Probability and Statistics

[Boof-head]

What if you had a cube, with three sides visible; what is the probability that you know what the three other sides colors are?

[CraigD]

What if you had a cube, with three sides visible; what is the probability that you know what the three other sides colors are?

The question translates to, I think, how many cube arrangements have the same appearance when only 3 sides are visible?

 

I’ve an answer, but want to know Boof-head’s answer and reasoning before I reveal it. He asks a fun question, but it’s not fair to ask questions and always leave the figuring of them to others! :)

 

Notes:

• 3 sides visible mean that 7 of 8 corner pieces are visible. This determines the color and orientation of the remaining corner piece.
  
• 2 pairs of side pieces are partially invisible, so can be exchanged, but can’t have their orientations changed
  
• 1 triple of side pieces are completely invisible, so can be exchanged, and their orientation changed
  

[Boof-head]

Well, I'm here again:

 

I've been reviewing some articles about this and that; there is a well-known mathematical formalism that applies to colored cubes and color-maps in general. At MIT, someone published a paper on Rubik's toy back in the '90s, about rotation groups and conjugates in, I suppose, a color phase-space.

 

Since there are 6 colors on 6 sides and you can see at most 3, there are 3 hidden sides; the dimension is 3+3 (like position and momentum), but of colors and their geometric relations. The R-cube is a general kind of mathematical device, in which abstract colors can be "anything physical" including two electrons, on the 'inner edges' of the cube - one might find a path to another face, etc.

 

You can think of the cube as a kind of representation of stable states (fixed colors) and how they can rotate (along paths) around the space of the 'outside' of the surface. Inside is a 'hidden' sphere.

Then any particular mapping or scrambling, is a sum of paths or a path integral.

Also you can perform rotations, then perform their inverse; in between you can 'sneak in' another rotation and then undo the first set. Or for a group of rotations that have an (Abelian) inverse, there's an anholonomy if a rotation is made before the inversion. If the rotation group is, say A, its inverse A', then a further group H of rotations can occur "inside of AA' " or AHA', which is conjugation.

 

In fact a 6-color cube with 'slice groups' is a color tensor in Euclidean space (with always classical outcomes). There is a proof somewhere that, given any scrambled cube state, there are no more than 23 moves required to solve it = find the initial color-map.

[CraigD]

I've been reviewing some articles about this and that; there is a well-known mathematical formalism that applies to colored cubes and color-maps in general.

You’re correct, BH, there’s a huge amount of literature about Rubic’s cubes, and it’s very well-understood in terms of some pretty old (early 1800s, the days of Abel and Galois, something of a major time for set theory and abstract algebra) mathematical formalism.

 

I’d recommend not referring to this formalism as having to do with “color”, or especially with “color maps”, because the first is usually associated with the physics theory of quantum chromodynamics the latter with problems in geometry like the four color theorem. The mathematical notation of group theory, which is what’s traditionally applied to puzzles like the Rubic’s cube, tends to label elements more abstractly, and be divorced from the geometry, physics, and practical mechanics of devices like the cube.

At MIT, someone published a paper on Rubik's toy back in the '90s, about rotation groups and conjugates in, I suppose, a color phase-space.

You might be thinking of David Singmaster’s 1981 “Notes on Rubic’s Magic Cube”. Appearing in bookshops within a year of the cube hitting seemingly every kind of store existent in 1980, this is to my knowledge the first well-known mathematical treatment of the cube, and something of a collectors item these days. In 1981, my school’s Math department somehow got a photocopied copy of it, which got copied and floated around campus promiscuously. :) Singmaster was praised for being a really quick writer with good instincts about popularity, though it turns out he had a bit of a head start at guessing how popular the cube would be, having gotten his hands on an early Hungarian model in 1978, and written an newspaper article about it in 1979.

 

As above, I’d avoid referring to the group of possible cube configurations as a “phase space”, as this term is usually associated physical systems, including the mass and velocity of their bodies. Since few people care much about questions about the physical mechanics of cube puzzles (eg: how far does a cube face move after you let go of it, how much heat is generated by solving a typical cube, how does a cube behave when you play baseball with one :), etc.), this term seems rather out-of-place.

 

“State”, rather than “phase”, is a more conventional, and IMHO a better term.

