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March 18, 2009
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Two Platonic solids are the cube and octahedron. These are dual to one another. For each corner of one, there is a corresponding face in the other. Keep cutting the corners from one and you get the other.
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:iconbear48:
bear48 Featured By Owner Feb 23, 2012  Professional
far freaking out
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:iconomartk:
OmarTK Featured By Owner Apr 9, 2011
Very interesting!
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:iconrndmodels:
RNDmodels Featured By Owner May 19, 2010  Professional Artisan Crafter
Touche'
I believe you have nailed it!
:)
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:iconmetahedron:
Metahedron Featured By Owner Mar 18, 2009
trunctasmal
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:iconhop41:
Hop41 Featured By Owner Mar 20, 2009
"trunctasmal": a combination of "truncate" and "infinitesmal"?

The cube, truncated cube, cuboctahedron, and truncated octahedron are indeed points on a continuum of truncations, exactly what I was trying to show. An appropriate comment.
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:iconmetahedron:
Metahedron Featured By Owner Mar 20, 2009
This is also a sound interpretation however, I was combining "phantasmal" rather than "infinitesimal."
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:iconmarkdow:
markdow Featured By Owner Mar 18, 2009
Very nice. It took me a long time to convince myself that it made sense for crystals to have octahedral patterns when their molecular arrangement was cubic -- how octahedrons can be built of cubes.

There's a natural 2-coloring of all facets, those that belong to cubic and to octahedral orientations. Your top figure would have an even blend.

This reminds me that octahedrons also have a hexagonal projection, orthogonal to each face. Easy to forget.
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:iconhop41:
Hop41 Featured By Owner Mar 20, 2009
Like the hexagonal projection of the cube, the hexagonal projection of the octahedron has a bistable convex/concave perception. But this is harder for us to see since we're in the habit of living in rectangular faced polyhedra.

Earlier you had mentioned the cuboctahedron, rather than the truncated octahedron, could be a bridge between my octet toy and Legos. My first thought was "truncated octahedrons stack to fill space but cuboctahedrons don't".

Then it occurred to me cuboctahedrons alternating with octahedrons stack to fill space. And the octahedron is one of the two species of bricks in my octet toy.

The cuboctahedron alternating with octahedrons would be a very nice bridge between cubic and octet lattice structures.

Now I know of two possible ways to splice my toy onto the Lego universe.
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:iconmarkdow:
markdow Featured By Owner Mar 20, 2009
Ah yes. I've played a lot with this projection of cubes and their bistability. Now I'll start thinking about the same with octahedrons. Mixtures of the two with other depth cues will be interesting.
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