regular polygons
a common backplate
The construction of a honey comb as two rows of garage boxes with a shared backwall is logic. By sharing the back plate of a cell you need less wax. At one side the cells are open. The opposite site is closed by a shared plate.
shared walls
It's beneficial to share the side walls of the cells to. So the question is: "With which regular polygons it's possible to create a tesselation?"
You can create a tesselation with equilateral triangles, squares and six angled regular polygons. With regular five angled and seven angled polygons you can't, nor with any regular polygon with more than six angles.
Why six angled polygons?
In the next table the area is formulated as a function of the perimeter for an equilateral triangle, a square, a regular six angled polygon and a circle.
The greater the number of sides the greater the area compared to the perimeter which requires less wax for the same area. The circle (as a limit of an n-angled polygon with increasing number of n) gets the best score, but you cannot create a tesselation out of circles. So the best choice is the six angled polygon.
| form | perimeter P | area A | A as a function of P | |
| triangle | 3s | | | |
| square | 4s | s² | | |
| 6-angled polygon | 6s | | | |
| circle | 2r | r² | | |