There is a proof somewhere that, given any scrambled cube state, there are no more than 23 moves required to solve it = find the initial color-map.

Singmaster hypothesized in 1982 that the maximum number of moves (“slice moves”, which count any movement of a single slice, including one like R2, as a single move) need to solve a cube in any state was “in the low 20s”. In 2007, a computer search proved that no more that 26 moves were needed, and in late 2008, another proved that no more than 22 were needed. To the best of my knowledge, nobody has published an elementary, human-readable proof, and whether any cube can be solved in fewer than 22 moves remains unanswered. (source wikipedia article “Rubic’s cube”).

What if you had a cube, with three sides visible; what is the probability that you know what the three other sides colors are?

The question translates to, I think, how many cube arrangements have the same appearance when only 3 sides are visible?

 

I’ve an answer, but want to know Boof-head’s answer and reasoning before I reveal it.

Have you an answer yet? A simple, integer number?

 

If so (or if anybody else has one) I’d like to see if it agrees with mine – though the theory and arithmetic is pretty simple, I may have made a mistake. If nobody want to give their answer, I’ll give mine, but I still contend that he who asks it has first rights (and a bit of a responsibility) to answer it. ;)

[Boof-head]

I’d recommend not referring to this formalism as having to do with “color”, or especially with “color maps”, because the first is usually associated with the physics theory of quantum chromodynamics the latter with problems in geometry like the four color theorem. The mathematical notation of group theory, which is what’s traditionally applied to puzzles like the Rubic’s cube, tends to label elements more abstractly, and be divorced from the geometry, physics, and practical mechanics of devices like the cube.

Well, the thing is that 'color' could be 'number' too, the vanilla cube puzzle illustrates literally that color-maps can be algorithmic. There's an obvious link to group theory, a not so obvious one from the cube to coloring a sphere.

 

The algorithms are based on rotations of color; hence any colored 'piece' which is in fact, a "color-quantum", since the cube is pre-quantized or 'built' over an inner space - which is really hidden, because it's the 'virtual' sphere inside the cube, over which the pieces rotate (in cycle-groups). Also, there's an algorithm that builds a 'sliced' cube like this particular puzzle. You have to "invent a personal science" in order to solve a scrambled Rubik's puzzle, particularly the ones with 4, 5 or 7 slice geometries. Note there isn't a 6-slice puzzle (yet); there may be a reason that building a 6-sided 6-sliced cube is hard (perhaps NP-hard).

 

It's a question of finding a minimum 'initial' state and how it changes - which means how ths color-map changes. The smallest map on the puzzle is a 2x2 face, the other 5 faces have a "color-dependence" on an initial 2x2 map, of any face.

If you fix one face as a color-axis, then the 3 outer [square's] colors change, not the central 'pole' color; this is the case for any 2x2 map over the 6 faces - there are 3 sets of 'north'south' color poles, at the centre color, which we might label the x,y, and z color-pairs; each pair is a plane and all pairs are equidistant; the distance is a 2-color difference.

 

The key is seeing what a rotation does to the surface in terms of 'color', which is already a kind of deformation in the surface (in terms of 2x2 squares of fixed color with a spatial relation); the fact that the corners reach a certain distance during a rotation, or quarter turn, is a clue here - the corners with 3 colors are in a different subspace than the edges in a color-map (graph); a path must exist for any corner with 3 colors to 'find' a position next to any other color-pole, so 1 of its 3 surfaces is 'flat' (diagonally spatially adjacent), and each corner 'presents' 3 colors to 3 poles, at any path around the cube.

 

I view the cube's edge pieces as a kind of 'gear', and the corners as 'tumblers' that rotate between the gears - you can't move more than one gear at a time, the moves are "t-connected", a particular sequence of such connections is made for each 'tumbler', the edges or gear-teeth can only have colors flipped, but corners can have color-twist, or helicity.

 

Solving or scrambling a cube is finding a color-separation, of zero per face for a 'solved' cube, or maximum for a scramble. The general algorithm is contrast is minimized per face, or maximized per edge.

In fact, the puzzles are a kind of color-calculator, they have large numbers of 'states' which we might call decoherent; it's hard to determine where to start applying any algorithm that moves toward a solution, this is "more true" as the number of squares per face increases. You use 'color-matching' to build rows and columns, you solve a small part of the puzzle first.

