Message #1345

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Other 4D puzzles
Date: Tue, 25 Jan 2011 22:53:11 -0600

Great stuff guys!

Special thanks for helping me to picture the nature of a cell-turning
24-cell puzzle. In trying to understand the extra cuts you described, I see
now that they are somewhat related to the
incidence<http://en.wikipedia.org/wiki/Incidence_(geometry)>properties
of adjacent cells. In particular, the unusual cuts come from
the adjacent cells with vertex-only incidences. (btw, "parallel to vertex"
rather than "perpendicular to vertex" seems like decent language, though
this wording does refer to the adjacent cell the cut is based on.) At first
I thought the 24-cell puzzle would also need cuts parallel to edges, but
there are no adjacent cells having incident edges which do not also
have incident planes. On the 16-cell, there are adjacent cells with all
three possible incidence types, and it looks like there will be three styles
of cuts on its tetrahedral cells. Both puzzles sound difficult! We’ve
never run into these kinds of situations before because the adjacent cells
on puzzles with simplex vertex figures all have incident planes.

I also thought I’d mention that I never felt fully comfortable calling
Magic120Cell a "4D Megaminx", due to some of the analogy ambiguities you are
discussing. Similarly, my personal preference leans towards not using terms
like "4D Skewb", unless all could agree on the most defining Skewb-like
properties. Since a 3D Skewb is a vertex-turning puzzle with slices halfway
between diametrically opposed vertices, it could be argued that the 4D Skewb
must have all these properties, with the only change being that the
properties are now applied to a hypercube (in other words, that the 4D Skewb
is the puzzle you described that has faces that look like Dino cubes). I
guess my point is that I prefer language like Nan used, explicitly
describing the polytope and the nature of the twisting. But I also agree
the naming is not the most important aspect (and I’ve never been good at
creating interesting puzzle names), so that’s all I will have to say about
that :)

Cheers,
Roice

P.S. Anyone know where you can buy the face-turning-octahedron puzzle? I’d
like to own one.

On Sun, Jan 23, 2011 at 2:54 PM, Galla, Matthew <mgalla@trinity.edu> wrote:

