Message #1353

From: Andrey <andreyastrelin@yahoo.com>
Subject: [MC4D] Re: Other 4D puzzles
Date: Wed, 26 Jan 2011 13:55:04 -0000

— In 4D_Cubing@yahoogroups.com, "Galla, Matthew" <mgalla@…> wrote:
>
> As far as the face-turning 16 cell goes, the number of pieces depends on
> your cut depth. I have to admit, I was too interested in the 24Cell to
> really consider any tetrahedral-faced puzzle but I certainly can now.
> Assuming a cut shallow enough that no deeper interactions have occurred, I
> believe you get a few more pieces than you have listed. Applying all 3 types
> of cuts, I THINK I am seeing:
> 4 corners [rotated by 3 face cuts, 3 edge cuts, and 1 vertex cut] (8C),
> 4 sub-corners [rotated by 3 face cuts and 3 edge cuts] (1C),
> 12 "offset faces" [rotated by 3 face cuts and 2 edge cuts] (2C),
> 12 "offset sub-edges" [rotated by 3 face cuts and 1 edge cut] (1C), (these
> are tricky - they are located between 2 "offset face" stickers and 1 edge
> sticker)
> 4 "sub-sub-corners" [rotated by 3 face cuts] (1C) (Dunno what else to call
> these - they are located exactly between a sub-corner sticker and a center
> sticker)
> 6 edges [rotated by 2 face cuts and 1 edge cut] (4C),
> 6 sub-edges [rotated by 2 face cuts] (1C)
> 4 faces [rotated by 1 facecut] (2C)
> 1 center [rotated by nothing but cell rotation itself] (1C)
>
> [it appears that rotation by x face cuts, y edge cuts, and z vertex cuts is
> impossible if x<y or y<z at shallow depths, and x=y only at vertices so I
> THINK this is all of the available interactions]
>
> That gives a total of 53 stickers per cell! [for 592 pieces total in the
> whole puzzle, I believe] That’s almost as bad as the 24Cell ;)

Yes, I’ve checked again - it’s really 53. I’ve missed intersections that are twice far from the vertex than vertex-cutting plane. So there are 9 stickers assotiated with the vertex, not 2 as I thought before.

> Also as near
> as I can tell, increasing depth changes nothing until the center sticker is
> squeezed out of existence (or, more accurately, the 3D surface of that
> cell). After that, things get complicated….

Yes, it will be at depth=1/4 of the cell height :)

>
> And yes a hypercube with "Skewb-like" (supposedly what you mean by
> diagonal?) cuts would be a 4D shape mod of a cell turning 16Cell *shudders
> at thought of 4D shape-modding….* Depending on the exact nature of the
> geometry (I haven’t looked into it yet), a few pieces may be lost or gained
> going from one puzzle to the other, but proper depths should preserve most
> piece types between the two puzzles.

Even when pieces are preserved, they may change coloring - from 1C to 8C and back. With all orientation problems :)