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| Noble Faceting of Snub Cube |
A faceting of the snub cube, meaning they share
the same vertices. Discovered by yours truly, Robert Webb, in 2008, this
polyhedron is special because it is noble,
meaning all faces are symmetrically identical (isohedral), and all vertices are
also symmetrically identical (isogonal). Max Brückner studied such
polyhedra (and compounds) in 1906, and it appears this may be the first
addition to the set since then (over 100 years ago!), aside from some
degenerate cases where faces visit the same vertex more than once while
traversing their edges.
The polyhedron also has the unusual property of being self-dual.
I discovered this noble polyhedron using new features of
Great Stella/Stella4D. These were not made
public at the time, but are now available as of version 5.0. Stella can now
find facetings for you, allowing you to step through them one by one by hitting
the Up Arrow key in the Faceting Preview window. The new "Faceting" menu
includes the "Faceting→Maximum Facet Types" option. Set this to "1" to
ensure you only find isohedral polyhedra and compounds. Now hitting the Up
Arrow with the snub cube loaded takes you straight to this new noble
polyhedron!
My model of this polyhedron uses 4 colors, the minimum required to ensure
that no faces of the same color intersect or share an edge.
I first announced this new polyhedron on my forums
here.
You'll also find a photo of Ulrich Mikloweit's amazing model of it in that
thread.
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The faces are irregular pentagons. This screenshot shows which facet
of the snub cube to create in order to generate the model.
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There are 3 different pieces involved, and you'll need 24 of each.
My model has 4 different colors, so 6 copys of each part are required
in each of the 4 colors.
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Start fitting the pieces together like this. Be careful with the
coloring.
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At this stage the model seemed a bit flexible, so I put some extra
support inside. I used two kinds. One was simply a straight piece of
card across the tabs inside the point where edges intersect each other
under the 4-fold symmetry axes. The other can be created as a facet as
shown here, using Stella's Faceting Mode.
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You can see the two kinds of internal support added here. This really
helped to make it a solid model.
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Getting near the end here, but time to think about how that last piece
is going to be put in! I decided it might be best to put the last
piece on as a cap.
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Instead of continuing as before, I printed some larger pieces to attach
as seen here.
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A separate cap, seen here, can then be glued down onto the larger parts
above to finish the model.
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Another photo of the finished model.
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