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?