Turing numbers and geometry
Here's some more stuff on the group of cube puzzles:
Stacking cubes together is where Erno Rubik started with his eventual design of an N[sup]3[/sup] object, which is cubical because it's a stack of smaller "cubes", assembled so that cubic symmetry is preserved.
Now restricting N to the numbers corresponding to the first two constructions - that appeared on shelves by the early '80s - so N = {2,3}, we see that N corresponds to the "stack depth" of a symmetrical stack, N high and N wide, so N deep. When the values 0 and 1 are included, by backwards induction we have a single symmetric object when N = 1, and none (the trivial step) when N = 0.
The table, up to N = 3, goes:
The M-number is a function of N as well. Continuing the table, up to the current real-time and real-world physical basis, and restricting N to be > 0:
[Ed: after some thought, I should correct this since the first 2 columns are "cubed" numbers, and the last column out by a factor of 3 in that case]
![[IMG]](proxy.php?image=http%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2Fthumb%2F9%2F9a%2FRubik%2527s_Cube_variants.jpg%2F250px-Rubik%2527s_Cube_variants.jpg&hash=ba9eb32d31f6b0163dac6d7346a4acdb)
We know a single "unsectioned" cube is the intersection of three pairs of parallel planes, equidistant and perpendicular, pairwise. Each of six faces of a cube has a parallel face opposite. When N = 1, E the edge # is 12N and F the face # is 6N; this is the "naive algorithm".
If you have two identical cubes and want to stack them there's an asymmetry, because "stack" implies two faces are joined. The F-number for two distinct cubes is 2xF, but two faces vanish at conjunction so this becomes 2xF - 2, or 12 faces become 10 outer visible sections (faces) of the pair. Up to 3 cubes in a stack only 4 faces can be subtracted from F[sub]tot[/sub]. With 4 cubes you get a degree of freedom, because 3 cubes can be stacked two ways, so a 4th can subtract 4 faces or just 2 depending on if the 3-stack is "inline" or "angled". Stacking 4 cubes in a layer hides the most inner faces or sections.
Ok, so there's a sort-of algorithm here for constructing a stack of "cubes with hidden faces", so you get a "middle-layer" at some point, then these inner layers multiply linearly with N the section #. The inner layers correspond to edge-sections, since they are independent of any corner sections of the N-cubes. So the group of "edgies" shows up when N goes from 2 to 3. There is a permutation map for the above layouts of sections and divided edges, the Pocket Cube page on Wikipedia has this, and there's another table of the sets of permutations organized according to quarter and "full" moves, which I have (the coset table), the aim is to map the N and M numbers to this table, by finding an algorithmic correspondence.
Unfortunately this coset table is the only complete one for the cube groups. That is, it doesn't accommodate M > 0. However it is a map of the corners subgroup which all of the examples have; this group can be used as a divisor of the larger groups, with group theory and some category definitions.
Stacking cubes together is where Erno Rubik started with his eventual design of an N[sup]3[/sup] object, which is cubical because it's a stack of smaller "cubes", assembled so that cubic symmetry is preserved.
Now restricting N to the numbers corresponding to the first two constructions - that appeared on shelves by the early '80s - so N = {2,3}, we see that N corresponds to the "stack depth" of a symmetrical stack, N high and N wide, so N deep. When the values 0 and 1 are included, by backwards induction we have a single symmetric object when N = 1, and none (the trivial step) when N = 0.
The table, up to N = 3, goes:
N | Edge sections | Face sections 0 | 0 | 0 1 | 12 | 6 2 | 24 | 24 3 | 36 | 54for N a sectioning number. The number of slices or divisions per edge, is N-1. There are "midsections" when N > 2, or inner slice-layers between outer slice-layers. The inner layers have fewer sections than the outer layers.
The M-number is a function of N as well. Continuing the table, up to the current real-time and real-world physical basis, and restricting N to be > 0:
[Ed: after some thought, I should correct this since the first 2 columns are "cubed" numbers, and the last column out by a factor of 3 in that case]
N | Edge sections | Face sections | Midsections (0)| ( 0) |( 0) |( 0) 1 | 12 | 6 | 0 2 | 24 | 24 | 0 3 | 36 | 54 | 1x3 4 | 48 | 96 | 2x3 5 | 60 | 150 | 3x3 6 | 72 | 216 | 4x3 7 | 84 | 294 | 5x3We see that N divides edges as N-1, and N-2 is the "M-number", when N-2 is zero or higher. The sections per-face are 6N[sup]2[/sup] numbers. These are the representative bases:
We know a single "unsectioned" cube is the intersection of three pairs of parallel planes, equidistant and perpendicular, pairwise. Each of six faces of a cube has a parallel face opposite. When N = 1, E the edge # is 12N and F the face # is 6N; this is the "naive algorithm".
If you have two identical cubes and want to stack them there's an asymmetry, because "stack" implies two faces are joined. The F-number for two distinct cubes is 2xF, but two faces vanish at conjunction so this becomes 2xF - 2, or 12 faces become 10 outer visible sections (faces) of the pair. Up to 3 cubes in a stack only 4 faces can be subtracted from F[sub]tot[/sub]. With 4 cubes you get a degree of freedom, because 3 cubes can be stacked two ways, so a 4th can subtract 4 faces or just 2 depending on if the 3-stack is "inline" or "angled". Stacking 4 cubes in a layer hides the most inner faces or sections.
Ok, so there's a sort-of algorithm here for constructing a stack of "cubes with hidden faces", so you get a "middle-layer" at some point, then these inner layers multiply linearly with N the section #. The inner layers correspond to edge-sections, since they are independent of any corner sections of the N-cubes. So the group of "edgies" shows up when N goes from 2 to 3. There is a permutation map for the above layouts of sections and divided edges, the Pocket Cube page on Wikipedia has this, and there's another table of the sets of permutations organized according to quarter and "full" moves, which I have (the coset table), the aim is to map the N and M numbers to this table, by finding an algorithmic correspondence.
Unfortunately this coset table is the only complete one for the cube groups. That is, it doesn't accommodate M > 0. However it is a map of the corners subgroup which all of the examples have; this group can be used as a divisor of the larger groups, with group theory and some category definitions.
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