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 Post subject: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 12:38 pm 
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I've been thinking recently about how to assign an "order" to a non-cubic puzzle, where "order" is a number that corresponds roughly with the order of a cube (2 for the 2x2x2, 3 for the 3x3x3, etc.). It appears that there are at least two different ways of making the correspondence, which results in two different series of puzzles.

The first way is perhaps fairly named the "surface" series. Here, the puzzle is shallowly cut parallel to each face, with the number of cuts per edge corresponding roughly the order number. Here, the order corresponds roughly with the number of subdivisions per edge. For example, the megaminx would be an order-3 puzzle in this series, the gigaminx would be order-5, and so on. The problem with this series is that it's unclear how exactly the order number ought to be assigned.

The second way is based on a different kind of generalization from the cube. Rather than trying to map the order to the number of subdivisions per edge, we observe that even-numbered cubes are bisected, and odd-numbered cubes have center slices that rotate around the center of the puzzle. Generalizing this, we say that every bisected puzzle has an even order, and if there are N cuts on either side of the bisection, then the order number is 2(N+1). For example, the 2x2x2 has only the bisection cuts, and no others, so the order number is 2. The 4x4x4 has one additional cut on either side of the bisecting cut, so N=1, and the order is 4. And so on. For non-bisected puzzles with a center slice, we count the number of cuts M on each side of the center slice. The order is defined to be (2M+1). For example, the 3x3x3 has a center slice which is delimited by 2 cuts, one on each side. So M=1, and the order is 3. The 5x5x5 has a center slice which has 2 cuts on each side, so M=2 and the order is 5.

The interesting part is when we apply this system to non-cubic puzzles. For octahedra, the puzzle of order 2 has only bisecting cuts (since N=0 if 2(N+1)=2). This is the Skewb Diamond. For order 3, there is a center slice delimited by 1 cut on each side. This is the Face-turning Octahedron. An order-4 puzzle in this series would be a Skewb Diamond with additional cuts on either side of the bisecting cut (has this puzzle been built yet?).

For tetrahedra, the order-2 puzzle requires only bisection cuts: the only possibility seems to be the Pyramorphi[n]x. The order-3 puzzle requires a center slice, which only happens on the Master Pyramorphinx. So for tetrahedra, this series yields only shape-mods of the cubes.

For dodecahedra, however, we get more interesting puzzles. The order-2 puzzle is restricted to only bisecting cuts; so the result is actually an elaborated version of the Skewb Ultimate, where more cuts are introduced so that each face has a pentagonal star pattern---it is a lookalike to the Megaminx, but turns very differently. Unlike Mèffert's Megaminx, the star pattern must be sharp-pointed, since each edge must be precisely bisected. The face centers would not be fixed, and can be permuted. (I'm not sure if this puzzle has been thought of/built before?) The order-3 puzzle has quite a complicated subdivision of each face, owing to the center slice. If the cuts are chosen such that they coincide with edges, then we get a puzzle with each face subdivided into 5-pointed stars, where the points are at the vertices of the dodecahedron rather than the edge centers. (Is this related to the Pen(t)ultimate puzzle? I'm not sure if the cutting depth corresponds, this might be a new puzzle.)

For icosahedra, the order-2 puzzle would be the bisected icosahedron, which is related to the Pentultimate. The order-3 puzzle, however, I'm not sure if anyone has contemplated before. Each face would be subdivided into 9 smaller triangles, and there would be 12 corner pieces (5 colors each), 30 edge pieces (2 colors each), and 60 face pieces (1 color), all permutable relative to each other. It would also be a bear to solve, since it is very deep-cut. :twisted:

I wish I knew how to design puzzle mechs. Higher order puzzles generated from the dodecahedron/icosahedron are non-trivial generalizations that may be very interesting to solve. An order-5 dodecahedron in this series, for example, would be significantly more complex than the Gigaminx.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 1:01 pm 
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quickfur wrote:
For dodecahedra, however, we get more interesting puzzles. The order-2 puzzle is restricted to only bisecting cuts; so the result is actually an elaborated version of the Skewb Ultimate, where more cuts are introduced so that each face has a pentagonal star pattern---it is a lookalike to the Megaminx, but turns very differently. Unlike Mèffert's Megaminx, the star pattern must be sharp-pointed, since each edge must be precisely bisected. The face centers would not be fixed, and can be permuted. (I'm not sure if this puzzle has been thought of/built before?)

For icosahedra, the order-2 puzzle would be the bisected icosahedron, which is related to the Pentultimate.


Something doesn't add up here. The order-2 dodecahedron is in fact the Pentultimate (gelatinbrain 1.1.7), right? And it doesn't have a star pattern on its faces. The order-2 icosahedron would be gelatinbrain 2.1.5. Nothing with a true icosahedral twist has ever been built, right?

One problem with trying to define order in this fashion is that there is more specificity required than simply the number of cuts. For cubes, it doesn't matter where you put the slices, just how many there are. But the Megaminx, the Pyraminx Crystal, and Aleh's Starminx would all be considered order-3 dodecahedra.

Also it's not clear to me whether you're still talking about face-turning puzzles... I guess your comments on dodecahedra suggest you are thinking of vertex-turning puzzles there? Anyway that's another way you have to break it down: face-turning, vertex-turning, edge-turning, other. Some of which are not mutually exclusive, e.g. the Pyraminx.

It would be great to have an ordered system like this for categorizing all twisty puzzles, but I'm not optimistic it can be done.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 1:13 pm 
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If we are using a cube as reference geometry for order, we cannot pass the limits of a cube when referencing order. For this reason, deep cut puzzles cannot be included in the same notation, as they offer an entirely new dimension to the problem. Order, if classified with a single number, must imply that each plane bisect only those directly next to it as occurs in a cube.

