geometry of a polyhedron

[Dick Fischbeck]

We know that 10f^2+2=n, the number of vertexes.

We know that 20f^3=volume in unit tets.

We know 2(n-2)=area in unit triangles.

What is area in terms of f?

[RC]

Making progress!

We know that 10f^2+2=n, the number of vertexes.

     (Icosahedrons only).

     Frequency denoted as the number of subdivisions of the edges of the triangular faces:

     Vertices = 10 * freq^2 + 2

We know that 20f^3=volume in unit tets.

     (Again, Icosahedrons only).

     Tets being tetrahedrons with one shared vertex at the center of the icosahedron.

     The triangular face opposite the commonly shared vertex being one of the triangular

     faces of the subdivided icosahedron of frequency (f).

     

We know 2(n-2)=area in unit triangles.

     This formula is for convex polyhedrons in general.  It just states that any polyhedron

     with (n) vertices can be be subdivided into (x) number of triangular faces.

What is area in terms of f?

     Is that for just icosahedrons, or for all convex polyhedrons in general?

[Dick Fischbeck]

Yes, there are different formulas for octahedral and tetrahedral structures.

And area for icosas is 20f^2. Octa is 8f^2 and tets are 4f^2.

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[RC]

And, this leads to . . . ??

Anxiously waiting for the big reveal.

[Chris Kitrick]

Can someone explain the utilization of Euler's formula that defines a valid 3-dimensional convex polyhedra? The base formula (V + F - E = 2) is a topological construct that explains connectivity of members. It is not a formula intended for an area evaluation.

Cheers

[RC]

The Euler polyhedra formula relates the number of Vertices, Faces, and Edges to each other in any given polyhedron.  It has no relation to area or volume. 

[Dick Fischbeck]

Triangles are areas and tetrahedrons are volumes. I'm just counting them.

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[RC]

To know the area of the triangles, you need to know the lengths and/or angles of each triangle.  This is information is not provided by just knowing the number of vertices.

I suppose that if you do figure out the area of each triangle, and you know the distance of all vertices from the center origin of the polyhedron, then it would be possible calculate the volume of each tetrahedron.

So, there are a lot of ifs and missing information needed to calculate the surface area and volume of a arbitrary polyhedron.

[Dick Fischbeck]

Fair enough.

Are you thinking in xyz or ivm?

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[Dick Fischbeck]

I think if all the triangles of this polyhedron are the same area, then 2(n-2) equals the area of the polyhedron, regardless of the original intentions of Euler.

Cheers

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[Dick Fischbeck]

As CJ put it, What are the constraints?