This is a distance table for the group (of pairwise adjacent operators), that forms the 2-generator set, which is all permutations of any pair, so a row or column of B is

There are 18 nodes which are a distance of 17 from the identity (N = 1); these 18 must each have at least one generator, which is a word like A.B. There is a set of words that will generate sufficient (or all) of the 18 nodes at r = 17, these will also have a distance from each other which is a function of the word length.

In fact here is a paper that analyzes this group of extreme nodes (antipodes). You can see what their opinion is of the logarithmic function expressed as a ratio in Table 1.

Note that the algebraic freedom to select any pair in the subgroup (which together form an adjacent pair) is a function that depends strictly on H_color.

Note also that the first transition from r = 0 to r = 1 for the subgroup yields 4 nodes at the same distance.

There are 2 directions and 2 "digits" X,Y that explain this, since 2x2 = 2+2 by way of 4 being a perfect square--hence the first "interval" is

There is a monotonicity, which is the appearance of prime factors larger than 2 or 3 in the subtotals T, indexed by n and an inner "mechanical" m, both natural numbers with countably infinite range, domain R the reals. The 4-color/2-color identity is the open "box" B with four sides colored. There are exactly two ways to leave two faces (A and B) black, one of which also leaves a dihedral element of the full group (of combined elements) completely black, this reduces the diameter of the Cayley graph, but not n or m. The twist and flip operators (which are compositions or combinations of more fundamental functions, that act on sets of elements, the 2-ary swap() and 1-ary invert() operations on words with "color indices"), leave this element unchanged after acting on it (any actions "from the left" are the same as "from the right" if the order is swapped around, or if they're inverted, the element is "at infinity" as far as B is concerned, the machine simply outputs "bb" for this element, which is at the apex--of the cone over the simplex)

**not**permuted (this implies a function like invert(permute,combine), say):
Code:

```
Number of vertices
at distance r
r log B(r) / log(r)
0 1 −
1 4 −
2 10 2.32193
3 24 2.46497
4 58 2.6427
5 130 2.84243
6 271 3.02772
7 526 3.19162
8 980 3.33333
9 1750 3.46023
10 2731 3.57449
11 3905 3.6604
12 5229 3.72191
13 5848 3.76469
14 4792 3.77948
15 2375 3.7576
16 508 3.70136
17 18 3.62837
Table 1: The number of vertices at distance r from the origin
```

There are 18 nodes which are a distance of 17 from the identity (N = 1); these 18 must each have at least one generator, which is a word like A.B. There is a set of words that will generate sufficient (or all) of the 18 nodes at r = 17, these will also have a distance from each other which is a function of the word length.

In fact here is a paper that analyzes this group of extreme nodes (antipodes). You can see what their opinion is of the logarithmic function expressed as a ratio in Table 1.

Note that the algebraic freedom to select any pair in the subgroup (which together form an adjacent pair) is a function that depends strictly on H_color.

Note also that the first transition from r = 0 to r = 1 for the subgroup yields 4 nodes at the same distance.

There are 2 directions and 2 "digits" X,Y that explain this, since 2x2 = 2+2 by way of 4 being a perfect square--hence the first "interval" is

*a perfect 4th*(think about what a*5th*,*7th*, etc, is represented by, then consider that there are 12 edges on a cube equivalent to 12 points on a sliced geodesic ball, which is the same as the number of spheres in the packing problem that Newton and one of his contemporaries never resolved or "Kepler's Spheres". Kepler believed the ratios of planetary orbits to the orbits of their moons was a harmonic equation that was a kind of music...)There is a monotonicity, which is the appearance of prime factors larger than 2 or 3 in the subtotals T, indexed by n and an inner "mechanical" m, both natural numbers with countably infinite range, domain R the reals. The 4-color/2-color identity is the open "box" B with four sides colored. There are exactly two ways to leave two faces (A and B) black, one of which also leaves a dihedral element of the full group (of combined elements) completely black, this reduces the diameter of the Cayley graph, but not n or m. The twist and flip operators (which are compositions or combinations of more fundamental functions, that act on sets of elements, the 2-ary swap() and 1-ary invert() operations on words with "color indices"), leave this element unchanged after acting on it (any actions "from the left" are the same as "from the right" if the order is swapped around, or if they're inverted, the element is "at infinity" as far as B is concerned, the machine simply outputs "bb" for this element, which is at the apex--of the cone over the simplex)

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