We consider order counts for only 1 intersection of each plane by another. Anything else is deep cut. So a megaminx would be 3^3, a gigaminx 5^3, a paraminx crystal 3^4, starminx 3^5, and pentualem 3^6. Power refers to the number of unique pieces (internal or external) that appear as cuts become deeper, base number the number of planes +1.

Power 1 is considered baby face; 2 is only truly re-creatable on a sphere, but it is the equivalent of, actually it involves less intersections than, a edges only; 3 involves 1 intersection of every plane; 4 and on are impossible to recreate on a cube because of 90 degree angles.

Even orders do not exist (but can be simulated), as at no level of cut can you reveal "corners only." Even on a cube you must travel through a 3x3x3 to make a 2x2x2. In the cube, however, you do not achieve a higher power when reaching a 2x2x2, you simply eliminate the center layers. This is why a kilo-minx or an imposiball is a 2x2x2 equivalent. These are geometrically impossible on a dodecahedron, and for this reason they do not exist. An equivalent model, however, can be made on icosahedrons.

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 1:21 pm 
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I should add that to me, one of the most tempting ways to generalize Rubik's Cube is as follows: make one infinitesimally thin shallow cut per face, at constant depth. Now, feel free to reposition that cut for aesthetics, as long as it doesn't add or remove any new pieces.

So to start with a cube, we make infinitesimally thin slices, then slide in further to get Rubik's Cube -- which mathematically is identical to the version with infinitesimally thin slices. For the dodecahedron we get the Megaminx. For the tetrahedron we get the Halpern-Meier pyramid. For the octahedron we get what gelatinbrain calls the "Dino Octa":

Image

For the icosahedron, curiously, we get something which does not even appear on gelatinbrain -- all the face-turning icosahedra there have deeper cuts. But the face pattern would look like this:

Image

And one can continue with nonregular polyhedra as well. The advantage of this approach is that there is no ambiguity in where the cuts go; there is a single "most natural" Rubik's Cube analogue for any given polyhedron. The disadvantage, of course, is that it doesn't generalize to nxnxn, or beyond face turns. But you can't have everything, I think.

Anyway, with this approach, you always have (non-moving) centers, edges, and corners, and you have additional off-corner bits when the vertex degree is higher than 3.


Last edited by bhearn on Sun Nov 01, 2009 1:27 pm, edited 2 times in total.

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 1:24 pm 
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Well, I'm not so much interested in covering all possible puzzles, as in exploring the relationships between them. So I admit there are many holes in my enumeration scheme, but I was interested in this particular series (the second series) because their solutions are "similar" to the solutions of the corresponding cubes, due to the way they move. Having center slices means that techniques used on the cubes can be carried over, with suitable modifications, to achieve similar results. Of course, not everything is the same, but that's where it gets interesting to study why they differ.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 1:31 pm 
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Quote:
I should add that to me, one of the most tempting ways to generalize Rubik's Cube is as follows: make one infinitesimally thin shallow cut per face, at constant depth. Now, feel free to reposition that cut for aesthetics, as long as it doesn't add or remove any new pieces.


I agree this is what most people mean when they refer to order.

What would be interesting is charting out all the apps puzzles on a 3d graph. (number of planes per face, depth of planes, refference shape)

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 1:35 pm 
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bhearn wrote:
[...]Something doesn't add up here. The order-2 dodecahedron is in fact the Pentultimate (gelatinbrain 1.1.7), right? And it doesn't have a star pattern on its faces.

It doesn't? Hmm, weird. I'm looking at my Skewb Ultimate right now, and it seems that, if the cuts of 4 more Skewb Ultimates were to be applied to it in the right geometry, you would get something that looks like a megaminx but is bisected. There are 20 planes of bisection, not 12, as in the Pentultimate.

Quote:
[...]Also it's not clear to me whether you're still talking about face-turning puzzles... I guess your comments on dodecahedra suggest you are thinking of vertex-turning puzzles there? Anyway that's another way you have to break it down: face-turning, vertex-turning, edge-turning, other. Some of which are not mutually exclusive, e.g. the Pyraminx.[...]

No, I'm throwing out face or vertex turning as being criteria, and instead consider bisections and middle slices. Of course, in retrospect, I realize that this is ambiguous, since there may be more than one way to bisect a puzzle, which would lead subsequent higher orders to be in distinct series.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 1:42 pm 
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quickfur wrote:
It doesn't? Hmm, weird. I'm looking at my Skewb Ultimate right now, and it seems that, if the cuts of 4 more Skewb Ultimates were to be applied to it in the right geometry, you would get something that looks like a megaminx but is bisected. There are 20 planes of bisection, not 12, as in the Pentultimate.

Yeah, but that's a vertex-turning puzzle, gelatinbrain 1.2.9:

Image

which is the same, I think, up to piece orientation as gelatinbrain 2.1.5, a deep-cut face-turning icosahedron:

Image

Quote:
No, I'm throwing out face or vertex turning as being criteria, and instead consider bisections and middle slices. Of course, in retrospect, I realize that this is ambiguous, since there may be more than one way to bisect a puzzle, which would lead subsequent higher orders to be in distinct series.


Ah, OK. But then, yes, there is ambiguity. The shape of the polyhedron cut is then not what really matters. If you require face turning for order 2, then you do get a unique categorization, I think. It just breaks down when you go higher than 2.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 3:01 pm 
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If you want to classify each and every individual puzzle, then simply stating the order is not enough... the cube for instance has at least 3 different order 2 puzzles, I believe the dodecahedron has at least 4 or 5 and the icosahedron would be even more varied.
But, the series you are speaking of is the vertex-turning series, which has not currently been physically made for any other polyhedra besides tetrahedron, cube and octahedron I believe...

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 3:14 pm 
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elijah wrote:
But, the series you are speaking of is the vertex-turning series, which has not currently been physically made for any other polyhedra besides tetrahedron, cube and octahedron I believe...

