# Message #1523

From: schuma <mananself@gmail.com>

Subject: Re: new hemi-cube and hemi-dodecahedron puzzles

Date: Mon, 07 Mar 2011 23:47:50 -0000

Roice,

I think you are definitely right. I think of the alternative hemi-dodecahedron because my interpretation is essentially 12-face but not 6-face. The hemi-cube works in either way because the consequence of a CW turn is always as same as that of a CCW turn.

Nan

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:

>

> >

> >

> >> 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

>