>
>
> Ah,
>
> schuma (Nan?) is quite right. A very "natural" (perhaps even the most
> "natural") extension of the FTO to 4D is a cell turning 16Cell. However, I
> was looking for a puzzle where each cell looks like an FTO, and this
> obviously cannot be the case for a 16Cell, which has tetrahedral faces.
>
> It seems that in 4D there are two ways of interpreting the analogue of some
> puzzles.
> On the one hand, you could construct a 4D puzzle where every cell looks
> like the 3D counterpart, in the case of all puzzles except tetrahedral and
> icosahedral, this unambiguously assigns the 4D shape. In the case of a
> tetrahedral puzzle, you can choose between the 5Cell, the 16Cell, and the
> 600Cell. In the case of icosahedral, this interpretation fails to produce an
> equivalent puzzle.
> On the other hand, you can analyze the construction of the 3D shape and
> construct the equivalent 4D shape. In the case of the octahedron, 4
> triangles meet at a point (triangle being the 2-simplex). Thus the 4D
> equivalent should have 4 tetrahedra (tetrahedron being the
> 3-simplex) meeting at an edge. This can unambiguously find analogues for all
> regular polyhedra (in the case of the FTO, this interpretation gives a
> 16Cell with pyraminx-like cells, the one schuma is referring to), and
> possibly more; however no puzzle will ever get mapped to the 24Cell (because
> the 24Cell has no 3D equivalent).
>
> I realize the first method given above is "artificial" in a sense. You do
> not design a 3D puzzle by first deciding what each face should look like and
> then repeating it over the rest of the puzzle. BUT YOU COULD! ;) As long as
> you pick a face that is cut in such a way that all cuts are parallel to the
> sides of the face-shape and at equal depths, the resulting puzzle should be
> "playable". (the 4D analogue for this is choosing a cell layout such that
> all cuts are parallel to the faces of the cell and at equal depths - but
> this is PRECISELY what allows the cell to alone be a 3D puzzle)
>
> In any case, it seems that both methods produce valid puzzles, and while
> some 4D puzzles can be obtained through either interpretation, there are
> some (like the 24Cell 4D FTO I described earlier) that can only be produced
> through one interpretation. I therefore think it is important that we
> consider both interpretations (plus I think a 24Cell would be more exciting,
> but maybe that’s just me ;) )
>
>
> Thanks for bringing that up schuma!
>
> -Matt Galla
>
> PS On TP my username is Allagem ;)
>
> On Sun, Jan 23, 2011 at 12:44 PM, schuma <mananself@gmail.com> wrote:
>
>>
>>
>> Hi Matt,
>>
>> Thank you for starting the discussions about other 4D puzzles.
>>
>> Can you explain more about why the 4D analogue of the FTO is a 24-cell
>> instead of a 16-cell? Although the faces of the 24-cell are octahedra,
>> 24-cell is a self-dual polytope that is not a simplex. From this point of
>> view, it has no 3D analog. In fact it has no analog in any dimension other
>> than 4D. However, the 16-cell belongs to the family of cross-polytopes,
>> which are the duals of hypercubes, and exist in any number of dimensions. (
>> http://en.wikipedia.org/wiki/Cross-polytope). In 3D, the cross-polytope
>> is 16-cell. Therefore I think a natural extension of FTO is a cell-turning
>> 16-cell, because they share more similarities.
>>
>> For example, you may know that in 3D, the FTO can be regarded as a
>> shape-mod of Rex Cube, a vertex turning cube (
>> http://www.twistypuzzles.com/forum/viewtopic.php?f=15&t=12659). If the 4D
>> FTO is a shape-mod of the vertex turning hypercube, it should be a
>> cell-turning 16-cell instead of a cell-turning 24-cell.
>>
>> No matter calling it 4D FTO or else, I believe what you have described in
>> the third paragraph is a cell-turning 24-cell. It should be an amazing
>> puzzle to solve. I have special feeling about it because of its uniqueness
>> in all the dimensions.
>>
>> Nan
>>
>>
>> — In 4D_Cubing@yahoogroups.com <4D_Cubing%40yahoogroups.com>, "Galla,
>> Matthew" <mgalla@…> wrote:
>> >
>> > Hey everyone,
>> >
>> > As I mentioned in my response about my solve of the 120Cell, I have been
>> > looking into some other 4D puzzles and have worked out how several of
>> these
>> > puzzles should work and even discovered some interesting properties.
>> Here is
>> > a snipet from my 120Cell solve message I sent Roice discussing this
>> subject:
>> >
>> > "I am still hoping for more complicated 4D puzzles and am willing to do
>> > whatever I can to help make them a reality. Coding a 4d space like you
>> have
>> > is quite intimidating, but perhaps I can try to build off a pre-existing
>> one
>> > with some guidance. I have already determined what the 4D analogue of
>> the
>> > FTO (face turning octahedron, invented some time last year if you have
>> not
>> > already seen it) would look like and how it would function as well as
>> the 4D
>> > analogue of the Skewb and Helicopter Cube (on that note I also have a
>> > suggestion as to how to make the interface for 4D puzzles that are
>> non-face
>> > rotating, like the Skewb and Helicopter Cube). I have also made some
>> > interesting discoveries like for example making a 4D puzzle out of a 3D
>> > puzzle can make some additional internal cuts without altering the
>> exterior
>> > of a 3D face (true for all three puzzle I mentioned so far) and how a 4D
>> > Skewb is not deepcut! (that is every cell looks like a Skewb and seems
>> to
>> > behave as such) The vertex turning deepcut hypercube has faces that
>> > externally each look like a dino cube. Is there anything I can do to
>> make
>> > help make these a reality? After spending 150 hours on the 120Cell, I
>> can
>> > honestly say that about 146 of the hours all feel exactly the same and I
>> am
>> > dying to find a more interesting 4D puzzle to explore :)"
>> >
>> > To expand a little on some of the things I mentioned above, the 4D FTO
>> would