The Pentultimate is the equivalent of the order-2 vertex-turning icosahedron.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 3:31 pm 
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grouping puzzles in vertex face and edge turning leads to an almost limitless number of puzzles. If we are classifying order it is best to reffer to base geometry. The pentumate hase planes parrallell to the faces of a dodecohedron that bisect it direclty through the middle.

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 3:36 pm 
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well, yes, it results in almost limitless puzzles, but these puzzles exist, so you can't simply ignore them when referring to order.
Some of the most common puzzles don't follow base geometry.

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 3:42 pm 
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I am saying that you would need a separate conversation for face vertex and edge turning, as it is often to hard to compare them. For example, a kilominx can be seen as impossible as a face turning dodecahedron, but it is easily accomplished on icosahedrons.

Also, what puzzle does not have base geometry? Jumbling puzzles oftentimes lack uniform geometry, but this is the only exception I can think of.

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 4:24 pm 
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There's a bit of a conflict here between the idea of the abstract mathematical structure of a puzzle, on the one hand, and the geometric, on the other. I think your idea of "base geometry" is intended to capture the abstract mathematical structure, but I don't think it's quite the same thing. What is the base geometry of a kilominx? A vertex-turning icosahedron, cut so that 5 faces turn as a unit? Maybe. But if it could be done with planar cuts on a dodecahedron, would that then be the base geometry instead?

Here's a (to me) very natural puzzle which doesn't seem to have a base geometry at all: something like a face-turning icosahedron, but with only center, edge, and corner pieces; this would sort of be the straightforward group-theoretic icosahedral analogue of a Rubik's Cube. If that can be done geometrically, let alone mechanically, I'd love to see it.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sun Nov 01, 2009 6:08 pm 
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If we're going to categorize all twisty puzzles, I say that we have to distinguish between its essential symmetry (i.e., "base geometry") and its outward shape. The essential symmetry determines how many axes of rotation there are, and we may perhaps add a parameter describing the number of parallel layers that may be independently rotated per axis, and a further parameter describing the level of intersection between adjacent layers.

Mathematically, there are several essential symmetries that may occur in symmetrical puzzles (less symmetrical puzzles like the Square-1 can still be included in this scheme, just that its particular type of symmetry would not recur in many other puzzles). Here is a list of the most common ones:

* T: tetrahedral: 4 axes of rotation, corresponding with the faces of the tetrahedron (or equivalently, the vertices).
* C: cubical: 6 axes of rotation in 3 colinear pairs, corresponding with the faces of the cube, or equivalently, the edges of a tetrahedron.
* O: octahedral: 8 axes of rotation in 4 colinear pairs, corresponding with the faces of an octahedron, or, equivalently, the vertices of a cube. Also equivalent to T, if the tetrahedral axes of rotation are extended to the opposite sides.
* Rd: rhombic dodecahedral: 12 axes of rotation in 6 pairs: corresponding with the edges of the cube or octahedron.
* D: dodecahedral: 12 axes of rotation in 6 pairs, corresponding with the faces of a dodecahedron, or equivalently, vertices of an icosahedron.
* I: icosahedral: 20 axes of rotation in 10 pairs, corresponding with the faces of an icosahedron or the vertices of a dodecahedron.
* Rt: rhombic triacontahedral: 30 axes of rotation in 15 pairs, corresponding with the faces of a rhombic triacontahedron, or equivalently, with the edges of a dodecahedron or the edges of an icosahedron.

Each of these symmetries have an intrinsic "order" (not to be confused with the order of a cube), which is the number of distinct orientations possible along each axis of rotation. The symmetries T, O, and I are "triangular": they have 3 possible orientations along each axis. The symmetry C is "square": it has 4 possible orientations, and the symmetry D is "pentagonal", having 5 possible orientations. The symmetries Rd and Rt are digonal, having only 2 possible orientations. Other values are possible for less symmetrical shapes, such as the n-prisms. A hexagonal prism would have a pair of axes that has order 6, for example.

The next parameter N describes how many planes of rotation are possible along each axis or each pair of axes. For example, the Pyraminx has 2 planes of rotation on each axis, one trivial one (the trivial tips) and one non-trivial one (the edge pieces). The 3x3x3 has 3 planes of rotation along each axis pair: a middle layer and two outer layers. This is probably the most consistent definition of "order" that we may have, since it counts the number of parallel layers along each axis rather than how they intersect the outer surface.

Given the symmetry symbol S and the number of planes of rotation along each axis(-pair) of rotation N, we still run into ambiguity concerning how close or far apart these planes are. A third parameter (and perhaps more) would be needed to distinguish between them. I haven't really thought it through enough yet, but I was considering assigning a number X describing how many intersections occur along the planes of rotation between adjacent axes.

In this way, we can classify each puzzle as a triplet (S,N,X,E), where S is a symbol indicating the underlying symmetry, N is the number of parallel layers of rotation along each axis (or axes pair), and X is some number describing how these layers intersect with each other, which hopefully makes the necessary distinctions. E is another symbol describing the external shape of the puzzle; e.g., all the Skewb puzzles have the same values for the first 3 parameters, differing only in the outer shape. All shape mods therefore share the same 3 parameters and differ only in E. I'm not sure how to come up with a consistent notation for E, since it can pretty much be any shape in any orientation as long as it is partitioned by the planes of rotation in S.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Mon Nov 02, 2009 2:55 am 
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bhearn wrote:
One of the most tempting ways to generalize Rubik's Cube is as follows: make one infinitesimally thin shallow cut per face, at constant depth. Now, feel free to reposition that cut for aesthetics, as long as it doesn't add or remove any new pieces.
...
And one can continue with nonregular polyhedra as well. The advantage of this approach is that there is no ambiguity in where the cuts go; there is a single "most natural" Rubik's Cube analogue for any given polyhedron.

