Andrea Ryan wrote:
> 
> I have made #declares of meshes for the tetrahedron, cube and
> octahedron.  I saw some vertexes of the dodecahedron and the icosahedron
> in another polyhedra include file.  They used the golden ratio.  Is
> there a reason for this?  I want to have the edges have lengths of one.

Yes!  If you have an icosahedron with edge lengths of one, you can
inscribe three mutually orthogonal golden rectangles.  With the
dodecahedron, you can inscribe three mutually orthogonal rectangles in
the ratio phi^2 (which incidentally is phi + 1).

-- 
David Fontaine  <dav### [at] faricynet>  ICQ 55354965
My raytracing gallery:  http://davidf.faricy.net/

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David Fontaine wrote:
> 
> Andrea Ryan wrote:
> >
> > I have made #declares of meshes for the tetrahedron, cube and
> > octahedron.  I saw some vertexes of the dodecahedron and the icosahedron
> > in another polyhedra include file.  They used the golden ratio.  Is
> > there a reason for this?  I want to have the edges have lengths of one.
> 
> Yes!  If you have an icosahedron with edge lengths of one, you can
> inscribe three mutually orthogonal golden rectangles.  With the
> dodecahedron, you can inscribe three mutually orthogonal rectangles in
> the ratio phi^2 (which incidentally is phi + 1).

Oh, of course, the edge length doesn't matter.  But the shorter sides of
the rectangles are opposite edges, so if the edge length is one, the
length of the other sides are just phi or phi^2.

-- 
David Fontaine  <dav### [at] faricynet>  ICQ 55354965
My raytracing gallery:  http://davidf.faricy.net/

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> Yes!  If you have an icosahedron with edge lengths of one, you can
> inscribe three mutually orthogonal golden rectangles.  With the
> dodecahedron, you can inscribe three mutually orthogonal rectangles in
> the ratio phi^2 (which incidentally is phi + 1).
> 

I looked at my paper models of the icosahedron and the dodecahedron and
saw the rectangles.  This should help with finding the coordinates of
the vertexes.  It's almost magic.  Thank you!  :-)

The golden ratio is one of those mysterious irrational numbers.  As you
said, its square is equal to the sum of one and itself.  Its reciproial
is equal to its value minus one.  I wonder if there's an undiscovered
equation that relates it to other constants like e^(i*pi) + 1 = 0.

I have seen the number called the golden mean.  It is because a golden
rectangle has a mean ratio between its sides and not extreme ones?
Brendan

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Andrea Ryan wrote:
> 
> I looked at my paper models of the icosahedron and the dodecahedron and
> saw the rectangles.  This should help with finding the coordinates of
> the vertexes.  It's almost magic.  Thank you!  :-)

Teehee

 
> The golden ratio is one of those mysterious irrational numbers.  As you
> said, its square is equal to the sum of one and itself.  Its reciproial
> is equal to its value minus one.  I wonder if there's an undiscovered
> equation that relates it to other constants like e^(i*pi) + 1 = 0.

If there is it must be something real obscure!  Hey, I read in the book
Chaos that some guy discovered another important irrational number, like
e and pi, that comes up as a sort of 'octave multiplier' (if you get
what i mean) in bifurcation diagrams and other chaosy stuff.

 
> I have seen the number called the golden mean.  It is because a golden
> rectangle has a mean ratio between its sides and not extreme ones?

I dunno.  Probably has to do with Fibonacchi.


-- 
David Fontaine  <dav### [at] faricynet>  ICQ 55354965
My raytracing gallery:  http://davidf.faricy.net/

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David Fontaine wrote:

> Hey, I read in the book
> Chaos that some guy discovered another important irrational number, like
> e and pi, that comes up as a sort of 'octave multiplier' (if you get
> what i mean) in bifurcation diagrams and other chaosy stuff.

Perhaps you are referring to the Feigenbaum constants?

http://www.mathsoft.com/asolve/constant/constant.html

http://pauillac.inria.fr/algo/bsolve/constant/fgnbaum/general.html

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Kevin Wampler wrote:
> 
> David Fontaine wrote:
> 
> > Hey, I read in the book
> > Chaos that some guy discovered another important irrational number, like
> > e and pi, that comes up as a sort of 'octave multiplier' (if you get
> > what i mean) in bifurcation diagrams and other chaosy stuff.
> 
> Perhaps you are referring to the Feigenbaum constants?
> 
> http://www.mathsoft.com/asolve/constant/constant.html
> 
> http://pauillac.inria.fr/algo/bsolve/constant/fgnbaum/general.html

I think that's it.  4.669 looks awfully familiar, I think that's the one
in the book.  Of course the book has to talk about technological
limitations and how these chaoticians would spend hours generating these
numbers to a whopping four significant figures on their HP calcs, at
least until they could sneak in a few minutes on the Los Alamos Cray...
;)

-- 
David Fontaine  <dav### [at] faricynet>  ICQ 55354965
My raytracing gallery:  http://davidf.faricy.net/

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David Fontaine wrote:

> these chaoticians would spend hours generating these
> numbers to a whopping four significant figures on their HP calcs, at
> least until they could sneak in a few minutes on the Los Alamos Cray...

It's kind of nice that times have changed, but some constants are still hard to
calculate to a high degree of accuracy.  Brun's Constant, for example, has only
been calculated to 11 or so significant figures.  I guess that the faster
computers get, the more people try to do with them.  Kind of like Pov.

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