# 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