Well what a coincidence... look what just appeared (twice in one day!). This new face-turning rhombic dodecahedron is exactly the "most natural" rhombic dodecahedron defined above:

New Puzzle Design - FT Rhombic Dodechedron


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Mon Nov 02, 2009 3:33 am 
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quickfur wrote:
Given the symmetry symbol S and the number of planes of rotation along each axis(-pair) of rotation N, we still run into ambiguity concerning how close or far apart these planes are. A third parameter (and perhaps more) would be needed to distinguish between them. I haven't really thought it through enough yet, but I was considering assigning a number X describing how many intersections occur along the planes of rotation between adjacent axes.


This is where I think it gets tricky; I'll be interested to see what you come up with.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Mon Nov 02, 2009 11:17 am 
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bhearn wrote:
quickfur wrote:
Given the symmetry symbol S and the number of planes of rotation along each axis(-pair) of rotation N, we still run into ambiguity concerning how close or far apart these planes are. A third parameter (and perhaps more) would be needed to distinguish between them. I haven't really thought it through enough yet, but I was considering assigning a number X describing how many intersections occur along the planes of rotation between adjacent axes.


This is where I think it gets tricky; I'll be interested to see what you come up with.

I looked at various possibilities yesterday, and decided that the cases would have to be treated separately depending on the angle A between two adjacent axes.

If A is 90 degrees, then no additional parameter is necessary, since each plane of rotation along and axis X1 will intersect with exactly N planes of rotation along an adjacent axis X2.

If A is less than 90 degrees, then we're dealing with a tetrahedral symmetry, and there are a number of possible cases. These cases can be adequately categorized by considering the pattern they make on a triangular face of the tetrahedron.

* The first case is a shallow cut corresponding with the Halpern Pyramid. Each plane of rotation intersects with 1 other plane of rotation on an adjacent axis.
* The second case is a deeper cut where each plane of rotation shares a single point of intersection with two other planes of rotation. This gives a Pyraminx without trivial tips.
* The third case is a yet deeper cut, where you get triangular face centers, 3 tetrahedral edge pieces per face, and 3 pentagonal corner pieces per face.
* The last case is a cut where the planes of rotation no longer intersect, giving rise to trivial tips.

A tetrahedral puzzle (symmetry T) would therefore have some combination of the above cases depending on the number N of planes of rotation per axis. Each plane of rotation can be categorized as one of the above cases by considering how they intersect with other planes of the same level (e.g., if N=2, then there are two sets of intersections to consider: the shallower cuts, and the deeper cuts). So a Pyraminx for example would have N=2, with case 2 for the first set of cuts, and case 4 for the second set of cuts.

When A is greater than 90 degrees, a lot more complexity arises. To deal with this, we need to consider not only a pair of adjacent axes, but 3 axes where the outer two axes are non-adjacent (i.e., the planes of rotation to consider are like 3 consecutive faces around a central pentagonal face on a dodecahedron). Planes on adjacent axes will always mutually intersect; what is of more interest is how many intersections occur between planes on two axes separated by another axis.

* If no intersections occur, we have a type 0 situation which gives us a megaminx in the case of dodecahedral symmetry.
* If 1 intersection occurs, we have two cases: a Pyraminx Crystal with face centers (shallow), and a Pyraminx Crystal without face centers (deeper). If we go yet deeper, we start getting more situations with face centers but now with additional pieces arising.
* This keeps going until we bisect the puzzle, after which going deeper just traverses the previous cases in reverse, so we only need to deal with the cases up to bisection.

These don't adequately cover all the cases for A > 90, though. I think I need to look into it more to see how many distinct cases there are. Of course, these only cover 1 set of rotational planes per axis; the full puzzle will have N sets of planes, each with their own depth characteristic. The puzzle should be fully described once each depth characteristic is defined. I hope. :)


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Mon Nov 02, 2009 1:38 pm 
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This discussion has went way to far for me to answer to something specific. Here are my 3 cents:

I have thought about classification as well and came up with this:
1. A "symmetry symbol" although different named than quickfur does. Here I have left out the TWO tetrahedral symmetries. But if you want them you have to include the edgeturning tetrahedron (or Rhombic hexahedron to name it consistently) with 6 "sides" and 180°-turns-only.
2. An order number where I count the number of cuts per throughgoing axis => 2x2x2, Pentultimate, BigChop etc. would have order=1. Megaminx would have order=2
3. A piece set. Look a the animation in this thread (Thanks again to Carl Hoff!):
viewtopic.php?f=1&t=14875
You see there that the faceturning dodecahedron of order=2 can have at most 8 different pieces. Each of them could be made visible in an appropriatly designed puzzle. Sadly not all at the same time.
4. Orientations needed:
All visible pieces not directly attached to the core need to be positioned correctly. But some have to be oriented, too.

The advantage I see in using piece sets (and orientation sets) is that you can cover easily every "shape mod" (compare 3.3.1 and 3.3.8 from Gelatibrain). Even non-planar cuts are no problem: The Rex cube differs from the Master Skewb by its piece set. Axis-configuration and order are the same.

A Megaminx therefore is a faceturning dodecahedron of second order with piece set [1,2,3] and orientations for [2,3].
The Pyraminx Chrystal is a faceturning dodecahedron of second order with piece set [3,4] and orientations for [3,4].
The Icosaminx would be a faceturning dodecahedron of second order with piece set [1,2,3] and orientations for [1,2].

In addition to that 6 (or 8) classes there are the hybrid puzzles like the SuperX. Think about faceturning archimedian solids and you get some inspirations.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Mon Nov 02, 2009 2:34 pm 
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This looks very promising to me. (For one thing, considering piece set allows the mathematically natural icosahedral puzzle I mentioned above, that has no obvious geometry.)

Are non-planar cuts really not a problem? They can't ever introduce piece types that wouldn't otherwise appear? (Suppose, to be perverse, you have coaxial cutting surfaces that intersect?) And is the total piece set insensitive to the type of solid cut?

