Message #3653

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
Date: Thu, 16 Feb 2017 16:09:37 -0600

>
>
> This is most clear when looking at Oskar’s lovely rendering
> <http://superliminal.com/cube/physical2x2x2x2oskar.jpg> of my previous
> concept. I was considering pieces as some sort of beads that would move
> along wire arcs, perhaps made out of some squishy material such as foam
> rubber. I don’t remember an "aha" moment but the current design probably
> resulted from staring at this picture and imagining various ways of
> squishing the parts while performing rotations.
>
>
Ah, thanks for sharing that again. I remember seeing it now, but had
forgotten!

> Using cubes for tetrahedra is possible because the tetrahedral group is a
> subgroup of the octahedral group. This doesn’t work in the lower
> dimensional case because a triangle group is not a subgroup of a square
> group (look up cyclic groups <https://en.wikipedia.org/wiki/Cyclic_group>
> if you want to research). That is why it was unnatural to deal with
> squares for the 2^3 analogue.
>
>
> So what does the 2D analog look like using triangles?
>
>
In my mind, it looks like a net of triangles of the octahedron (which can
take a number of jagged shapes
<https://www.google.com/search?q=octahedron+net&num=100&espv=2&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwiC5cPuz5XSAhUK-2MKHciWCqAQsAQIGw&biw=1659&bih=802>).
Each triangle has 3 lines that start at the triangle’s edge midpoints and
meet at the triangle center. These define 3 colored areas, rather than the
4 colored volumes in yours. Each triangle would have 3 magnets, one at
each edge midpoint.

This analogue is not nearly as clean since the octahedron net is jagged.

> 1. If we were to allow interim jumbling, I think we can get 90-degree
> twists of the blue and orange faces. Instead of performing a
> 180-degree rotation maneuver here, you would take the 4x2 block and
> translate it a step. The end result would be a 3x2x2 with two 2x1 blocks
> protruding off of it on opposite sides. But then you could just do a
> reorientation by rolling one of those protrusions around to meet the other
> and recover the original 4x2x2 block. Hope this is clear.
>
>
> I think the shearing step is clear enough though I’m not sure what
> orientation you intend the 2x1 block to end up in. Regardless, that would
> look like an even stranger result than anything I’ve seen so far because
> half of the pieces would have single stickers at the corners, and half
> would have 3-color junctions. I had been calling such oddly reoriented
> pieces "inverted", but is "jumbled" the more correct term? It would be nice
> if the terms jumbled and bandaged are the correct analogies.
>
>
I’m using "jumbled" to mean something else, like this
<http://twistypuzzles.com/articles/other-twistypuzzlesthatjumble/>. IOW, I
was using jumbled to mean allowing moves that don’t result in the 4x2x2
block. Sounds like there are 3 separate terms here.

> I want one Melinda! When are you going to set up a shop? :D
>
>
> Well the first question is which version do you want? The bandaged
> (current) version or Matt’s more general arrangement? I’m working with the
> dice guy right now to see if and how the 24 magnet pieces might be made and
> what they will cost. The magnets will likely need to be recessed so that
> it’s not too hard to turn. That means either an extra step to fill in the
> gaps, or just sticker over them. The nice thing is that none of the
> diagonal sticker cuts will cross any magnets.
>
> I’m happy to produce these for group members at cost, at least until it
> becomes too much work. It took me nearly 4 hours to sticker this one
> (nearly 200 triangles!) but I’m sure that with practice, I can soon do it
> in under 2 hours.
>

I think I’d like to buy whatever version evolves from this thread that we
know will support the full 2^4 permutation group (though it would be cool
to have both). I’d be happy to sticker it myself. I recently bought a
Tuttminx <https://en.wikipedia.org/wiki/Tuttminx> that came unstickered,
and stickering it took me a similar amount of time.

Roice