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Wikipedia says (link)that cartesian coordinates of icosahedron are:

(0, ±1, ± φ) (±1, ± φ, 0)  (± φ, 0, ±1) 

Where φ = (1 + √5) / 2 is golden ratio ≈ 1.618.

I found on the internet this code:

// vertex position and color information for icosahedron             vertices[0] = new VertexPositionColor(new Vector3(-0.26286500f, 0.0000000f, 0.42532500f), Color.Red);             vertices[1] = new VertexPositionColor(new Vector3(0.26286500f, 0.0000000f, 0.42532500f), Color.Orange);             vertices[2] = new VertexPositionColor(new Vector3(-0.26286500f, 0.0000000f, -0.42532500f), Color.Yellow);             vertices[3] = new VertexPositionColor(new Vector3(0.26286500f, 0.0000000f, -0.42532500f), Color.Green);             vertices[4] = new VertexPositionColor(new Vector3(0.0000000f, 0.42532500f, 0.26286500f), Color.Blue);             vertices[5] = new VertexPositionColor(new Vector3(0.0000000f, 0.42532500f, -0.26286500f), Color.Indigo);             vertices[6] = new VertexPositionColor(new Vector3(0.0000000f, -0.42532500f, 0.26286500f), Color.Purple);             vertices[7] = new VertexPositionColor(new Vector3(0.0000000f, -0.42532500f, -0.26286500f), Color.White);             vertices[8] = new VertexPositionColor(new Vector3(0.42532500f, 0.26286500f, 0.0000000f), Color.Cyan);             vertices[9] = new VertexPositionColor(new Vector3(-0.42532500f, 0.26286500f, 0.0000000f), Color.Black);             vertices[10] = new VertexPositionColor(new Vector3(0.42532500f, -0.26286500f, 0.0000000f), Color.DodgerBlue);             vertices[11] = new VertexPositionColor(new Vector3(-0.42532500f, -0.26286500f, 0.0000000f), Color.Crimson); 

Let's forget the code and focus only on coordinates of vertices.

When I look on coordinates from wiki and divide φ/1 its ≈ 1.618.
When I do same with coordinates from the code above 0.42/0.26 ≈ 1.615
So when I compare this two sets of coordinates I can say that 1 corresponds with 0.26 and 0.42 with φ.
If lets say that k = 1/0.26 = 50/13 ≈ 3.84, so if I multiply all coordinates from second set by k, I can write them as:

(±1, 0 ,±φ)  (0, ±φ, ±1) (±φ, ±1, 0)  

So for conclusion:
Wiki coordinates:

(0, ±1, ± φ) (±1, ± φ, 0)  (± φ, 0, ±1) 

Second set coordinates:

(±1, 0 ,±φ)  (0, ±φ, ±1) (±φ, ±1, 0) 

Why does this happens? Why does not corresponds the placement of φ and 0 and 1 in x,y,z position in coordinates?

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    So, one is a rotated version of the other...2012-05-06
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    What is "the icosahedron" ?2012-05-06
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    @J.M thanks, and can you say me please around what axis? Or better also how can I find it?2012-05-06
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    @Phira http://en.wikipedia.org/wiki/Icosahedron2012-05-06
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    I can't look at it now, but have you tried plotting your points in some computing environment and comparing? That might help you figure out what rotations to do.2012-05-06
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    Incidentally, to explain the "magic numbers" in the code, notice that $$0^2+0.262865^2+0.425325^2=0.25,$$ that is, for whatever reason, the person who wrote that code wanted to scale the icosahedron so that its vertices are at a distance of $\frac14$ from the origin.2018-02-28

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