One thing I'd be tempted to change is the order convention: if it's # of cuts per axis + 1, then you recover the intuitive idea of a Rubik's Cube as order 3. Or you could think of it as the number of pieces each coaxial set of cuts chops space into.

It does seem to me that there's a challenge here when dealing with higher orders, because then there is more than one slice-depth parameter that is varying. Enumerating all the possible piece types in a well-defined, unique order would seem to be a problem, even for, say, an order 4 (or 5, depending on convention) dodecahedron (Gigaminx family). And the same issue arises for hybrid puzzles of order 2 (3), because there then are axes of different types. Even if you pick some scheme for enumerating the piece types consistently, there's still no connection between piece type number (e.g. 1-8 for the order 2 (3) dodecahedron) and piece function.

I would love to see a taxonomy of all existing twisty puzzles using a scheme such as this. But I am still inclined to divorce the classification scheme a bit more from the geometry. In the end, you have a set of permutations of some number of piece types, satisfying some symmetries. I'm not sure how much is to be gained from tying the notion of piece set directly to geometric cuts which generate it.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Mon Nov 02, 2009 2:56 pm 
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I like the idea of piece sets too. It's definitely easier to manage than attempting to derive what pieces exist in a puzzle given just the symmetry and the position of the cutting planes. :-)

Note, however, that so far we only covered puzzles that have a single Platonic solid symmetry. It is possible to mix them; e.g. a 2x2x2 (cubic symmetry) with the addition of edge-turning axes (rhombic dodecahedral symmetry). I forget the name of this, I've seen it before somewhere on this forum. Each of the Archimedeans give rise to a number of possible mixings of the basic Platonic solid symmetries, and many more are possible if you consider the compounds (e.g., the compound of 5 cubes with icosahedral symmetry, if you fill it out appropriately to be more-or-less spherical, you can potentially embed, say, five 5x5x5's into a single puzzle and have pieces that can move along any of the 5x5x5's axes).

There are also the prisms and antiprisms, although so far the only antiprism I know of is the FT Octahedron, which also happens to be Platonic. For prisms, there are two kinds of axes, the N-gonal vertical axis, and N axes for the square faces along the sides. For antiprisms, there are 2N axes for the triangular faces girding the top/bottom in addition to the vertical axis.

The Square One may be regarded as a bandaged dodecagonal prism with 3 cuts vertically and 12 cuts radially, with 8 pairs of edges in the top/bottom layer bandaged into wider corner pieces, and the middle layer pieces bandaged into two large pieces.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Tue Nov 03, 2009 12:37 pm 
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My method of defining the order is just my taste. I should stop speaking about "order" and start talking about "number of cuts per axis". That will make it clear to everyone.

So far no virtual puzzle and especially no physical I know has tempted my belief that piece sets would suffice so far. I could be wrong but I won't believe it until somebody shows an example.

bhearn is right about his concerns regarding puzzle with more than 3 cuts per axis. 3 cuts shouldn't be a problem because one is fixed in the puzzles center. Because my approach in this field had a different direction I haven't tried higher numbers of cuts.
bhearn wrote:
Note, however, that so far we only covered puzzles that have a single Platonic solid symmetry. It is possible to mix them; e.g. a 2x2x2 (cubic symmetry) with the addition of edge-turning axes (rhombic dodecahedral symmetry). I forget the name of this, I've seen it before somewhere on this forum.
This is the SuperX I mentioned.

bhearn wrote:
Each of the Archimedeans give rise to a number of possible mixings of the basic Platonic solid symmetries, and many more are possible if you consider the compounds (e.g., the compound of 5 cubes with icosahedral symmetry, if you fill it out appropriately to be more-or-less spherical, you can potentially embed, say, five 5x5x5's into a single puzzle and have pieces that can move along any of the 5x5x5's axes).

Look again at the compound you have mentioned. If you want a non-shape-changing puzzle in the shape of the compound of 5 cubes you are restricted to 180°-twists. This shape leads you once again to the edgeturning dodecahedron!

The SquareOne, the Masterball, the Pucks, ... can be classified as dihedral. Look at Jaaps page for verification.

I will ignore the puzzles with more than 2 cuts per symmetry and the hybrids until I have got the piece sets for the edgeturning dodecahedron ready.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Tue Nov 03, 2009 1:25 pm 
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Andreas Nortmann wrote:
bhearn wrote:
Note, however, that so far we only covered puzzles that have a single Platonic solid symmetry. It is possible to mix them; e.g. a 2x2x2 (cubic symmetry) with the addition of edge-turning axes (rhombic dodecahedral symmetry). I forget the name of this, I've seen it before somewhere on this forum.

This is the SuperX I mentioned.

To clarify, that was quickfur's comment, not mine. I was including the SuperX in the hybrids I mentioned problems with.

Andreas Nortmann wrote:
I will ignore the puzzles with more than 2 cuts per symmetry and the hybrids until I have got the piece sets for the edgeturning dodecahedron ready.

Ah, so you're actively working on enumerating the piece sets; great!


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Tue Nov 03, 2009 3:36 pm 
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Andreas Nortmann wrote:
[...]
quickfur wrote:
Each of the Archimedeans give rise to a number of possible mixings of the basic Platonic solid symmetries, and many more are possible if you consider the compounds (e.g., the compound of 5 cubes with icosahedral symmetry, if you fill it out appropriately to be more-or-less spherical, you can potentially embed, say, five 5x5x5's into a single puzzle and have pieces that can move along any of the 5x5x5's axes).

Look again at the compound you have mentioned. If you want a non-shape-changing puzzle in the shape of the compound of 5 cubes you are restricted to 180°-twists. This shape leads you once again to the edgeturning dodecahedron!
[...]

