Message #1350

From: schuma <mananself@gmail.com>
Subject: [MC4D] Re: Other 4D puzzles
Date: Wed, 26 Jan 2011 10:19:37 -0000

Hi guys,

I guess the shallow-cut 600-cell is MUCH MUCH harder than the 120-cell rather than "almost exactly as hard". The reason is that there are too many small pieces due to "incidence" (I don’t know the exact meaning of this term, but I do have some kind of intuition).

My guess comes from the experience in 3D. We all know the neat structure of a megaminx. But what about the shallow-cut icosahedron? Does it have a neat shape? It looks like this:

http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/icosa_f3.gif

Even using the shallowest cuts, we inevitably have many small pieces. It’s because around each vertex, five cutting planes intersect each other to create them. I guess this is the vertex-incidence properties that Roice and Andrey talked about. As a result, solving such a puzzle is much harder than solving a megaminx. The number of steps to solve it is usually an order of magnitude more than that for a megaminx.

I think the 600-cell puzzle has a similar issue. At each vertex there is a 12C pieces. And around it, 12 hyperplanes intersect, producing numerous small pieces. The number of pieces in a shallow-cut 600-cell must be several times more than that of 120-cell. It’s just horrible.

Anyway to make it better?

(1) Simply drop some small pieces.

(2) Make the cuts curvy to avoid intersection. 3D Examples for icosahedron:
http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/icosa_f9.gif
and physical puzzle:
http://www.puzzleforge.com/images/photos/twistypuzzlesposts/radiolarian/stickered/Picture1%20079.jpg

(3) Make a vertex turning 120-cell rather than cell-turning 600-cell. Although it sounds like nothing is changed, it actually makes things much easier. The shallowest vertex turning 120-cell only contains trivial tips. If we slowly make the cuts deeper and deeper, we are slowly introducing new types of pieces. We can always find a depth that produces the right number of pieces.

– Nan

— In 4D_Cubing@yahoogroups.com, Melinda Green <melinda@…> wrote:
>
> Thanks for the links, Nan! I’ve ordered one from the first link along
> with a floppy cube since that is the closest thing to MC2D. :-)
>
> Oh, and a slightly sarcastic "thanks" to Andrey for mentioning a puzzle
> based on the 600 cell. I’ve sort of thought of that monster as "The Name
> We Must Not Say Aloud". Now that the spell is broken I suppose that at
> some point someone is going to implement one and someone else will solve
> it. I shudder to imagine how many hundreds of hours that solution will
> require. OTOH, maybe since it is the duel of the 120 cell, it will be
> almost exactly as hard. Predictions anyone?
>
> -Melinda
>
> On 1/25/2011 9:55 PM, schuma wrote:
> > Hi,
> >
> > This message is dedicated to answer Roice’s question about where to buy a face turning octahedron. Here are some suggestions:
> >
> > http://www.hknowstore.com/item.aspx?corpname=nowstore&itemid=058fbd01-4a44-4e6f-99ec-71ae3bd9eb23
> >
> > http://www.witeden.com/goods.php?id=174
> >
> > or ebay sellers, for example
> >
> > http://cgi.ebay.com/Magic-Octahedron-Rubiks-Cube-Star-Puzzler-Transparent-/160515420233?pt=LH_DefaultDomain_2&hash=item255f76f049
> >
> > This puzzle has been mass-produced twice, so they are pretty cheap now, for around $10+shipping. Shipping might take two weeks because the sellers are usually in Hong Kong or China. Just make sure it’s a face-turning one before you buy it, because the vertex turning octahedron is also mass produced, which looks almost the same.
> >
> > Nan
> >
> > — In 4D_Cubing@yahoogroups.com, Roice Nelson<roice3@> wrote:
> >> 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@> 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@> 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
> >>>>>
> >>>>
> >>>
> >>>
> >
> >
> >
> > ————————————
> >
> > Yahoo! Groups Links
> >
> >
> >
> >
>