Message #3456

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] 3^4 solved
Date: Sat, 16 Jul 2016 12:44:47 -0500

Hi Dave,

Maybe there are Archimedean twisty puzzles that are the "duals" of uniform
> twisty puzzles in the sense that one puzzle can be thought of as a sliced
> version of either polyhedron?
>

I’m not exactly sure what this question will lead to, but it is
interesting. I know that you can throw an identical slicing over dual
spherical polyhedra in MagicTile, to get very similar puzzles. For
example, compare this Rubik’s Cube <https://goo.gl/photos/3SHwsjHajjcZb4yB7>
to this vertex-turning Octahedron <https://goo.gl/photos/corQnW7oUcstZ9R8A>.
The slicing is the same on the sphere, and the puzzles are similar, though
if you start solving the latter you’ll see they do have differences (it’s
like solving a picture cube because the "centers" have orientation).

Is this kind of identical slicing on dual polyhedra what you were thinking
about? If so, the duals to the Archimedeans are Catalan solids
<https://en.wikipedia.org/wiki/Catalan_solid>, so your question has me
picturing vertex-turning puzzle for the dual to the truncated icosahedron
<https://en.wikipedia.org/wiki/Pentakis_dodecahedron>. That’s cool and
something I’ve never considered because that polyhedron is not uniform.
Let me know if you are thinking about something different though - maybe
there is something else to explore here!

Cheers,
Roice