Message #1351

From: "Galla, Matthew" <mgalla@trinity.edu>
Subject: Re: [MC4D] Re: Other 4D puzzles
Date: Wed, 26 Jan 2011 04:52:00 -0600

Hey guys,

Thanks for really running with this idea! I hope you are starting to feel
some of the excitement I’ve had for a long time :)

Roice:
I think I will take your suggested terminology of parallel to vertex,
thanks! And yes, I think you are associating the extra cut to the right
phenomenon. Now there may be a more mathematical reason for this, but my
impression is that these extra cuts are caused by the number of cells
meeting at a edge and at a vertex in the 4D shape. In the case of the
24Cell, 3 cells meet at every edge (and 2 faces meet at every edge in an
octahedron). When one of these cells is rotated, it also moves one layer of
each of the other 2 cells, while the pieces "in common" of these two layers
are pairs of stickers that represent the same group of pieces. However, 6
cells meet at every vertex (and 4 faces meet at every vertex in an
octahedron). When one of these cells is rotated, it also moves one layer of
4 of the other cells (see the correlations in the numbers?). However, the
remaining cell is also affected and it is from this interaction the
additional cut is born, parallel to the vertex (and it must line up with the
cut depth of the cuts parallel to the faces of the cell). On the 16Cell, 4
cells meet at an edge, leaving one that receives a new cut parallel to the
edge; and 8 cells meet at a vertex: moving one moves 3 via face cuts, and 3
via edge cuts. The remaining 1 receives a new cut parallel to the vertex. So
yes, a cell-turning 16Cell exhibits all 3 cut types. I also agree with your
hesitation on the ambiguous names for these monsters. I was merely using 4D
Skewb and 24Cell FTO in lieu of a better name for discussion purposes.
Though I am having trouble seeing how 4D megaminx can possibly refer to
anything but the Magic120Cell…. Perhaps I need to ponder that longer.

Andrey:
Yes, you got the 24-cell I had in mind exactly right. There would be 57
pieces in one cell: 6 corners (6C), 6 sub-corners (1C), 12 edges (3C), 24
"offset faces" (2C), 8 "sub-faces" (!!!!!!!2C!!!!!!!), and 1 center (1C)
[grand total of 672 visible pieces for the whole puzzle, I believe]. For
this puzzle the sub-corners are not mathematically connected in any way, so
each one acts independently and would naturally have only one color.
However, what I am calling the "sub-faces" (attempting to use your
terminology) ARE mathematically connected in pairs (in fact I believe they
would be one physical piece if the 4D puzzle could be constructed in
physical form). Because of this, each piece has 2 colors (stickers),
making their orientation partially distinguishable: only 3 of the 6 possible
orientations would be accepted as correct. Adding extra colors to make the
orientation (and permutation in the case of the sub-corners!) of these
pieces visible is of course an option, just as it was on the 4D cube and
120Cell with the centers. As neither of those puzzles used them, I was sort
of leaning toward not adding "super" colors, but there is definitely
something missing without them. Maybe it could be optional to add "super
stickers"? :)

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 ;) 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….

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.

As for a 600Cell puzzle, I starting pursuing this in order to find more
interesting 4D puzzles without getting as big and repetitive as the 120Cell.
Don’t you think the 600Cell is headed in the wrong direction? ;) The
symmetry guarantees there will be no more than 14400 of any chiral pieces
and no more than 7200 of any non-chiral pieces (the same guarantee we have
for cell-turning 120Cell puzzles), but still, the numbers can get ridiculous
really fast on a 600Cell :)

Also, for shallow-cut 600Cell puzzles, it appears you are right: there are 5
different cuts appearing. In addition to the standard face cut, there are a
pair of angled, mirrored cuts due to edge interactions (that can probably be
considered only one type of cut) and 4 types of cuts due to vertex
interactions (of which 2 are mirror images of each other and probably best
considered as only one type). This gives types of cuts that are not set
parallel to the faces of each cell in addition to the one type that is set
parallel to each face. Not only would this puzzle have an obscene number of
each type of piece. There will be dozens of different piece types in the
most natural form! Maybe we should stick to the simpler polytopes for now :)

Melinda:
When I started typing this message, I wrote your reply first. However I see
since then, Nan has already addressed your question. I might as well back
him up: he is absolutely right. The 120Cell has a large number of vertices
compared to cells, while its dual, the 600Cell has a large number of cells
compared to vertices. Assuming we are talking about cell turning puzzles in
each case, the number of cells determines the "complexity" of each puzzle.
The cut intersections will be more numerous on the 600Cell, less symmetric,
and produce a great deal many more pieces than the 120Cell. However, as I
hinted at above, there is something to be said for the duality. Each of the
120Cell and 600Cell has 7200-fold symmetry. Since a given piece can only
occupy a space where it holds the same geometrical relationship to the cells
of the puzzle, there can only be a maximum of 7200 instances of a single
type of piece on either puzzle (if the piece displays the minimum amount of
symmetry while remaining achiral: chiral pieces may exist in separate orbits
where each orbit consists of 7200 identical pieces and the argument could be
made that there are 14400 instances of that piece type, although half are
mirror images of the other half). Anyway, a 600Cell face turning puzzle
would be quite formidable indeed, and much more difficult than the 120Cell.

Nan:
Your intuition serves you well. I can already tell a cell-turning 600Cell
puzzle would be nothing short of apocalyptic ;)
And if you start looking at deeper cuts, we’re no longer talking about
thousands of copies of dozens of types of pieces. We are talking tens of
thousands of copies of hundreds of types of pieces. I would like to state
right now that I will NOT be attempting a cell-turning 600Cell solve without
use of a programmable interface that can locate and execute hundreds of
setups and algorithms for me! (I have no problem with identifying the
algorithms though!)

Thank you all for such an interesting discussion
Keep it up!
-Matt Galla

PS My spellchecker does not recognize the word achiral —– that’s
acceptable
My spellchecker also does not recognize the word vertices —— REALLY?!?!
sigh…

PPS Following Melinda’s example, I also deleted the quoted sections to cut
down on length, which I know for me can be an issue… :)