You can envisage the puzzles as a kind of 6-part 'memory' in which each part is connected (via rotations) to each other part; each part has only 1 independent way to move that keeps its 'memory' unchanged.

 

P.S. the cubic symmetry in QCD depends on assigning mass to the leptons with charge - electrons/positrons, and not to the neutrinos. Then 'mass' is the upper vertex in the cubic color-graph; distances or 'edges' are fractional charges of quarks, etc.

[Boof-head]

As to the statistical/probabilistic aspects of the cube, this is in the problem domain of: what states correspond to what probabilities, etc.

 

You could think up any number of experiments, like say, two people start with solved 3x3x3 cubes, and perform moves one at a time, exchanging cubes each move; after a set number of turns, what expectation is there of seeing, say, only 1 particular color on any face (the central color)? How many faces should have 3 or more squares with the same color as the central square? etc.

 

Although the puzzles are fixed geometries, you can imagine a cube (or a sphere) with a large number of divisions; an 'infinitely' sectioned surface would mean an infinite amount of time to rotate a single section - of a cube, or to find the 'uncolored' area on a sphere. Obviously the puzzles are finite, but what questions do they answer (or ask), about spatiality. Why does coloring a surface 'change' it?

 

Obviously, two adjacent edges (joined to the same vertex) on the cube are 'linearly independent'. and obviously there's a flat surface between them. If you want to color this surface, what do you need to know about the adjacent two? Can you color the projected surface without seeing the others? In what sense? Can you imagine the color is a kind of mixture (a sum or difference) of the other two 'hidden' colors? What does rotating the cube, to project more than 1 surface, actually do? Is there an 'orthonormal basis', even though the colors, and surfaces are all fixed?

Does the last question correspond to finding a vector space with fixed eigenvalues (a 6-color basis)?

 

P.S. Still trying to connect all this to statistical measures and Hadamard's transform - why is it general and why does a vector space have to generate -1, from +1...?

Is this connected to a successor function, and where or how is it generated in the space?

[Boof-head]

OK, I've had time to think about this and now I remember I've seen it before - it's used in image compression as a lossy algorithm, which means certain information is discarded by the encoding - there's a way to pack the 'color-map' more efficiently so sufficient structure remains and the 'visual' image is preserved.

 

MPEG and JPEG encoding produces a map M of discrete involutions, over a digital nxm image, n,m are 'color' registers with a fixed width. The involution 'map' is generated by a recursive Hadamard transform, of a 2x2 sub-matrix over nxm.

(the Hadamard itself has a recursive define)

 

Rubik's cube is in the same domain here - you transform a 2x2(x2) subspace, which is connected reversibly to a larger 3-d space (the cube); A pattern over the 'R-cube', is a map of rotations in Euclidian space (a digitized color-map), it's a model in some sense, of the JPEG/MPEG compression algorithm.

 

Successor functions are the domain of algorithmics; you 'find' them by mathematical induction - a recurrence is induced. This is generally a static 'initial' state at t(0), with a sequence of successor states at t(1,2,3,... ,n); the latter is the dynamic part of the algorithm, but you need a generator for it, that is, a time-dependent transfer function for the computational flow.

 

There are 3 kinds of computational dynamic spaces: classical, probabilistic, and quantum. There is a topological connection between them; 'C-space' only has two dynamic states, 1 and 0 (ultimately any classical machine can be represented by a sum of products of 1 and 0) which are 'real' numbers.

I.x is also an arbitrary width binary register x_n => x in {0,1}, you can consider the connection between 1 and 0 in C-space is an arbitrary 'dotted line'. The simplest "machine" or circuit simply inverts either state but fairly complex machines can be 'built between' these two orthogonal machine-states in practice.

 

'P-space' has these two states connected by an abstract cylinder, which represents a set of values (real or complex) generated by a probability measurement. In QM this is either 1 or the empty set.

Q-space is the Bloch sphere, a transform of the cylinder, with 2 poles 1 and 0.

[Boof-head]

Another observation about the title "probability and statistics"; the color-map over a cube that the SU(3) symmetry group represents, has colored vertices - the quark colors red, green, blue appear when a 'lepton symmetry' is the abstract axis z|z', with a positron or electron at the top of a 'quark tree'.

