better understood why the wax is so valuable. Bees make the most of the smallest candle crumbs and make the most of the wax. It was even observed that when they had to leave a hive completely, they used a method such as carrying wax from the old hive instead of consuming honey to produce wax. The German scientist Dr. N. Koeniger found a bee colony that left the old hive to make a new hive elsewhere. The next day, Koeniger observed that worker bees returned to the hive and found that the bees gnawed wax from the old cells and carried them to their new nests. The reason for this frugal behavior of bees is that it requires a lot of energy in the production of wax.

Bees use wax wisely formed from pin-sized pieces and construct the most honeycomb with minimum wax. For example, it was found that bees used only 40 grams of wax for a honeycomb of 22.5×37 cm. This honeycomb with an empty weight of 40 grams can carry approximately 2 kg of honey.

### Honeycomb Making Is a Unique Craftmanship

As the world of bees was examined, the astonishment of scientists increased. Surprisingly, the calculations of mathematical shapes such as hexagon, trapezoid, rhombus, and details of which of these shapes will be found in the honeycomb, are complete by bees. For example, in the book The World of Bees, one of the most important works on bees, the researcher Murray Hoyt summarizes the construction of honeycomb as follows:

It is surprising that many different bees have the same thickness and shape after leaving the wax in their mouths where necessary. Of all these, you conclude that each of the tens of thousands of insects is a self-skilled engineer.

Each bee adds a small wax to its area in the honeycomb. And yet each honeycomb cell is the same size and shape as the others. When you look at the work of the bees, you think that each of them runs randomly from one place to another. The honeycomb process has dimensions and widths, just like an engineer’s great program. Hundreds, thousands of bees work from every point. Optimal gaps, optimal cell sizes are revealed.

It should also be noted that the shapes that are tried to be drawn on paper are two-dimensional. Bees form three-dimensional hexagonal prisms. In the construction of these three-dimensional prisms, there are very precise calculations such as the thickness and elasticity of the walls. In addition, since the honeycomb is bi-directional, the problem of joining the bases of the cells on both sides will also arise. In addition, all honeycomb cells have a 13-degree slope that prevents honey from flowing out.

Above all, the honeycomb is a structure that is created by bringing together the individual parts. In other words, honeycomb is not formed as a small piece grows and grows. In honeycombs, the pieces produced by each bee independently are added end to end. Even when combing honeycomb slices produced in different regions at the same time, there is no trace. The hexagons corresponding to the junction points of the cells do not remain halfway or because they are of different sizes, there are no problems such as the formation of incompatible cells at different heights. Bees connect the cells to each other in such a perfect way that it is not possible to identify the joints after the honeycomb construction is finished.

One question may come to mind as to why bees do not start making honeycomb from one side. If the bees started producing honeycomb from one side, it would take a long time to build the honeycomb. Because the area built would be narrow, but only as the number of cells increased, new bees could begin their mission. When we begin to weave honeycomb on a few sides, as all bees do now, honeycomb is completed very quickly as more bees work. As you can see, the details of honeycomb construction are extremely high. It is obvious that honeycomb is a specially designed structure.

## Why Do Bees Make Their Honeycombs as Hexagons?

Bees are the natural architects of nature. We all know that when you say honeycomb it is hexagonally shaped but most of us don’t know why it is hexagonal. The honey that they lay hexagonal to the laths placed vertically in the beehives is positioned at a fixed angle. An intermediate wall is encountered when cross-sections of honeycombs having an interesting shape are taken. This intermediate wall made of beeswax shows a common ground feature. Pits that prevent honey from flowing are related to the stability of the inclination angle.

The first stage in honey production is the production of wax. With the clamping of many worker bees in groups, a certain temperature value is reached and secretion starts. While the first secretion is white, the color turns yellow with secretion and pollen. The waxes produced as small as the head of the needle are realized with very high energy consumption. As a result, approximately 22 kg of honey is consumed to produce 1 kg of wax. Honeycomb production is the next stage. The combs obtained from the patchless image are an engineering marvel.

Scientists and mathematicians have proved that the most efficient storage method is hexagonal. Bees that start at this position based on maximum use are complemented by flawless hexagons in the middle. This awesome math scheme is not possible with other geometric shapes. Circles, pentagons, and octagons, such as spaces will certainly have gaps. Hexagonal is the most ideal type of shape to optimally divide an area with minimum material, as the square and triangle will have more wall circumference to fill the same volume.

And also; each hexagon has a depth of 3 centimeters and a wall thickness of five percent of a millimeter. Honey production is quite easy for bees that need this kind of work. Thousands of different bees always produce honeycombs of equal length. From the point of view of the offspring, they achieve everything from honeycomb construction, finding nutrients to honey production. For this reason, honey and beeswax are very valuable for us.

## Honeycomb Architecture and Geometry Behind It

Bees are the real architectures of nature. Therefore, the architecture of the honeycomb and the geometry behind it have attracted the attention of people throughout history. This structure, which consists of side-by-side hexagons, is extremely sensitive and has an average wall thickness of 0.1 mm. The deviation from this average value is up to 0.002 mm. It is necessary to have a mathematical perspective in order to understand the ideal of the geometry bases used in the construction of the combs.

A circle is a geometric shape with the shortest edge length surrounding a fixed area. For example, when the circumferential lengths of the square and circle are compared, the circumference of the circle is shorter. However, this is not exactly the case in the construction of the honeycomb. Here, the wide frame of the honeycomb will be divided into equal and smaller areas and the shape with the least circumferential length will be used in the division process. If we wish to divide the frame into small circles with equal areas, the shortest edge property will be provided as described above, but more candles will be spent for the spaces between the edges of the circles.

