File:FS S dia.png

The hexagon as base element.

The side length of the hexagon ${\displaystyle a_{0}}$
Since the regular hexagon can be disassembled into six equilateral triangles around ${\displaystyle S_{0}}$ we have:
${\displaystyle |AS_{0}|=|BS_{0}|=|CS_{0}|=|DS_{0}|=|ES_{0}|=|FS_{0}|=a_{0}}$
So the diameter between two corners is ${\displaystyle |AD|=|BE|=CF|=2\cdot a_{0}}$.

Since the distance between ${\displaystyle S_{0}}$ corresponds to the height of the triangle we have:
${\displaystyle |GS_{0}|={\frac {\sqrt {3}}{2}}\cdot a_{0}}$

Perimeter of base hexagon: ${\displaystyle P_{0}=6\cdot a_{0}}$

Area of the base hexagon ${\displaystyle A_{0}=6\cdot {\frac {\sqrt {3}}{2}}\cdot a_{0}\cdot {\frac {a_{0}}{2}}=6\cdot {\frac {\sqrt {3}}{4}}\cdot a_{0}^{2}={\frac {3{\sqrt {3}}}{2}}\cdot a_{0}^{2}}$

Centroid positions of the base hexagon: ${\displaystyle x_{0}=0\quad y_{0}=0}$

Positions of the point A: ${\displaystyle x_{A}=-{\frac {a_{0}}{2}}\quad y_{A}=-{\frac {\sqrt {3}}{2}}\cdot a_{0}}$

In the normalised case the area of the base hexagon is set to 1.
So ${\displaystyle A_{0}={\frac {3{\sqrt {3}}}{2}}\cdot a_{0}^{2}=1\quad \Rightarrow a_{0}^{2}={\frac {2}{3{\sqrt {3}}}}\quad \Rightarrow a_{0}={\sqrt {\frac {2}{3{\sqrt {3}}}}}}$

Segment of the base hexagon ${\displaystyle a_{0}={\sqrt {\frac {2}{3{\sqrt {3}}}}}\quad \approx 0.62}$

Perimeter of base hexagon ${\displaystyle P_{0}=6\cdot a_{0}=6\cdot {\sqrt {\frac {2}{3{\sqrt {3}}}}}={\sqrt {\frac {36\cdot 2}{3{\sqrt {3}}}}}={\sqrt {\frac {24}{\sqrt {3}}}}\quad \approx 3.7224194}$

Area of the base hexagon is by definition ${\displaystyle A_{0}=1}$

Centroid positions of the hexagon: ${\displaystyle x_{0}=0\quad y_{0}=0}$

Apart of the base element there is no shape allocated. Therefore the integer part of the identifying number is 0.
The decimal part of the identifying number is the decimal part of the sum of the perimeters and the distances of the centroids in the normalised case.

${\displaystyle decimalpart(3.7224194+0.0)=decimalpart(3.7224194)=.7224194}$

So the identifying number is: ${\displaystyle 0.7224194}$

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