Hmm, you are right. So compounds don't really add to the list of possibilities, although the Archimedeans do. The spherical Archimedeans are essentially just unions of the Platonic solid symmetries; e.g., the (small) rhombicuboctahedron has 6 square faces with 4 orientations each, 12 square faces with 2 orientations each, and 8 triangular faces with 3 orientations each. Essentially just the union of face-turning, edge-turning, and vertex-turning cubes.

The snubs give an interesting twist to the mix: some of the triangular faces have only a single orientation, but you can pair them up into an effectively rhombus-shaped face with 2 orientations. In this case, they are reduced to the parent cubic/dodecahedral symmetry, with squares/pentagons with face symmetry, the triangle pairs with edge symmetry, and remaining triangles with vertex symmetry.

The remaining Archimedeans are prisms and antiprisms, which give two infinite series of symmetries.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Tue Nov 03, 2009 3:41 pm 
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quickfur wrote:
IThere are also the prisms and antiprisms, although so far the only antiprism I know of is the FT Octahedron, which also happens to be Platonic. For prisms, there are two kinds of axes, the N-gonal vertical axis, and N axes for the square faces along the sides. For antiprisms, there are 2N axes for the triangular faces girding the top/bottom in addition to the vertical axis.


But antiprisms (in general) don't work, do they? Each triangular face has two adjacent triangular faces, and one adjacent larger polygonal face. It has no rotational symmetry.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Tue Nov 03, 2009 3:59 pm 
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bhearn wrote:
quickfur wrote:
IThere are also the prisms and antiprisms, although so far the only antiprism I know of is the FT Octahedron, which also happens to be Platonic. For prisms, there are two kinds of axes, the N-gonal vertical axis, and N axes for the square faces along the sides. For antiprisms, there are 2N axes for the triangular faces girding the top/bottom in addition to the vertical axis.


But antiprisms (in general) don't work, do they? Each triangular face has two adjacent triangular faces, and one adjacent larger polygonal face. It has no rotational symmetry.

Hmm, you're right. Antiprisms (in general) don't let you mix the top and bottom pieces, so they're pretty worthless for making twisty puzzles with. :(


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Wed Nov 04, 2009 5:44 am 
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Another thing people need to realize, is that cubes are unique in that they can be produced using a tiling of smaller cubes. So if you consider the 2x2x2 cube and the 3x3x3 cube, their corresponding dodecahedral analogs are non-comparable. The 3x3x3 cube is obviously analogous to the megaminx, whereas the 2x2x2 is deep cut and the kilominx isn't. The 2x2x2 is a corners-only 3x3x3, also the kilominx is a corners-only megaminx. But the deep cut dodecahedron is a Pentultimate, which has more in common with a skewb than a 2x2x2. Yet the skewb has 4 axes and the cube has only 3! Likewise, there are no cubic analogs of a Pyraminx Crystal or a Starminx. The reason is because on a cube, all non-adjacent planes are also opposite and thusly also parallel. The dodecahedron has non-adjacent, non-paralell faces, which make the Pyraminx Crystal and the Starminx possible. As for comparing the Pentultimate (which is face-turning) with the Skewb (which is vertex-turning), one needs to consider that of all the Platonic solids, the cube is the only one with faces of even sides. With each polyhedra, excepting the tetrahedron, two opposite faces are congruent when translated and rotated 180 degrees. The square/cube, however, is the only regular solid in which the 180 degree rotation is not required when translating from one face to the other. This is what gives the unique qualities of the even cubes, which behave differently than other magic polyhedra.

The face-turning Octahedron is also in a different lot. It has 4 faces that meet at ever vertex, an even number. Thus, pieces cannot be translated between adjacent or opposite faces, they can only be translated between opposite faces. This is why octahedrons are happy with only four colors, and in fact, with the eight-color face-turning octahedron, no matter how badly you scramble it, and even though all pieces on even- and odd-numbered sides are congruent, you will never get more than four colors on a face, and the same two sets of four colors will always be grouped together. The even numbered vertices also necessitate that the rhombis on the corner pieces be split into two separate triangles on the Face-turning octahedron. The tetrahedron, being self-dual, has the identical geometry to that of the octahedron, with the exception that every other face is disabled. The corners on a skewb octahedron behave in a similar fasion to the edges on the Tetraminx/Pyraminx, despite one being deep cut and the other not. As the Pyraminx prooves, a face-turning Tetrahedron is also a vertex-turning Tetrahedron.

And just like the face-turning octahedron has split corners, the icosahedron corners would require two splits across every side creating Lord knows how many fragmetns, making the mechanism, if a practical model exists, exceptionally complex.

Another observance with the tetrahedral/octahedral symmetries is that space can be tiled with a matrix of all cubes, and can likewise be tiled with a matrix of tetrahedrons/octahedrons. Just look at the Pyraminx (4 octahedrons, ten outer tetrahedrons, and a tetrahedral core) and Rubik's Cube (26 outer cubes plus inner cubic core), the original twisty puzzles, for inspiration!

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Wed Nov 04, 2009 11:15 am 
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stardust4ever wrote:
[...snipped very enlightening discussion about the differences between twisty puzzles made from different Platonic solids...]

