- View SourceI think:

Vertexes - 2 = 2*triangles

Vertexes - 2 = 3*edges

or just,

n-2=2*area

and

volume= [rt ((2/5)(n-2)^3]/10

for icosahedral shells, or at least, that's how many tetrahedrons it has.

This is true for all Euler #2 shell structures.

So, if triangle is counted as area and tetrahedrons are counted as volume, we have a new and maybe powerful generalized counting principle at our finger tips.

No school. Big snowstorm. Hanging by the wood stove. Beautiful day. - View SourceSorry, I got that backwards. 2(n-2)=area, 3(n-2)=edges

--- In antiprism@yahoogroups.com, "richardfischbeck" <dick.fischbeck@...> wrote:

>

> I think:

>

> Vertexes - 2 = 2*triangles

>

> Vertexes - 2 = 3*edges

>

> or just,

>

> n-2=2*area

This parts is okay.

>

> and

>

> volume= [rt ((2/5)(n-2)^3]/10

>

> for icosahedral shells, or at least, that's how many tetrahedrons it has.

>

> This is true for all Euler #2 shell structures.

>

> So, if triangle is counted as area and tetrahedrons are counted as volume, we have a new and maybe powerful generalized counting principle at our finger tips.

> - View SourceHi Dick and John

On Tue, 1 Feb 2011, richardfischbeck wrote:

> Fwd from another group:

>

> "I find it interesting that all tetrahedra with the same base area and height

> have the same volume no matter what their shape is." - JB

Cavalieri's Principle applies to this

http://en.wikipedia.org/wiki/Cavalieri%27s_principle

A horizontal slice through the two tetrahedra being compared

always interesects them in two triangles of the same area,

as can be seen by considering that these sections are the base

triangles scaled down by the same factor.

Adrian.

--

Adrian Rossiter

adrian@...

http://antiprism.com/adrian - View Source
--- In antiprism@yahoogroups.com, "richardfischbeck" <dick.fischbeck@...> wrote:

> Ta-Da.

> Fwd from another group:

> "I find it interesting that all tetrahedra with the same base area and height

> have the same volume no matter what their shape is." - JB