 

The map doesn't imply an electron is "made out of 3 quarks", the quarks are colored; leptons are colorless. The particle at the vertex of 3 up quarks (an electron/antielectron) represents quark interactions. That is, the cube's corners are 3 quark pairs (up/down + color/anticolor) and an axis - the paired lepton-antineutrino for an electron, and antilepton-neutrino for a positron, both are a sum of charges, +1 or -1; mass is the 'background', edges are 'gluons'.

 

A neutron is 'udd' and decays to a proton 'uud'; the down transforms to an up quark (actually an antidown becomes an up because of the color-exchange). This is a path over the SU(3) graph where the cube is sectioned differently to the puzzle-cube, along 1/3 'height' planes, when the cube has 3 (the maximum) faces projected.

 

 

Neutron decays, as helium nuclei, are possible because of the weak vector bosons which are also abstract vertices in Feynman diagrams.

[Boof-head]

The idea that probabilities are 'governed' as it were, by the structure seen when quark colors are distributed over the vertices of a cube, so that colorless leptons have a vertex and mass and 'electric' charge are a part of this lepton-quark structure, is a little hard to grasp at first.

But 'decays' like the neutron -> proton process that transforms a quark into an antiquark are definitely probabilistic.

However, Howard Georgi's diagram (of the Georgi-Glashow collaboration), in an old SciAm issue from back before the top quark was found, illustrates as I mentioned and still am convinced of, something fundamental about reality, and so do Rubik's cubes (and now the sphere, a 4-colored icosahedral map, with all manner of regular features). They are, in one view, static descriptions of a symmetry group, and they both embed 'internal' structure - as rotation groups in the puzzle, and as interactions along edges in the SU(3) cubic color-space.

 

Georgi's diagram encodes a 'particle' at each vertex; these are 3-colored around each 'equatorial region' where there are two Euclidean planes intersecting the cube through each 3-colored vertex (at the 3 quark vertices called "up", and "down"). These two 'quark color-planes' are at 1/3 and 2/3 of the overall colorless charge distance (from the massless neutrino at the lowest, to the highest vertex, a positron, or an electron).

 

Since the +1 and -1 charges are 'obtained' by exchanging up quarks with their antiquarks, and likewise with the antidown to down quarks on each plane, the cube is a hypersurface. You 'copy' it to get a positron-electron exchange, and connecting 8 vertices from one copy to the other is a hypercube with 4 dimensions.

 

The 4th dimension is the quark colors/anticolors and the colorless lepton/neutrino exchanges. The map explains that 'mass' (which has to be determined empirically and 'plugged in' to the theories), is also an abstract vertex (or edge exchange) in a graph; in the SU(3) picture, we find the SU(2), and the U(1) symmetries. The quark-lepton cube embeds electroweak symmetry and the exchanges that mean SU(2)xU(1) is 'broken' in 3d space + 1 time dimensions.

 

As Mr Georgi goes on to explain in his 1981 article, there are a lot of symmetries in a 4-dimensional cube. The quark colors have a direction - the upper triangular plane points 'up' for up quarks, and 'down' for anti-up quarks; each color has a direction and a place at the vertices of a 3-color subgraph. This is a parallel/antiparallel tangent space, but the lepton-neutrino axis is always antiparallel (both 'null-colors' point towards the quark-planes, i.e. 'inward'), colors are never 'seen' beyond the hyperspace.

 

Weak interactions are mediated by vector bosons, the W and Z particles as exchanges between leptons and (anti)neutrinos. You need to define exchanges between 'electric' charged particles as well, which are massless photon exchanges - here though, the strong force also means color charges have to be accounted for. It's 'simpler' as the article shows, to construct a 'weak charge' and assign handedness to massive particles. However the U(1) symmetry is not chiral - photons can't have a 'spin direction', only massive particles can.

This explains why or how SU(2)xU(1) is a broken symmetry, it breaks because of the handedness of matter particles and weak interactions.

[CraigD]

Boof-head, you seem to me to have been writing a weird sort of flow-of-consciousness blog in this thread for the last few weeks. While – speaking for myself only – not uninteresting to read, I think it’s more nonsense than not.

