# 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