Message #1049

From: Andrew James Gould <agould@uwm.edu>
Subject: Re: [MC4D] Re: definition of a twist
Date: Mon, 19 Jul 2010 23:06:18 -0500

Just my first thought, but I’d be with Matt on counting 180 degree twists as one twist–it’s what I do on a 3D cube. The 37 twist example you mentioned, Melina, I’d also count as 1.

Matt,
I play enough video games to keep the visual part of my brain in shape to answer this. One way a 4D person could do the A1 twist is to put his right hand on the 3 brown stickers and the 3 opposing green stickers that will move, and his left hand on the 24 brown stickers and the 24 opposing green stickers that don’t move. He can then twist. Anything you can do to a 3^3 Magic cube, a 4D person could do to the -t face of the 3^4 Magic Cube. That includes all x-y, x-z, and y-z twists, and he would never have to touch the stickers on the -t face to do those twists.

Note: of those 54 green and brown stickers, he wouldn’t just be touching the surfaces that we see in MC4D, he’d be touching every molecule inside. (Here, I use the ‘going from 2D to 3D’ analogy to explain going from 3D to 4D.) If you’re a stick figure in 2D, you can’t get inside a solid 2D square sticker to cover the whole thing without a 3rd dimension. But on a 3^3 Magic cube in 3D with its 9 yellow stickers at z = 1.5, placing your hand on top of the 9 yellow stickers is like placing your hand at z = 1.50001 at the same x & y coordinates of the cube–touching all the yellow stickers on their +z side. Similarly in 3D, you can’t get inside a solid 3D cube sticker to "cover" the whole thing in 3D. But on a 3^4 Magic Cube in 4D with its 27 blue-gray stickers located at t = -1.5 in an (x,y,z,t) coordinate system, he could "cover" all molecules of those stickers by placing his 4D hand at t = -1.50001 at the same x, y, & z coordinates of the tesseract. His hand is therefore touching all the blue-gray stickers everywhere on their -t side. Now THAT’s what I call…an up-grayed.

–
Andy

—– Original Message —–
From: "Melinda Green" <melinda@superliminal.com>
To: "4D Cubing" <4D_Cubing@yahoogroups.com>
Sent: Monday, July 19, 2010 3:11:59 PM
Subject: Re: [MC4D] Re: definition of a twist

Matthew,

I know it’s been discussed too but I’ll just give my current opinion
which is that I like the idea of defining a twist as any combination of
currently supported twists on a single face that can be reduced to a
single one with a single slice mask. So a twist with one slice mask
followed by the same twist with a different mask should only count as a
single twist and represented in the log file with their combined mask.
Likewise three 90 degree twists should only count as one -90 degree
twist. And of course any combination that leaves the puzzle unchanged
shouldn’t count at all. I currently cancel pairs of twists that are the
inverse of each other, so long as they don’t cross macro boundaries. I’d
love to support more such cases.

The one case that I’m not currently ready to accept are combinations of
face twists that can be represented by a single transformation but which
are not reachable by a single existing twist. I’m talking about moves
like double 90 degree twists. At first blush that seems reasonable to
represent as an atomic move but look at the Onehundredagonal Duoprism.
It’s not at all clear to me that 37 single twists around the cylinder
should count as a single twist. maybe, maybe not.

Even though these sorts of changes will affect the twist counts of
previous solutions, there is the possibility of adding more twist
compression logic to MC4D or to a standalone program that will factor
out any redundancies so that comparisons will always be reasonably fair.
At the moment I don’t have plans to do any of this work but I think it’s
great to have these discussions in the hope that we can come up with the
prefect definitions to base future work on.

-Melinda

matthewsheerin wrote:
> […] I feel that it’s worth mentioning the old problem in MC4D. A
> face can be moved to most its possible positions in one move, but the
> three positions reached by two 90 degree twists require two moves. I’m
> sure it has been discussed before but a quick look didn’t find it.