Message #1347

From: schuma <mananself@gmail.com>
Subject: [MC4D] Re: Other 4D puzzles
Date: Wed, 26 Jan 2011 05:55:13 -0000

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
> >> >
> >>
> >>
> >
> >
> >
>