>> > be a 24Cell with faces that look like an exploded version of this
>> puzzle:
>> > http://www.jaapsch.net/puzzles/octaface.htm
>> > with one big difference, in addition to every cut on the 3D analogue of
>> the
>> > puzzle, the 4D version has and additional cut perpendicular to the
>> vertices
>> > of each face that line up with first cut down. :/ Sorry, I know that
>> wasn’t
>> > very well worded and I’m not sure how well sending a picture would work
>> > through a yahoo group. Let me try again: these extra cuts would
>> essential
>> > cut off the vertex pieces of each cell. Removing the pieces that are
>> > affected by this new unexpected cut will result in cells that have an
>> > exterior that matches this puzzle:
>> > http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=451
>> > (If you can follow my inadequate descriptions above, the 4D FTO would
>> have 6
>> > distinct visible pieces, not just the 5 present on an exploded 3D FTO -
>> the
>> > extra comes from splitting each of the vertex pieces of the 3D Fto in
>> half)
>> >
>> > A similar phenomenon occurs on both the 4D helicopter cube (3D:
>> >
>> http://www.puzzleforge.com/main/index.php?option=com_content&view=article&id=49:hcannounce&catid=1:latest-news&Itemid=50
>> )
>> > and 4D Skewb (3D: http://www.jaapsch.net/puzzles/skewb.htm) [by
>> analogue, I
>> > mean each cell looks like the respective puzzle and moves in a similar
>> > manner]. In each of these puzzles, the new cut clips off the corners.
>> > Remembering that to truly express the 4D nature of these puzzles, each
>> cell
>> > must be "exploded", so what used to be he vertex pieces for each of
>> these
>> > puzzles have now been cut in half resulting in an internal piece that
>> > behaves as one might have expected the single original piece to act and
>> an
>> > external piece that in addition to moving every time the internal piece
>> > moves, can also be affected by a non-adjacent face.
>> >
>> >
>> > As to a nice interface for non-face rotating 4D puzzles, my suggestion
>> is to
>> > display the wireframe of a 3D solid that displays all the symmetries
>> implied
>> > by the rotation between the faces and perform clicks not on the puzzle
>> > itself, but only on this wireframe. For example, on a 4D Skewb,
>> rotations
>> > are made around the "corners" of each cell. These rotations are all
>> > equivalent to some rotation on a face turning 16Cell. So, in the
>> Hypercube
>> > shape, we could display wireframes of tetrahedrons that "float" between
>> the
>> > appropriate corners of 4 hypercube cells. When the user clicks on a face
>> of
>> > this floating wirefram tetrahedron, both the tetrahedron and the pieces
>> > affected by the corresponding "vertex twist" all rotate. Clicking on the
>> > actual stickers of the puzzle does nothing; all rotations are executed
>> by
>> > clicking on these "rotation polyhedra". In the case of the 4D Helicopter
>> > Cube, the appropriate wireframe shape would be a triangular prism -
>> > rotations around both the triangle faces and the rectangular faces are
>> > possible moves on the 4D Helicopter Cube, and each of these rotations
>> can be
>> > executed unambiguously by clicking on the appropriate face of the
>> triangular
>> > prism wireframe floating between the cells of the puzzle.
>> >
>> >
>> > As to the deepcut comment, attempt to visualize a 4D Skewb puzzle, that
>> is -
>> > a hypercube consisting of exploded skewbs (with additional cuts clipping
>> off
>> > the corners). Now identify all the pieces affected by one particular
>> > rotation and try to identify the move that is on the opposite side of
>> the
>> > puzzle. Identified correctly, this opposite move does not affect any of
>> the
>> > same pieces. However, not every piece is affected by these two moves!
>> There
>> > is a band of pieces remaining untouched, much like the slice of a 3x3x3
>> left
>> > untouched by UD’. This means the puzzle is not deepcut! If we push the
>> 3D
>> > hyper cutting planes deeper into the 4D puzzle, we get cells that look
>> like
>> > Master Skewbs. Continuing to push, certain pieces of these Master Skewbs
>> get
>> > thinner and thinner until they vanish at the point when opposing
>> hyperplanes
>> > meet. This is the deepcut vertex turning 8Cell puzzle. Each cell looks
>> like
>> > an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cells
>> that
>> > look like dino cubes that is shallower cut. Although these puzzles are
>> > visually identical, a single move on the shallower cut puzzle affects
>> pieces
>> > on only 4 cells while a single move on the deepcut puzzle affects pieces
>> on
>> > all 8 cells. Also of interest is the series of complicated looking
>> puzzles
>> > that appear at cut depths between the 4D Skewb and each of these dino
>> cell
>> > puzzles, although there are only 3 slices per axis in these puzzles
>> (same
>> > order as 3x3x3), each cell is an exploded Master Skewb!
>> >
>> > Although I have explored several other ideas, the three puzzles (4D FTO,
>> 4D
>> > Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
>> > candidates for the next run of 4D puzzles, they implement some complex
>> piece
>> > interactions without becoming too large or too visually crowded.
>> >
>> > These puzzles are of an incredible interest to me, because the
>> interactions
>> > of the pieces are so much more intricate than the 120Cell or any of the
>> > simplex vertex puzzles possible in the current MC4D program! As I
>> mentioned
>> > in my message to Roice, I have a good idea of how each of these puzzles
>> look
>> > and function and would gladly assist anyone (Roice? haha) who wants to
>> > attempt to program it. In the meantime, I will take a look at the code
>> Roice
>> > has provided me and try to do some work myself, but I highly doubt I
>> will
>> > have success without an experienced programmer’s help ;)
>> >
>> > I would love to hear others’ thoughts on these!
>> > -Matt Galla
>> >
>>
>>
>
>
>