However, when we apply the principles of geometry to solve this problem with the shortest edge length and the least material, it will be seen that polygons should be used to divide the combs. Consider polygons with the same area with the number of edges n. Of these, the shortest perimeter has a smooth n-edge. By uniform, all edges and inner angles are equal. Such a polygon can always be drawn into a circle and the corners of the polygon are on the circumference of the circle. Since such a structure is close to the ideal circle shape, the circumference is minimal.

For example, the shortest circumference in equilateral triangles is obtained in the equilateral triangle and the shortest circumference between the quadrangles is obtained in the square. Similarly, if pentagons and hexagons are compared between each other, the shortest circumference can be obtained in smooth pentagons and hexagons. The first question that can come to mind is which smooth polygon should we use when dividing a particular area. If we examine a circle and a regular polygon with n edges drawn into it, we can see that an internal angle of the polygon is 180-360 / n degrees.

When we want to divide a given large area into smaller areas, the neighboring polygons must fit together and have no space between them. For this to happen, the sum of the inner angles of adjacent polygonal corners leaning against each other should be 360 degrees. In other words, the integer of an inner angle should be a multiple of 360 degrees. N represents the number of adjacent angles, so we can write the following equation (N is an integer):

N x (180 – 360 / n ) = 360

If N is solved from here, we can only get n = 3, 4 and 6 as integer values, and no integer can be obtained for any number greater than 6. So if we want to divide an area without spaces, we must use either triangles, quadrangles or hexagons. It is not possible to make a gap-free section with a smooth polygon with more than 6 edges. Similarly, straight pentagons are not a suitable solution. By placing three straight pentagons side by side, a 360-angle free space is created. However, hexagons can be placed side by side without spaces.

In addition, when equal-area triangles, rectangles, and hexagons are compared to each other, the minimum line length is in the hexagon. Therefore, the minimum wax consumption can be achieved by using the chamber in this way. Mathematicians also investigated whether polygons with curved edges are better or not. When the edge is curved, a convex shape is obtained in a polygon while inevitably a concave shape is obtained in a neighboring polygon. The advantage obtained with the convex curve (because it is more similar to the circle part) eliminates the further disadvantage from the concave curve and clearly does not yield any gain.

Thomas Hales of the University of Michigan put an end to the debate in 1999, and when we wanted to divide an area into equal small areas, it proved to be the ideal hexagon. Although it has long been stated that the hexagonal shape is an ideal shape, it has not been able to prove solid mathematics. So far we have looked at the problem in two dimensions. However, honeycomb is a three-dimensional object and has a hexagonal prism shape. The hexagonal prism-shaped combs are in two layers with one end open and the other closed end-to-back.

When the frame is placed perpendicular to the ground, the prisms are constructed with a horizontal inclination angle of 13 °, which is the smallest angle sufficient to prevent honey from flowing. What kind of geometry should be used at the closed end of the honeycomb for minimum wax consumption? In 1964, mathematician Fejes Toth showed that the ideal closure could be achieved by two hexagons and two squares. Bees, on the other hand, are closing it with three rhombuses. The inner angles of the rhombuses are 70.5 degrees and 109.5 degrees, giving the ideal mathematical solution for the shape of the three rhombus roofs.

Apparently, there was a very small loss of 0.035% in the application of bees compared to two hexagons and two squares. However, there was one point that was overlooked, and the calculations were taking the wall thickness extremely thin. Researchers used liquid air foam to experience Toth’s mathematical model. They pumped two layers of a detergent solution with bubbles of a 2 mm diameter between two glasses. The bubbles in contact with the glass turned into hexagonal structures. At the border of the two layers in the middle, two hexagonal and two square structures formed by Toth were formed. When the bubble walls were slightly thickened, an interesting situation emerged and the structure suddenly became three rhombus structures, just like bees.

## A Scientific Breakthrough: The Amazing Secret of Honeycombs Is Revealed

The regular hexagonal structure in honeycombs, once seen as an unbelievable achievement of mathematician insects, was illuminated by a very simple mechanism. For years, scientists have looked at the angular perfection of honeycombs in amazement; however, none have fully explained this mechanism. Engineers in the UK and China have taken some first and important steps towards solving this mystery. The team showed that this hexagonal shape, which is pleasing to the eye in honeycombs, was originally a circular structure and that they took a hexagonal structure in seconds. Researchers published their findings on July 16 in the Journal of the Royal Society Interface.

Bhushan Karihaloo, an engineer and author of the study at the University of Cardiff; Recalling that two great geniuses like Galileo Galilei and Johannes Kepler were fascinated by this phenomenon, he says, people have speculated a lot about how honey bees make their honeycombs. After adding that there are many interesting, even funny explanations based on the fact that bees use mathematics, Karihaloo states that the explanation is much simpler, contrary to popular belief.

The team set up a facility in Beijing where they could easily observe the construction of honey bees’ honeycombs. In their studies, they documented the various stages of the construction of the honeycombs. As a result of the observations, it was noticed that the honeycombs were circular in the first stage. After this first cellular structure of the honeycomb was formed, team observers noticed that honeybees were heating the walls of the honeycomb (in fact, this was described in previous studies but not yet explained in detail) and they think that this is the most important step in the hexagonal formation.

The bees provide the heat they use in this process from their own bodies. When they heat the honeycomb they make the candle fluid and then the honeycomb becomes the most stable form of a hexagon. But the process is not yet fully elucidated. The question the team is currently working on is whether the bees heat a honeycomb completely or partially. “My guess is that the honeycomb is only partially heated at certain points, but it is easier to heat a honeycomb at one time,” says Karihaloo. The team calculated how long both scenarios would take. This means that when you heat a honeycomb cell completely into hexagons, it takes 6 seconds, and when you heat it at certain points, it takes 36 seconds. Researchers in the team say they will use this data later in the study.

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