How to Find the Coordinates of an Icosahedron's Faces?

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In summary, the speaker is seeking help in obtaining the coordinates for the faces of an Icosahedron without having to manually enter the data. The data structure for the information is complex, with vertices being shared by adjacent faces. The speaker is also looking for an efficient way to sort the faces, with the idea of selecting a "north pole" node and working around the icosahedron. The latitude of non-pole nodes is mentioned as well. The speaker suggests looking up "Icosahedron - Cartesian coordinates" on Wikipedia for more information.
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BJRowlett
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Ive posted this on gamedev.net, and figured Id post here aswell.

What I am trying to figure out, is how to get the coordinates (X, Y, Z) of the faces of an Icosahedron, without having to manually enter all the data.

This wouldn't be as hard, but the data structure for the information is as follows:
GeodesicSphere::Faces::Vertex::Coordinates

An example of this is:
(GeoSphere.FaceIndex[0].A.X, GeoSphere.FaceIndex[0].A.Y, GeoSphere.FaceIndex[0].A.Z)
(GeoSphere.FaceIndex[0].B.X, GeoSphere.FaceIndex[0].B.Y, GeoSphere.FaceIndex[0].B.Z)
(GeoSphere.FaceIndex[0].C.X, GeoSphere.FaceIndex[0].C.Y, GeoSphere.FaceIndex[0].C.Z)
Which holds the X,Y,Z values of the all the vertices on the first Face.

With this, I will have to assign the values to multiple indices coordinates, because adjacent faces share vertices. And I haven't found an easy way to do so.

Also, finding a efficient way of sorting the faces, my idea was to select one node to be a 'north pole' and select any face, then work around the icosahedron, moving down a level, and repeating. This way, I can always use two of the vertices from the previous face, and just calculate the new one.

If you have any ideas that would work better, it would be greatly appreciated.


~BJRowlett
 
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  • #2
I do recall the latitude of the non-pole nodes are arctan( 1/2 )
 
  • #3

FAQ: How to Find the Coordinates of an Icosahedron's Faces?

What is an icosahedron?

An icosahedron is a three-dimensional geometric shape with 20 triangular faces, 30 edges, and 12 vertices. It is one of the five Platonic solids, which are regular and convex polyhedrons.

Why is it important to find icosahedron vertices?

Finding icosahedron vertices is important in various fields such as geometry, crystallography, and computer graphics. It helps in understanding the structure and properties of icosahedrons, which have applications in chemistry, physics, and engineering.

How can one find the vertices of an icosahedron?

There are various methods to find the vertices of an icosahedron, but the most common one is using its formula. The coordinates of the vertices can be calculated using the formula:

x = ±1, y = ±(1+√5)/2, z = ±(2+√5)/2

Another method is by constructing an icosahedron using paper or cardboard and labeling the vertices accordingly.

What is the significance of the icosahedron in nature?

The icosahedron has a significant presence in nature, appearing in various forms such as the icosahedral viruses, carbon molecules, and crystals. It also has a close relation to the golden ratio, which is seen in many natural phenomena, making it a symbol of beauty and harmony in nature.

Can an icosahedron have more than 12 vertices?

No, an icosahedron can only have 12 vertices as it is a Platonic solid with a fixed number of faces, edges, and vertices. Any additional vertices would result in a different shape, not an icosahedron.

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