<s>[INST] Co-ordinates of the vertices an icosahedron relative to its centroid. 
This is a picture of an icosahedron. I need to know the coordinates of the vertices of the icosahedron relative to it's centroid in order to programme a projection of one on a three dimensional plane. (By the way, it has twelve vertices, so it is going to be a very long process)
Unfortunately, I have not been able to find one vertices. 
 [/INST] PolyhedronData["Icosahedron", "VertexCoordinates"]

$$
\left(
\begin{array}{ccc}
 0 & 0 & -\frac{5}{\sqrt{50-10 \sqrt{5}}} \\
 0 & 0 & \frac{5}{\sqrt{50-10 \sqrt{5}}} \\
 -\sqrt{\frac{2}{5-\sqrt{5}}} & 0 & -\frac{1}{\sqrt{10-2 \sqrt{5}}} \\
 \sqrt{\frac{2}{5-\sqrt{5}}} & 0 & \frac{1}{\sqrt{10-2 \sqrt{5}}} \\
 \frac{1+\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & -\frac{1}{2} &
   -\frac{1}{\sqrt{10-2 \sqrt{5}}} \\
 \frac{1+\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & \frac{1}{2} &
   -\frac{1}{\sqrt{10-2 \sqrt{5}}} \\
 -\frac{1+\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & -\frac{1}{2} &
   \frac{1}{\sqrt{10-2 \sqrt{5}}} \\
 -\frac{1+\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & \frac{1}{2} &
   \frac{1}{\sqrt{10-2 \sqrt{5}}} \\
 -\frac{-1+\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & -\frac{1}{2}
   \sqrt{\frac{5+\sqrt{5}}{5-\sqrt{5}}} & -\frac{1}{\sqrt{10-2 \sqrt{5}}}
   \\
 -\frac{-1+\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & \frac{1}{2}
   \sqrt{\frac{5+\sqrt{5}}{5-\sqrt{5}}} & -\frac{1}{\sqrt{10-2 \sqrt{5}}}
   \\
 \frac{-1+\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & -\frac{1}{2}
   \sqrt{\frac{5+\sqrt{5}}{5-\sqrt{5}}} & \frac{1}{\sqrt{10-2 \sqrt{5}}}
   \\
 \frac{-1+\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & \frac{1}{2}
   \sqrt{\frac{5+\sqrt{5}}{5-\sqrt{5}}} & \frac{1}{\sqrt{10-2 \sqrt{5}}}
\end{array}
\right)
$$
</s>