 

There’re so many claims in your posts, it’s difficult to know where to begin answering them, so I’ll try to focus on ones that I can understand that appear obviously wrong.

 

First, the only significance of “coloring” the faces of a Rubic cube-type puzzle is to make it pretty. No matter how one represents the cube – for example, by labeling each cubie (the conventional name for an actual physical piece of a cube that make up the faces) with a number on a single face, to indicate both the cubie and its orientation, or, as in the scheme I give in post #16, label each face of each cubie independently (which allows “illegitimate” operations analogous to peeling off and moving the sticky labels – 54 of them on a ordinary cube) – it can be represented as an ordered list, and define everything it can do with a small number of permutation operations (in the case of a 2x2x2 or ordinary 3x3x3 cube, no more than 3, for a 4x4x4 or 5x5x5, no more than 4, etc.) which can be represented in exactly the same manner as the state of the puzzle. This holds true for any puzzle based on one of the 5 platonic solids (4, 6, 8, 12, or 20 faces), as well as many (arbitrarily, I think) puzzles that would be impossible to physically build.

 

It’s not much more difficult to write algorithms to solve cubes with other than the usual 6 faces and 3 slices.

 

There is in principle no limit to the number of “slices” a physical cube-type puzzle can have, though when the number becomes large, the puzzle becomes prone to falling apart, especially as it physically wears.

Note there isn't a 6-slice puzzle (yet)

Yes there is. Verdes Inovations sells a 6x6x6 cube puzzle, as well as a 7x7x7 one. This company has designed, and possibly prototyped, cubes with up to 11 slices.

 

there may be a reason that building a 6-sided 6-sliced cube is hard (perhaps NP-hard).

This is an inappropriate use of the term “NP hard”. The term refers to the way the amount of time, or equivalently, operations, required for a particular kind of computing machine, required to solve a problem.

 

Building physical cubes is a mechanical engineering problem, non-trivial only if the desired product must meet some market expectations. For example, if a customer is willing to be very careful, a cube puzzle of any size can be made using a stack of little cubes held together by nothing but gravity and friction. Most customers, however, want a puzzle where the slices turn like the ordinary puzzle.

 

I don’t think there’re any useful analogies between particle physics and cube puzzles. Because a puzzle has discrete states does not mean that it’s analogous to some or all branches of quantum physics.

 

Finally, cube puzzles are purely deterministic. If you manipulate them using random processes, you can ask probability questions with them. This is also true of mechanically simpler objects, such as coins and dice. There is nothing inherently random or probabilistic about any of these physical objects.

[Boof-head]

This is an inappropriate use of the term “NP hard”. The term refers to the way the amount of time, or equivalently, operations, required for a particular kind of computing machine, required to solve a problem.

It also refers to the way a problem, like designing a device of some kind, can be hard in terms of the time and thinking involved.

 

 

Building physical cubes is a mechanical engineering problem, non-trivial only if the desired product must meet some market expectations. For example, if a customer is willing to be very careful, a cube puzzle of any size can be made using a stack of little cubes held together by nothing but gravity and friction. Most customers, however, want a puzzle where the slices turn like the ordinary puzzle.

You're making the observation that it's trivial to construct a colored cube, so you can stack colored cubes into a 3x3x3 'stack of cubes'. Archimedes probably understood that one.

 

I don’t think there’re any useful analogies between particle physics and cube puzzles. Because a puzzle has discrete states does not mean that it’s analogous to some or all branches of quantum physics.

Except that the cube puzzles have quark and meson states according to cubists at MIT. Why do they call these cube settings the same as particles in high-energy physics?

Finally, cube puzzles are purely deterministic. If you manipulate them using random processes, you can ask probability questions with them. This is also true of mechanically simpler objects, such as coins and dice. There is nothing inherently random or probabilistic about any of these physical objects.

And here you're confirming that probability is something mechanical, in the sense we measure probabilities with devices, such as coins and dice - why is there nothing inherently random about SU(2)xU(1)?

