Message #1522

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: new hemi-cube and hemi-dodecahedron puzzles
Date: Mon, 07 Mar 2011 16:34:42 -0600

>
>
>> Another natural way to construct a hemi-cube or hemi-dodecahedron is that,
>> opposite faces turn clockwise simultaneously or counter-clockwise
>> simultaneously. The physical meaning is that opposite faces are not bandaged
>> but connected using differential gears. Such a hemi-cube is basically a gear
>> cube/caution cube<http://www.youtube.com/watch?v=UDVb9NExsA8>, where L
>> and R always go together if you take the middle slice as the reference. I
>> think the hemi-cube of this kind is more non-trivial than the current
>> hemi-cube (I’m only talking about length-3).
>>
>>
> This does sound interesting, but note that in this case we are no longer
> talking about hemi-puzzles. A puzzle where opposite faces would be coupled
> like you are describing would necessarily be 6-faced for the cube and
> 12-faced for the dodecahedron. You are no longer able to identify opposite
> faces as one and the same after a twist. You could color opposite faces the
> same, but they would still behave differently. A hemi-cube is an abstract
> polytope with 3 faces, and a hemi-dodecahedron one with 6 faces.
>

I was wrong on the hemi-cube. It does still work there if you twist as you
described (which is why we were already getting the hemi-cube behavior with
the other 3-colored orientable puzzles). But for the hemi-dodecahedron, I
think what I wrote is correct, and that the new puzzle would have have 12
faces.

On Mon, Mar 7, 2011 at 3:09 PM, Andrey wrote:

>
> If you twist opposite sides in same direction (relative to sphee surface),
> you’ll get orientable puzzle. I don’t know, if there are sphere paintings
> that are invariant to this transformation (i.e. order of adjacent colors of
> all intances of one face is the same). There is 5-color painting if
> icosahedron (all faces of outscribed tetrahedra have the same color), but
> I’m not sure that it will work.

So I think the above answers the question of whether their are sphere
paintings which work in an orientable fashion. Yes. But the reason it
works in the hemi-cube case is that each face has both left-handed and
right-handed piece versions. Maybe that is always necessary - I’m not sure.

seeya,
Roice