Bees keep the honey they produce in honeycombs which have special structures. Honeycomb is the place where honey bees obtain their flower pollen and store their eggs. The shape of the honeycomb is 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.

## Information About Honeycombs

Each honeycomb is prepared in collaboration with many individuals and is built up from the bottom. The rhombic section is the first base section, followed by two adjacent honeycomb walls, the second rhombic base being added to the base and two more honeycombs formed. The third rhombic section and two wall hexagons complete the hexagon. Honey bee honeycombs are hung vertically, the honeycombs on both sides are separated by a wall in the middle. The workers knead and soften the wax and place it on the walls of various thicknesses so that their thickness is no more than 0.02 mm.

In the case of 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 probably 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.

If you stand near a honeycomb, you will see nothing but bees flying around. However, every flying bee is a superior mathematician who knows at which angle he should stick the wax he carries. Is it possible for a bee to be so skillful and talented? Does a bee have the ability to calculate better than humans? Of course, the bee has neither knowledge of mathematics nor superior abilities. Did he learn to create this perfect honeycomb by chance? Could every honeybee, for millions of years, be born by chance? Of course, it is not possible for a bee to have this ability by chance.

## Wax and Honeycomb Making

Beeswax is a soft yellow or darker substance that bees secrete from the wax glands found in the abdominal rings (segments) to make their combs. Generally, beeswax means the wax in the honeycomb prepared by the bee. In addition, the substance prepared in the industry is called wax as well.

The wax production process starts with the attraction of bees by brightly colored flowers found in a garden or forest. Bees seek these flowers to produce pollen and nectar. Bees convert most of the nectar which has low sugar levels to the honey which have high sugar content and use some of this nectar to produce wax. The bee colony cannot survive without beeswax. The reason for this is that the bees are making honeycombs using this wax. It is essential to keep the honey in honeycombs because new generation bees will be fed with this honey.

Thus, the continuity of the colony is ensured. The bees carefully store the honey they do not consume into the honeycombs. These combs are hexagonal cylinders and are constructed on the first side by side and then one after the other. Bees keep both honey and pollen in these honeycombs. At the same time, these honeycombs are where the bees raise their young.

When the worker bees return to the hive with the nectar they collect, they give it to the bees responsible for making wax. These bees eat both honey and nectar and process these materials in their bodies to produce wax. In general, bees consume 6 to 8 units of honey to produce one unit of wax. Naturally, the beeswax that bees need to maintain the colony is produced by bees consuming large amounts of nectar.

Immediately after eating the nectar and honey, the bees remove the wax they produce in their belly in small scales. The bees then chew this wax to make it soft and shapable. When the wax thickens, the bees start to make honeycombs. After filling the hexagonal combs with honey, they cover the top again with wax.

### Wax Production

In the segments of the bee called abdomen, a right-left pair of wax secretions (wax mirror) thickens during the beeswax-making period of worker bees and gains the ability to secrete wax. The waxes secreted on the mirrors as liquids solidify into the wax pockets and become small flakes. The bees cling to each other in chains and leak the wax with special movements. The bee chews the wax flakes he takes to his mouth with the help of his feet, softens them and kneads them and thus uses them in the making of honeycomb eyes. Wax glands degenerate and turn into a row of cells in worker bees who have completed the waxing period.

### Chemical Structure of Wax

The beeswax is a mixture of high molecular weight saturated or unsaturated acids, alcohol, and ester. These are alkali esters (72%), free fatty acids (14%), hydrocarbons (11%), free alcohols (1%) and unknown substances (2%). The density of wax is close to one (0.966). Melting point 62-65 ° C.

### Use of Wax

In beekeeping, the wax obtained from the old combs can be reused. To obtain the wax, first, the combs are separated from the honey. Then it is thrown into boiling water. The wax collected on the surface is removed. The melting temperature of the wax is 62-65 ° C. Therefore, it is collected on the water when melted. Since beeswax may carry certain diseases that may be harmful to bees, the wax to be used in the foundation of honeycomb should be sterilized at 110 ° C for 12 hours according to the standards.

When substances such as chalk, oil, and cola are added to the wax, they are very difficult to detect. Yellow wax is used in parquet polishing, sculpture, some paint and candle making. Purified, cleaned white wax is used in the pharmaceutical industry, creams. Since candles burned in churches must have at least 32% wax, they are used in the candle making industry in significant amounts.

## Mathematical Aspects of Honeycombs

Bees are the real architectures of nature. Therefore, the architecture of honeycomb has 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 circumferential length is uniform n-gene. 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.

An inner angle of a circle and a smooth polygon with n edges drawn into it 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 internal 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 internal angles, in this case we can write the following equation (N is an integer): N. (180 – 360 / n) = 360 If N is solved here, we can only get n = 3, 4 and 6 as integer values, and no integer for any number greater than 6 can be obtained. So if we want to divide an area without spaces, we must use either triangles, quadrangles or hexagons. With a smooth polygon with more than 6 sides, no gap is possible. Similarly, straight pentagons are not a suitable solution. 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 by 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 a 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 be proved with 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, close with three rhombuses. The inner angles of the rhombuses are 70,5O and 109,5O, giving the ideal mathematical solution for the shape of the three rhombic 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 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 arose and the structure suddenly turned into three rhombus structures like bees.