 

The point, is that we have to build something to be able to measure anything at all; this includes a certain colored cube which is also what SU(5) looks like: two sets of quark planes (equilateral Euclidean triangles, with one color at each vertex), aligned perpendicular to a lepton axis. The planes are in SU(3), the axes in SU(2); add exchanges or 'swapping' and you're back in Rubik's cube again, because you can swap edges, in pairs over the cube. A general swap operation is in fact essential to any kind of calculation we can get a machine to do.

[CraigD]

there may be a reason that building a 6-sided 6-sliced cube is hard (perhaps NP-hard).

This is an inappropriate use of the term “NP hard”. The term refers to the way the amount of time, or equivalently, operations, required for a particular kind of computing machine, required to solve a problem.

It also refers to the way a problem, like designing a device of some kind, can be hard in terms of the time and thinking involved.

I’ve never heard or read the term “NP hard” used to describe a mechanical engineering problem. Boof-head, can you back up your claim by offer any reference to such a use of this term?

 

Such a use appears to me to almost completely contradict the meaning of NP hard. Not only can mechanical problems such as creating a 6-slice Rubic cube be solved using an easy, deterministic approach, simply by slightly redesigning a 4-slice one, the verification that the design is successful – easy to turn, sturdy, etc – is subjective, and requires actually building a prototype, and thus harder than solving the problem – that is, designing the cube. This verification can’t even be objectively defined, let alone shown to require no more than some number of seconds that some polynomial.

 

Further, you’re dodging the more obvious evidence I gave that your claim

Note there isn't a 6-slice puzzle (yet)

is simply wrong.

Yes there is. Verdes Inovations sells a 6x6x6 cube puzzle, as well as a 7x7x7 one.

In short, you appear to be writing nonsense in the hope that it won’t be criticizes because nobody can understand it. I don’t see how this furthers anybody’s knowledge of either Rubic’s cube puzzles or quantum physics.

 

You seem to be purposefully being wierd just for the sake of being wierd. :)

[Boof-head]

I think you're just being less abstract than you need to be to see what I'm trying to say.

 

The difficulty of designing and constructing a 6-slice cube is in a problem domain, which might not be NP-hard, maybe it's 'simple' as you say, in which case there's a simple way to do it. What I've seen of the actual innards, on that website suggests there are several 'inner' surfaces involved; I imagine there was a bit more to it than extending the 4- or 3-slice design.

 

What I'm saying in not so many words is that you can abstract all kinds of things, there are examples of such abstractions everywhere. The 'other' cube that has quark and meson states is the 3-colored SU(3), SU(2)xU(1) model that emerges when electric charge is added to the pairs of quark planes, the u and d quarks are a simplified form of the group; QED and electromagnetism are 'in' U(1); QFT is 'in' SU(3), and Pauli's algebra emerges in SU(2) weak interactions which are 'in' infinite-dimensional Hilbert spaces. There's a regularity in these symmetry groups (which is 'broken' symmetries), the abstraction of mass and time, and so what probability is, also emerge from the same structure - you could say they are independent parameters, or that the model doesn't account for either, probabilities evolve in unitary Hilbert spaces. SU(5) emerges from the 2 lepton axes + 3-color quark planes to get the 5 dimensions in this GUT.

 

And abstracting is fun, it's how someone figured out how to build a certain puzzle with a structure that 'embeds' a lot of math; topology depends on how you look at it.

[CraigD]

The difficulty of designing and constructing a 6-slice cube is in a problem domain, which might not be NP-hard, maybe it's 'simple' as you say, in which case there's a simple way to do it.

The point I’m trying to communicate is that the problem domain of building a cube puzzle is very different than that of describing the state of a cube. In particular, what I meant by

Not only can mechanical problems such as creating a 6-slice Rubic cube be solved using an easy, deterministic approach, simply by slightly redesigning a 4-slice one, the verification that the design is successful – easy to turn, sturdy, etc – is subjective, and requires actually building a prototype, and thus harder than solving the problem

is that you can’t apply terse, formal techniques such as use in post #16, in which you know exactly when you have verified a solution (for the usual “solve the cube” puzzle, when the cube’s representative tuple equals the identity element), but can only subjectively conclude that the physically built puzzle “feels good enough”. One person’s favorite build of the same puzzle may differ from another – something that can never happen concerning the formally described state of a cube, where disagreement about such questions as whether a cube is or is not solved is effectively impossible.