All this discussion makes me wish we had ready access to a space of 4 dimensions, where we can explore the symmetries of the unique 24-cell, the only non-simplicial self-dual polytope above 2D, and the awesome complexity of 120-cell symmetry. The 24-cell also tiles 4-space, so I wonder what manner of interesting puzzles we can get out of it, and what properties they would share with the 4D cubes.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Wed Nov 04, 2009 11:43 am 
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bhearn wrote:
To clarify, that was quickfur's comment, not mine. I was including the SuperX in the hybrids I mentioned problems with.
Oops! Wrong button!
bhearn wrote:
Ah, so you're actively working on enumerating the piece sets; great!
Kind of. Ask me again 28 days later. :shock:
stardust4ever wrote:
[...]Pentultimate vs. Skewb.[...]
I needed a long time to understand why somebody would relate the Skewb to the Pentultimate. Now I know it is because of the apperance but I for myself have always related the Pentultimate to Gelatibrain 1.2.9 as mentioned above.
You haven't mentioned another property of the cubes: They allow cuboids => Subgroups with 180°-turns on some faces. I don't know anything similar on the Megaminx.
And because of the faceturning cubes unique behavior this thread was started.
quickfur wrote:
All this discussion makes me wish we had ready access to a space of 4 dimensions, [...]
Well. If 6 classes (derived from the platonics), the dihedrals and prisms aren't enough, look at Gelatibrains spherical puzzles. Take the known (or additional ones?) axis configurations and don't use cycles of length 2,3,4,5 but 6, 8 or 13 and see what you get.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Fri Nov 06, 2009 8:29 am 
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Hi,

I don't have much time to contribute to twisty puzzle forums anymore, but I noticed this thread and figured it would be good to reference a previous attempt at the topic (made by me of course :) )

I attached an essay that I wrote about a year ago regarding a method of classifying puzzles. (I was going to point you towards the link in my signature, which at one time contained solutions to alot of puzzles using those classifications, but it appears geocities has closed! :( )

In summary, my idea is that often, too many concessions are made to rank twisty puzzles by order. My classification scheme is based on idenifying what I call "fundamental" puzzles which are distinctly different from each other, and whcih all other puzzles can be extrapolated by ie a) ordering (ie, 2x2x2 -> 3x3x3) or b) hybridization (ie, 2x2x2 + dino = superX).

It's not perfect but you might want to give it a look :)

Oh btw, I noticed a comment about antiprisms. Just wanted to point out that they can be cut about their edges so they can make some pretty interesting puzzles after all :)


Attachments:
Classification_of_Cubes.pdf [265.21 KiB]
Downloaded 86 times

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Fri Nov 06, 2009 10:42 am 
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Looks intersting. I have to take a second and third look.
AndrewG wrote:
Oh btw, I noticed a comment about antiprisms. Just wanted to point out that they can be cut about their edges so they can make some pretty interesting puzzles after all :)
Yes! I have almost overlooked that.
Sadly this lateral edgeturning puzzle don't present something new. I think they represent the same class as the faceturning prisms.
And since we ar speaking about these: edgeturning dipyramids and trapezohedrons belong into the same class.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Fri Nov 06, 2009 11:45 am 
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Andreas Nortmann wrote:
You see there that the faceturning dodecahedron of order=2 can have at most 8 different pieces. Each of them could be made visible in an appropriatly designed puzzle. Sadly not all at the same time.


Not easily all at the same time. With holes or transparent parts I'm sure there is some way all 8 pieces could be made visible at the same time. In the thread you link to I've already shown how it could be done with a new applet at Gelatinbrain.

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Sat Nov 07, 2009 3:00 am 
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wwwmwww wrote:
Not easily all at the same time. With holes or transparent parts I'm sure there is some way all 8 pieces could be made visible at the same time. In the thread you link to I've already shown how it could be done with a new applet at Gelatinbrain.
Acknowledged.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Mon Nov 09, 2009 2:57 pm 
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Andreas Nortmann wrote:
[...]
quickfur wrote:
All this discussion makes me wish we had ready access to a space of 4 dimensions, [...]
Well. If 6 classes (derived from the platonics), the dihedrals and prisms aren't enough, look at Gelatibrains spherical puzzles. Take the known (or additional ones?) axis configurations and don't use cycles of length 2,3,4,5 but 6, 8 or 13 and see what you get.

Actually, the spherical puzzles don't add anything beyond the platonics and the prisms. The symmetries of the platonics represent the only possible finite rotation groups in 3-space, and therefore no matter how hard you try, all finite rotation groups (in 3-space) will be derived in some way from the platonics. The prisms represent the rotation groups of the polygons, which are inherited from 2D finite rotation groups. Because of this, as long as we remain in 3-space, and are limited by puzzles of finite precision, it is not possible to transcend the platonics (and prisms), not because we lack the skill or imagination to do so, but because it is a mathematical impossibility.

Furthermore, cycle lengths are directly dictated by the symmetries of the platonics. You cannot, for example, build a puzzle of finite precision with cubic symmetry (whether it's spherical or not is irrelevant, since spherical puzzles reduce to the platonics when analysed closely) that has, say, 5-fold rotation or 13-fold symmetry around the face. It is possible to attempt to do this, of course, but you will either end up with an infinite number of possible positions (i.e., the puzzle cannot be physically built), or with some restrictions in movement (aka "bandaging" of some kind) that reduce it to one of the platonic or prism symmetries, or some combination of them.

The closest thing to transcending the platonics/prisms is to use intersecting tracks on a sphere in which ball-bearings or marbles may move, being able to switch tracks at the intersection points. You may then attach another layer of pieces over the marbles to make an outer surface (hiding the mechanism, so to speak), but if the configuration of the tracks and marbles do not follow the platonic symmetries, then you will either be unable to make a tight-fitting puzzle (some surface pieces will have to be loosely hung because otherwise they cannot be permuted into some positions that the marbles can), or some positions possible to reach in the marbles will be impossible to reach without the surface pieces colliding. This is because the marbles will be in non-equivalent positions, and it will be impossible to make surface pieces of equivalent shapes that still permit the same permutations the marbles can make. Furthermore, the marbles themselves will not be tight-fitting unless they (and the tracks) follow one of the platonic symmetries, for the same reasons.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Tue Nov 10, 2009 10:27 am 
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I have revisted Gelatibrain and seen:
You are right.
Sad, but true.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Wed Nov 11, 2009 12:05 am 
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I agree with this strategy, it's easily adopted.

Andreas Nortmann wrote:
My method of defining the order is just my taste. I should stop speaking about "order" and start talking about "number of cuts per axis". That will make it clear to everyone.


