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The Ideal of Geometry Rules in the Construction of Honeycombs

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The interesting architecture of honeycombs has attracted the attention of people throughout history. This structure consisting of side by side hexagons is extremely sensitive and the average wall thickness is 0.1 mm. The deviation from this mean value is up to 0.002 mm. In order to understand the ideal of geometry rules in the construction of honeycombs, it is necessary to have a mathematical point of view.

The circle is a geometric shape with the shortest side length surrounding a certain fixed area. For example, when the perimeter of the square and the circle of 10 cm2 are compared, it is seen that the circumference of the circle is shorter. However, this is not the case in the construction of honeycomb.Here, the large frame of the honeycomb will be divided into equal and smaller areas, and the shape with at least circumferential length will be used in the division process. If we want to divide the frame into small circles with equal areas, the shortest side feature will be provided as described above, but more candles will be spent for the spaces between the edges of the circles. However, when we refer to the principles of geometry to solve this problem with the shortest side length and least material, it will be seen that polygons should be used in dividing the combs.

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Let’s consider polygons with the same area as the number of sides. Of these, the shortest length of the periphery is the proper n-gene. By straight, all edges and inner angles are equal. A polygon of this type 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 length is the least. For example, the shortest circumference length in triangles with equal area is obtained in equilateral triangle, while the shortest circumference in quadrilaterals is obtained in squares. Similarly, if the pentagons and hexagons are compared among themselves, the shortest circumferential length can be obtained in the proper pentagon and hexagon.

The first question that can be imagined is what regular polygon we should use when dividing a certain area. A part of a smooth polygon with a circle and n-drawn edges is shown on page 2. As can be seen from the figüre, an inner angle of the polygon is 180-360 / n degrees. When we want to divide a given large area into small areas, adjacent polygons must fit into each other and there should be no spaces between them. For this to happen, the sum of the internal angles of adjacent polygonal corners that are justified must be 360 degrees. In other words, an integer of an internal angle must be a solid 360 degrees.

In order to represent the number of neighboring internal angles, we can write the following equation in this case (N is an integer):N. (180 – 360 / n) = 360 If N is solved here, as integer values, we can obtain only n = 3, 4 and 6 and no integer can be obtained for any number greater than 6. So if we want to divide an area without spaces, we should either use triangles, quadrangles or hexagons. A smooth polygon with a number of edges greater than 6 is not possible without space. Similarly, smooth pentagons are not a suitable solution. A three-sided rectangular area with 36O angles is formed. However, hexagons can be brought together side by side.In addition, the triangular, rectangular and hexagonal equal areas are compared with each other. Thus, at least the amount of wax consumption can be obtained by dividing 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 the neighboring polygon will necessarily have a concave shape. The advantage obtained by the convex curve (because it is more similar to the circle part) destroys more disadvantages from the concave curve and no net gain is obtained.

Thomas Hales of the University of Michigan put an end to the debate in 1999, and when we wanted to divide an area into even smaller spaces, it proved the ideal shape to be a hexagonal one. Although the hexagonal shape has been described as an ideal shape for a long time, it has not been proved to be a solid mathematical proof.If the honeycomb construction techniques of the bees had evolved since their first creation, the fossil record had to have other geometric shapes other than hexagons. However, there is no clue that another form of honeycomb is used. Charles Darwin himself described honeycomb as an engineering wonder that perfectly economizes labor and wax.

So far we’ve looked at the problem in two dimensions. However, honeycomb is a three-dimensional object and is in the form of hexagonal prism. Hexagonal prism shaped honeycombs are in two layers, one end is open, the other closed ends are placed back to back. When the frame is placed perpendicular to the ground, the prisms are constructed so as to make an inclination angle of 13 ° horizontally and this angle is the smallest angle that is sufficient for the honey not to flow.

What kind of geometry should be used for the minimum wax consumption at the closed end of the honeycombs?

In 1964, the mathematician Fejes Toth showed that the ideal closure could be achieved with two hexagons and two squares. The bees were slightly different with three rhombuses. The internal angles of the rhombuses are 70.5O and 109.5O, 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 field according to two hexagons and two frames in the application of bees. However, there was a point to be overlooked, and the wall thickness in calculations was taken extremely thin. The researchers used liquid foam to experience the mathematical model of Toth. Between the two glasses, they pumped the detergent solution with bubbles of 2 mm diameter so as to have two layers. The bubbles in contact with the glass turned into hexagonal structures. At the boundary of the two layers, two hexagonal and two square shaped structures were formed by Toth. When the bubble walls were slightly thickened, an interesting situation arose and the structure was transformed into three rhombus structures, just like in bees.


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