The geometry of honeycomb fractal

The nests that bees knit for reproduction and where they store their honey are called honeycomb. The bees produce the geometry of the honeycomb by designing and working as a mathematician, as an engineer or as an architect. The regular hexagon was selected for the honeycomb as a basic geometric shape. The following truth of a mathematical property "when any plane is divided into the parts with equal area, the part with the smallest perimeter is a regular hexagon" is known from long time ago. Therefore, the bees gave a mathematical decision instinctively and chose a regular hexagon. To do this, the least material, the minimum area with the minimum dimension and the maximum volume were taken as a basis. The honeycombs knitted in this way are extremely orderly and sturdy structures. Although every bee starts from different places and directions, they progress by knitting the regular hexagons, which are the copies of each one, with their wax secretions towards a single point without any trial or error. During this progress, they continue to knit the hexagons. At the end they join in the middle. The meeting points are not noticeable. Also a copy of the hexagons fits the place from where it was separated in such an incredible way which makes it hard to believe. Moreover, actions like trials or errors are not acceptable in this process. Each bee honeycomb is a fractal. Every step in these fractals has similar appearance and structure in itself. If we take a hexagon as a first step and put on every side of this hexagon other hexagons as a second step, then the number of hexagons becomes N=7. If we continue this iteration, the number of hexagons in the n step will be N=1 + 3•n•(n+1) Since the bee honeycombs are fractal then this fractal must have a dimension. If we calculate this dimension by using the Cantor's Triple Method, then the operations from first step to n step should be in the following way: If r1 = 1/6 then N(r1) = 12 and dimension d_1=(logN(r_1 ) )/(log(1/r_1 ))=log12/log6 =1,386 If r2 = 1/18 then N(r2) = 30 and dimension d_2=(logN(r_2 ) )/(log(1/r_2 ))=log30/log18 =1,767 If r3 = 1/30 then N(r3) = 48 and dimension d_3=(logN(r_3 ) )/(log(1/r_3 ))=log48/log30 =1,138 If r4 = 1/42 then N(r4) = 66 and dimension d_4=(logN(r_4 ) )/(log(1/r_4 ))=log66/log42 =1,1209 . . . If r_n=1/((2n-1)•6 ) then N(r_n )=(3n-1)•6 Thus the dimension d_p=lim┬(n→∞)〖logN(r_n )/(log 1/r_n )〗=lim┬(n→∞)〖log((3n-1)•6)/(log((2n-1)•6) )〗=1 . Key Words: Geometry; Bee; Honeycomb; Honey; Fractal; Dimension; Hexagon; Angle; Motif; Iteration