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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... –  J. M. May 6 '12 at 13:50
    
What is "the icosahedron" ? –  Phira May 6 '12 at 13:51
    
@J.M thanks, and can you say me please around what axis? Or better also how can I find it? –  user1097772 May 6 '12 at 14:10
    
@Phira en.wikipedia.org/wiki/Icosahedron –  user1097772 May 6 '12 at 14:11
    
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. –  J. M. May 6 '12 at 14:14
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1 Answer 1

There are a number of transformations you can perform to make the two sets of coordinates match. For instance, if you exchange the $x$ and $y$ axes, the first quadruple of points of the first set becomes the first quadruple of the second set, the second quadruple becomes the third quadruple and the third quadruple becomes the second. Since all the non-zero coordinates occur with both signs, you can do this either by just swapping the axes (by a reflection in a suitable plane) or by rotating them into each other (by a rotation through $\pi/2$ in either direction about the $z$ axis).

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