 

The term “NP” is a very specific term referring to formal problems, and shouldn’t be used a synonym for “a tricky problem in mechanical engineering”. Further, using “NP hard” to describe a physical construction problem where one can’t even precisely define “verifying the solution”, let alone describe a simple proof of it, isn’t even good use metaphor, because it makes a metaphorical connection between two ideas that have almost opposite meanings. :QuestionM

What I've seen of the actual innards, on that website suggests there are several 'inner' surfaces involved; I imagine there was a bit more to it than extending the 4- or 3-slice design.

I think Boof-head’s referring to this graphic of a disassembled 7x7x7 Vcube-7:

Folk with experience taking cube puzzles apart and putting them back together will likely see the similarity between it and the 3x3x3 and 4x4x4 cubes of the 1980s - 2-faced side pieces with two flanges like both 3x3x3s and 4x4x4s, “free floating” single face pieces supported by a central sphere like 4x4x4s, and a 6-faced “armature” - and a couple of key difference: rather than being connected to their neighbors via flanges like 1980s cubes, its 3-faced corner pieces are connected to the central sphere. The 2-faced edge pieces of its middle slice also appear to connect to the central sphere, which consists of 16 pieces of 2 shapes.

 

The first of these difference, I think, is the key design breakthrough its designer, Panagiotis Verdes, found to solve the “big even-number-of-slices cubes are too lose and prone to falling apart” problem. Where a 1980s cube is “locked tight” only by its 6-sided armature piece (which 4x4x4 and other “even” cubes don’t have), Verdes’s Vcubes are also locked tight by their corner pieces, which when assembled is effectively a single 24-faced piece. For Vcubes with even numbers of slices (eg: 6x6x6), the corners are the only pieces connected to a central sphere or armature.

 

Another distinct feature of the large Vcubes is that their faces aren’t flat, making them more spheroids with edges than true cubes. This addresses the decreased ability of adjacent slices to prevent cubes from slipping sideways when their slice is turned as the number of slices increases.

[CraigD]

I think you're just being less abstract than you need to be to see what I'm trying to say.

I’m of the “pragmatic” school of thought about math and mathematical physics that holds that you’ve got to actually do the formalism of a thing to understand it. Hence, I like examples – actually doings of a formalism – and, also being of the “less is better” school, like to strive for terse examples of things, such as the description of a 2x2x2 Rubic’s cube as a formal system of 24-tuples consisting of an identity member and 3 postulated theorems I give in post #16. I like it because it’s immediately good for something – in this case, as the basis of a general computer program that can simulate any cube-like rotation puzzle (even physically impossible ones, such as ones in more than 3 dimensions and even discontinuous spaces).

 

I don’t think I’m being insufficiently abstract, but rather appropriately explicit – though, paradoxically, even talking about being “abstract”, or “appropriate” is an example of being more informal than I like

 

Concerning the connections being implying between various unitary group and the group of the states of a Rubik’s cube, such as in

This is a path over the SU(3) graph where the cube is sectioned differently to the puzzle-cube, along 1/3 'height' planes, when the cube has 3 (the maximum) faces projected.

I just can’t see how you get from one to the other. That is, I don’t see how one can define an isomorphism between the group of the states of a Rubic’s cube and a unitary group, special or general.

 

Though post #16, which represents 2x2x2 cube states as 24-tuples, uses a multiplication operation less conventional than the unitary group operation of matrix multiplication, it’s easy to transform a N-tuple [math]A[/math] into a NxN matrix [math]B[/math], via the following function

[math]B_{i,j} =

\begin{cases}

1, & A_i = j \\

0, & A_i \not= j

\end{cases}[/math]

 

For example, the face-rotating tuple {1,2,8,6,5,21,7,22,11,9,12,10,3,14,4,16,17,18,19,20,15,13,23,24}

is isomorphic to the matrix

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

So the group generated by any collection of “initial generator theorems” is a linear group (linear groups are a superclass of unitary groups which are a superclass of special unitary groups such as SU(3)). Checking the 3 theorems from post #16 shows that their associated 24x24 matrixes are unitary (and, since none of their 0 and 1 entries have an imaginary part, orthogonal). However, their determinants, and the determinants of the states of a cube that they generate, are 1 and -1, while a special unitary group may consist of only matrixes with determinants 1.

