Message #1521

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: new hemi-cube and hemi-dodecahedron puzzles
Date: Mon, 07 Mar 2011 16:15:57 -0600

Hi Nan,

inlines below :)

On Mon, Mar 7, 2011 at 2:20 PM, schuma wrote:

> I just got time to solve the hemi-dodecahedron (length-3). It’s a beautiful
> puzzle. During the solve I ran into the situation that I need to twist a
> corner by +120 deg and its antipode -120 deg. It’s a surprise for me and it
> took me a while to figure out why it happened. It’s a nice twist!
>
>
That’s really cool to hear. What this means for my intended interpretation
of this puzzle (as a hemi-dodecahedron, more on that below) is that you can
have a single, isolated twirled corner. Nice! And nice pun too ;)

> For me this puzzle is a bandaged Megaminx, that is, a Megaminx painted with
> six colors, and with the modification that opposite faces are bandaged so
> that they move together. I don’t know an internal physical structure to make
> such a bandage, but logically it’s like that. Because of this bandage, when
> you look downward at the top of Megaminx and turn the top face CW, the
> bottom face is also turned CW viewed from top, that is, CCW viewed from
> bottom. Therefore when the Megaminx is projected to a plane, the two faces
> of the same color always turn in opposite directions.
>
> Since I view this puzzle as a bandaged version of the familiar Megaminx, I
> don’t think in the fancy way that it’s a non-orientable puzzle on a
> projective plane.
>

Either are valid interpretations of what is going on. In the view where
opposite faces are identified with each other (that is, they abstractly
represent the same actual face), the puzzle only has 6 faces, rather than 12
coupled ones in the case of a bandaged Megaminx.

> 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.

> Another question is, is it possible to construct puzzles on the Klein’s
> bottle? Because of their periodic boundary condition, the hexagonal puzzles
> can be viewed as puzzles on a torus, can’t they?
>
>
I see Andrey responded as well, but you are absolutely correct. And in
fact, investigating what a "Rubik’s Torus" analogue "should be" was the
initial motivation for MagicTile. I have been planning Klein bottle
variants.

> Finally, I found the hemi-cube (3 color) length-4 buggy. The second/third
> layer turning of the central face doesn’t work properly.
>

Thanks for the problem report on the length-4 hemi-cube (looks like the
other even-length ones have the same issue). I’ll try to get that fixed
tonight. I perhaps shouldn’t even include these though, since as
hemi-puzzles, the even length variants must move in a way that makes them
combinatorially equivalent to the odd length ones (length-4 is just length-3
in disguise). This is because what looks like equatorial slices are
actually glued to themselves, and no real twisting can happen there. The
length-2 hemi-cube is degenerate for this reason (the only thing you can do
is rotate the entire thing around). I was the first to solve that one!
Hope I am making sense.
Roice