GeoGebra Klaslokaal

# 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.
 ﻿form ﻿perimeter P ﻿area A A as a function of P﻿ ﻿ ﻿triangle 3s ﻿ ﻿ ﻿ ﻿square ﻿ 4s ﻿s² ﻿ ﻿ 6-angled polygon ﻿ 6s ﻿ ﻿ ﻿ ﻿circle ﻿ 2r ﻿r² ﻿ ﻿
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.