# Message #1519

From: schuma <mananself@gmail.com>

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

Date: Mon, 07 Mar 2011 20:20:42 -0000

Hi,

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!

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.

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).

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?

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

Nan

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

>

> Greetings,

>

> I added two new sets of puzzles to MagicTile, the first non-orientable

> puzzles which appear so visually in the program.

>

> The hemi-cube puzzles are combinatorially the same as the 3-colored puzzles,

> which Andrey has pointed out. At first I thought the even-length ones might

> be different, but those end up behaving like odd-length puzzles anyway.

>

> The 6-color hemi-dodecahedron puzzles are something new to investigate. If

> you discover unusual behaviors during a solve, please do share :) I have

> the feeling the length-3 puzzle will turn out to be a favorite for me.

>

> These both have the topology of the real projective

> plane<http://en.wikipedia.org/wiki/Real_projective_plane>,

> so now we just need a Boy’s

> surface<http://en.wikipedia.org/wiki/Boy%27s_surface>display of them

> :D

>

> I think with the right mechanism, these puzzles might be candidates for

> actual physical variants. Perhaps I should write experts like Oskar or Bram

> about this.

>

> At the same time, I went ahead and fixed the loading/saving to persist out

> the "slicing expansion" setting, so that people can record their progress

> with deep-cut puzzles. Disclaimer: I know making the circles too big can

> produce unintended weirdness, so it will be up to you to decide if the

> setting you’ve chosen has produced a valid puzzle. I still want to make

> this setting less experimental in the next version.

>

> Enjoy!

> Roice

>

> Some Fine Print… To get hemi-puzzles to work right, I needed to fix the

> twisting behavior on Rubik’s Cube/Megaminx for the faces inverted by the

> projection (which Matt Galla suggested I do early on). The new behavior is

> better, but it has the downside of breaking any log files that were

> generated for these puzzles. I rev’d the version number in the log files to

> mark this, but things aren’t backward compatible for loading. I’ll keep

> around the last version, so people will be able load those if necessary.

>