The 12 vertices of an icosohedron, kindly provided by Mathematica.
            x                      y                        z
 0.0000000000000000000	 0.0000000000000000000	 1.1755705045849462583
 1.0514622242382672121	 0.0000000000000000000	 0.52573111211913360603
 0.32491969623290632616	 1.0000000000000000000	 0.52573111211913360603
-0.85065080835203993218	 0.61803398874989484820	 0.52573111211913360603
-0.85065080835203993218	-0.61803398874989484820	 0.52573111211913360603
 0.32491969623290632616	-1.0000000000000000000	 0.52573111211913360603
 0.85065080835203993218	 0.61803398874989484820	-0.52573111211913360603
-0.32491969623290632616	 1.0000000000000000000	-0.52573111211913360603
-1.0514622242382672121	 0.0000000000000000000	-0.52573111211913360603 
-0.32491969623290632616	-1.0000000000000000000	-0.52573111211913360603
 0.85065080835203993218	-0.61803398874989484820	-0.52573111211913360603
 0.0000000000000000000	 0.0000000000000000000	-1.1755705045849462583