 

My knowledge of matrix algebra is pretty rudimentary, having been applied mostly to practical tasks like 3-d graphics, optimizations (ie: linear programming), and solving systems of equations, so special unitary groups are outside of my knowledge and comfort zone, so I’m not sure if this issue is a showstopper in mapping the group of the states of a rubic’s cube to some SU(n). Until I’ve seen someone actually, formally, make such a mapping, however, I’m not sure it isn’t, so am skeptical of the claim that such an isomorphism is possible.

[Boof-head]

The term “NP” is a very specific term referring to formal problems, and shouldn’t be used a synonym for “a tricky problem in mechanical engineering”. Further, using “NP hard” to describe a physical construction problem where one can’t even precisely define “verifying the solution”, let alone describe a simple proof of it, isn’t even good use metaphor, because it makes a metaphorical connection between two ideas that have almost opposite meanings

See, I think you're being a tad pedantic with the objection to my use of 'NP-hard' in reference to an engineering problem; if engineering problems are outside the problem domain NP, or the overlap NP-hard, then "we can build anything we like, it's only an engineering problem, all NP and NP-hard problems are computer programs".

 

The anecdotal version of 'hard' in regard to designing something can be NP-hard; can you think of a way to design an algorithm that can design a cube-puzzle with arbitrary numbers of divisions, from 2, up to 7, then keep going up to any number? Is that problem in the NP-hard domain (maybe not, but I haven't seen any proofs)?

 

Since the 6-slice cube puzzle was designed recently and took this long to appear, is that an indication that it was a tricky problem (perhaps they used a computer program to help out?), or that it just didn't occur to anyone until decades after the original?

 

As to the connection to SU(2)xSU(3) here's an article on the mathematical groups in the 3x3x3 puzzle: Rubik's Cube group - Wikipedia, the free encyclopedia

 

P.S. I've seen a bit of marketing hype on the site by the Ideal toymakers, about using the cube puzzles to explain particle physics - the model does appear to reflect certain properties of unitary groups - we already know conjugation is in there, and there are subgroups (2x2 inner matrices), and 'vectors' as pairs of matching adjacent colors; put 2 vectors together with a rotation and you have 2x2 squares the same color. Since edges can match but have their outer colors reversed, this represents disjunction - you can have either edge in the same place from the pov of 1 projected face: e1 OR e2 = 1; you get conjunction: e1 AND e2 = 0, when they're oriented properly - there's only 1 correct placing that matches; this correspondence to Boolean algebra is also part of the algebraic structure, which depends on fixed colors. However alternate colorings are possible as the 1981 article I started with demonstrates (it helped me develop my own personal solution strategy, an algorithm).

 

The cubes are sectioned differently and colored in a way that means they aren't in a special unitary group like fundamental particles (as I've mentioned the SU(3)xSU(2) symmetry depends on equilateral triangular sections and an axis), but there is a relationship and the cubes are all rotation or symmetry group representations. The 3-slice is S(48), as the wiki claims - it looks like a pretty detailed analysis. Special groups are symmetry groups too, aren't they?

 

P.P.S. Sorry if the above seems a little polemical at all - I've had some coffee, and now remember that i forgot to mention (although you've uncovered it) that a Rubik's cube is not in any unitary group, except U(1) is a subspace - trivially the circle group preserves rotations of any single face, but not of the edge colors of the face which are permuted. This illustrates there is nothing 'special' about U(n) symmetry.

 

To be a unitary group, Rubik's cubes need skew-symmetry and they don't have it; a computer model could though.

[Boof-head]

Oh yeah - how does all this relate to an involutional transform in unitary Hermitian terms, in a Hilbert space of vectors? The Hadamard is still hovering vaguely on my algebraic horizon here; I think I need to get a handle on the different ways it 'works' in unitary/non-unitary vector spaces.

 

Does the colored, sliced cube have a norm? It isn't a cube made out of complex numbers, but fixed colors. Still, it's kind of mind-boggling that absolute libraries of math are 'in there'.

 

Even if you don't know about the mathematics, you still go through it when you solve a cube; writing a program that solves a cube, given any scrambled state, would certainly require some math and some thinking about "what you do" when you rotate the faces and permute color-states.