# 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… :)