While morphing between a snubcube and a cuboctahedron 
I spotted a nice arrangement of diagonals forming these 
interlocking triangles.

Jim Buddenhagen

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James Buddenhagen wrote:
> While morphing between a snubcube and a cuboctahedron 
> I spotted a nice arrangement of diagonals forming these 
> interlocking triangles.
> 
> Jim Buddenhagen
That is very nice!  I was playing around with the same shapes (cube and 
octahedron), a while ago, but never noticed the diagonal vertices.  What 
I did notice, was that rotating the faces of the cube 45 degrees and 
out, gives the same shape as rotating the faces of the octahedron 60 
degrees and out -- and the vertices are identically those of your 
interlocking triangles (what is it, a snubcube or a cuboctahedron?)  But 
the triangles -- that's really fascinating.

Dave Matthews

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"Dave Matthews" <n@nn.n> wrote in message news:4212d2f4@news.povray.org...
> James Buddenhagen wrote:
> > While morphing between a snubcube and a cuboctahedron
> > I spotted a nice arrangement of diagonals forming these
> > interlocking triangles.
> >
> > Jim Buddenhagen
> That is very nice!  I was playing around with the same shapes (cube and
> octahedron), a while ago, but never noticed the diagonal vertices.  What
> I did notice, was that rotating the faces of the cube 45 degrees and
> out, gives the same shape as rotating the faces of the octahedron 60
> degrees and out -- and the vertices are identically those of your
> interlocking triangles (what is it, a snubcube or a cuboctahedron?)  But
> the triangles -- that's really fascinating.
>
> Dave Matthews

Interesting pictures, but I'm not sure I see the same vertices.  In any case the
vertices of 'my' three triangles were vertices of the cubeoctahedron, namely:

triangle 1:  <-1, 1, 0>, <0, -1, 1>, <1, 0, -1>
triangle 2:  <-1, -1, 0>, <0, 1, -1>, <1, 0, 1>
triangle 3:  <-1, 0, 1>, <0, -1, -1>, <1, 1, 0>
triangle 4:  <0, 1, 1>, <1, -1, 0>, <-1, 0, -1>

I put a crude wireframe animation of the morphing of cubeoctahedron to snubcube
temporarily here:  http://www.buddenbooks.com/jb/misc/cubeoct_snubcube_anim.gif
(306k)

Jim Buddenhagen

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"James Buddenhagen" <jbu### [at] REMOVEtexasnet> wrote:
> Interesting pictures, but I'm not sure I see the same vertices.  In any case the
> vertices of 'my' three triangles were vertices of the cubeoctahedron, namely:
>
> triangle 1:  <-1, 1, 0>, <0, -1, 1>, <1, 0, -1>
> triangle 2:  <-1, -1, 0>, <0, 1, -1>, <1, 0, 1>
> triangle 3:  <-1, 0, 1>, <0, -1, -1>, <1, 1, 0>
> triangle 4:  <0, 1, 1>, <1, -1, 0>, <-1, 0, -1>
>
> I put a crude wireframe animation of the morphing of cubeoctahedron to snubcube
> temporarily here:  http://www.buddenbooks.com/jb/misc/cubeoct_snubcube_anim.gif
> (306k)
>
> Jim Buddenhagen


The gold balls on my pictures (the vertices of the yellow wireframe shapes)
form the vertices of your cubeoctahedron.  The 4 at the top correspond to
the 4 vertices at the top of your 4 intersecting triangles, etc.  In fact,
the yellow frames in my pictures look identical to the start of your .gif

To see my "morphing" in reverse, take your initial gif cubeoctahedron, look
at the square faces, rotate each of them by 45 degrees and collapse them
in, until they fit snug -- you get a cube.  Now take the triangular faces,
rotate them each by 60 degrees and collapse them in -- you get an
octahedron.

Anyway, I noticed the cubeoctahedron shape of the vertices of your triangle,
and recognized the resulting shape of my outward twists of cube and
octahedron (which would make sense.)

Your morphing .gif is interesting, also.  Thanks.

Dave Matthews

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"Dave Matthews" <dma### [at] wrmnwestmnscuedu> wrote in message
news:web.42139c1c7cff8ff58c7259570@news.povray.org...
> "James Buddenhagen" <jbu### [at] REMOVEtexasnet> wrote:
> > Interesting pictures, but I'm not sure I see the same vertices.  In any case
the
> > vertices of 'my' three triangles were vertices of the cubeoctahedron,
namely:
> >
> > triangle 1:  <-1, 1, 0>, <0, -1, 1>, <1, 0, -1>
> > triangle 2:  <-1, -1, 0>, <0, 1, -1>, <1, 0, 1>
> > triangle 3:  <-1, 0, 1>, <0, -1, -1>, <1, 1, 0>
> > triangle 4:  <0, 1, 1>, <1, -1, 0>, <-1, 0, -1>
> >
> > I put a crude wireframe animation of the morphing of cubeoctahedron to
snubcube
> > temporarily here:
http://www.buddenbooks.com/jb/misc/cubeoct_snubcube_anim.gif
> > (306k)
> >
> > Jim Buddenhagen
>
> The gold balls on my pictures (the vertices of the yellow wireframe shapes)
> form the vertices of your cubeoctahedron.  The 4 at the top correspond to
> the 4 vertices at the top of your 4 intersecting triangles, etc.  In fact,
> the yellow frames in my pictures look identical to the start of your .gif
>
> To see my "morphing" in reverse, take your initial gif cubeoctahedron, look
> at the square faces, rotate each of them by 45 degrees and collapse them
> in, until they fit snug -- you get a cube.  Now take the triangular faces,
> rotate them each by 60 degrees and collapse them in -- you get an
> octahedron.
>
> Anyway, I noticed the cubeoctahedron shape of the vertices of your triangle,
> and recognized the resulting shape of my outward twists of cube and
> octahedron (which would make sense.)
>
> Your morphing .gif is interesting, also.  Thanks.
>
> Dave Matthews

Here is a pic that shows how 'my' triangles fit into a cubeoctahedron.
The 4 vertices at the top, as you say, are the top vertices of
the 4 triangles.

Jim Buddenhagen

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