## Re: The light comes on !

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• Hi John, ... Yes. Not practical. Dropping. Roger
Message 1 of 8 , Feb 1, 2011
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Hi John,

> I must say, only if all elements shared one base area kept the same height,
> during the flexing....

Yes. Not practical. Dropping.

Roger
• I 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
Message 2 of 8 , Feb 2, 2011
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I 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.
• Sorry, I got that backwards. 2(n-2)=area, 3(n-2)=edges ... This parts is okay.
Message 3 of 8 , Feb 2, 2011
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Sorry, 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.
>
• Hi Dick and John ... Cavalieri s Principle applies to this http://en.wikipedia.org/wiki/Cavalieri%27s_principle A horizontal slice through the two tetrahedra
Message 4 of 8 , Feb 3, 2011
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Hi 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.

--
• ... height ... * Pyramid (geometry) Volume * Cone (geometry) Volume
Message 5 of 8 , Aug 16, 2012
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--- 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

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