Starting off from this way, we are ready to make it clear how the cuts were placed when necessary, especially when there's a deep cut.

Given the axes of rotation, in order to state cuts clear, a series of coordinates can be used [..., x(n), x(n-1), ..., 0, y(1), ..., y(m), ...].

For example, a standard Rubik's cube can be 'order'-ed like [-1/6,1/6]x[-1/6,1/6]x[-1/6,1/6], the 'order' of a standard(equally divided layers)3x5x6 = [-1/6,1/6]x[-3/10,-1/10,1/10,3/10]x[-1/3,-1/6,0,1/6,1/3], etc. The number of square brackets equals the number of independent axes. Value 0 states there's a deep cut perpendicular to the current axis.

But why bother?
1)mXnXp cubes requires order be given to each independent axis of rotation, i.e., each axis of rotation has an order;
2)Pyramid of three layers requires two directions of an axis be distinguished as one of the cuts is a deep cut, i.e., each direction of an axis has an order;
3)The series of puzzles based on dodechedron requires that depths of cuts be stated, at least relatively;

Now, we have an 'order' for the usual petaminx like this: [-.9,-.8,-.7,-.6,.6,.7,.8,.9]x[-.9,-.8,-.7,-.6,.6,.7,.8,.9]x[-.9,-.8,-.7,-.6,.6,.7,.8,.9]x[-.9,-.8,-.7,-.6,.6,.7,.8,.9]x[-.9,-.8,-.7,-.6,.6,.7,.8,.9]x[-.9,-.8,-.7,-.6,.6,.7,.8,.9], or simply 9x9x9x9x9x9, or simply 9.

In most occasions, splitting every independent axes is not necessary due to transitivities between face/edge/corner/partial body cut from platonic solids. However, when we consider other types of polyhedrons, things will be different.

Sometimes we are easily confused by different layers(FT/ET/VT, etc), but they really have nothing to do with the order, nor does the overall shape of a puzzle. These two aspects are there for practical operation symmetries. A rotation is generally just a guarantee of 360 centigrade conservation while interesting puzzles usually accept a division of 360.

May this clear a bit of things.

Leslie

Andreas Nortmann wrote:
My method of defining the order is just my taste. I should stop speaking about "order" and start talking about "number of cuts per axis". That will make it clear to everyone.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Wed Nov 11, 2009 12:15 am 
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plus, what order do you call things like the jumble prism???

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Wed Nov 11, 2009 12:45 am 
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elijah wrote:
plus, what order do you call things like the jumble prism???


Made-to-order.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Wed Nov 11, 2009 2:39 am 
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@elijah
I'm not so much aware of jumble puzzles like 24-cube, so the following statements can be flawed.

The order of a puzzle is always determined by a set of axes of rotation together with some cuts. If any jumble was allowed, there must be a rotation hence an axis with certain cuts.

However, in most situations, jumblability(spell?) are custom made. The famous triple circle puzzle(planar) may help you understand this point. Especially when it comes to the case when any exhaustion won't stop the jumble, there must be practically an end to this nightmare(jumblability).

With all available rotations together with strange layers, follow the definition in my previous post. So, in order to answer your question, I have to know the practical jumblability of the jumble prism, right? But I don't know. I see it a customized thing.

Well, if no new axes of rotation as well as cutting planes were introduced in jumble states, the order remains constant.

@Jared
Hey Jared, as I see, you've got it! :mrgreen:

Jared wrote:
elijah wrote:
plus, what order do you call things like the jumble prism???


Made-to-order.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Wed Nov 11, 2009 2:46 am 
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I don't believe jumbling makes axes change so much as they just rearrange cutting planes, but I wasn't actually refferring to the jumbling itself as much as just the jumble prism... It has 2 crossing non-perpendicular cuts on some sides, on others the cuts go under the faces, and the cuts are placed ON the edges of the face...
I don't see under a set system what order this would be considered.

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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Wed Nov 11, 2009 3:24 am 
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Sorry for my off-topic wandering :D

Judging from this source:
http://www.shapeways.com/model/31866/jumble_prism.html

I don't see any new axes of rotation(surely) or new cuts(sure, otherwise there will be new parts created during rotation). So in *my* opinion, the order remains unchanged.

[EDIT]
I don't completely understand its geometry, so here are just the steps to get its order in my way.
1) draw all axes of rotation;
2) group similar axes with identical cuts together to name an order to each group of axes;
3) to figure the order of a particular group of axes: pick a representative, put all cuts along the axis, remember to identify the origin(intersection of all axes) as well as coordinates of other cuts;
4) practical simplification: 1)~3) provides an abstract order, to simplify it, use common rules.

Hey elijah, I'm not intentionally going off-topic. Looking into jumblability is to draw the conclusion that EVEN IF jumblability(which is customized) makes things worse, the conclusion remains, i.e., it doesn't change the order of a puzzle.

In any case, with a solid of unusual symmetries, the notation of order is not as friendly as that of platonic solids unless some convention was adopted.

Leslie


elijah wrote:
I don't believe jumbling makes axes change so much as they just rearrange cutting planes, but I wasn't actually refferring to the jumbling itself as much as just the jumble prism... It has 2 crossing non-perpendicular cuts on some sides, on others the cuts go under the faces, and the cuts are placed ON the edges of the face...
I don't see under a set system what order this would be considered.


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 Post subject: Re: The "order" of a non-cubic puzzle
PostPosted: Wed Nov 11, 2009 10:49 am 
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elijah wrote:
plus, what order do you call things like the jumble prism???

I ignore them!
Sounds unfair but looking a the "Fairly Twisted" from Oskar I think you have infinite possibilities if you allow jumbling. Maybe an axis configuration has to be added to a description / classification scheme.
I think I would make more sense to classify the puzzles with 4+ cuts per axis first. Not that I have started with that yet.


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