# 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

> >>>>>

> >>>>

> >>>

> >>>

> >

> >

> >

> > ————————————

> >

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> >

> >

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