Message #1523

From: schuma <>
Subject: Re: new hemi-cube and hemi-dodecahedron puzzles
Date: Mon, 07 Mar 2011 23:47:50 -0000


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.


— In, 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<>, 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