My snake!

From Quanta:

To take an example from topology, imagine covering a ball with stretchy netting that partitions the surface into shapes such as triangles and rectangles. The number of shapes will, of course, depend on the netting you use, as will the numbers of edges and corners. But mathematicians figured out centuries ago that a certain combination of these three numbers always comes out the same: the number of shapes plus the number of corners minus the number of edges.
If, for example, your netting partitions the sphere into a puffed-out tetrahedron (with four triangles, four corners and six edges), this number works out to 4 + 4 − 6 = 2. If your netting instead forms the pattern of a soccer ball (with a total of 32 hexagons and pentagons, 60 corners, and 90 edges), you again get 32 + 60 − 90 = 2. In some sense, the number 2 is an intrinsic feature of sphere-ness. This number (called the sphere’s Euler characteristic) doesn’t change if you stretch or distort the sphere, so it is what mathematicians call a topological invariant.

The umber two is the Markov condition including about ten scientists and their constants.  It says that in any equiparitition there is a Morkov triple with enough N to make the partitions with two curvatures to a decreasing finite error bound.

One can see that in the Maclaurin series for ln(1+x) which only converges for x=1. But the bigger point is Quanta, one of the great magazines of all time.  They are the best at following this line of logic, and the results are astounding, proves Lord Kelvin right.

One thing to mention, in my snake the color curvature overlaps along the Markov centroid.  That is where path elimination occurs and makes this a continued fraction problem. It is the point of throwing away the older, bad estimates, ther 3xy-z jump in a Markov, separating out one layer like a tensor. This is the volume to surface area relationship.   

The color operators at each Markov N-tuple node have jags on the boundary, variance in deviation. The jags show up as ticks on the Euclidean axis for fractional approximation. The tick marks are the geodesic form which allows us to dump the complex plane, they are resolutions about the unit circle. We are using the unit circle as a marked yard stick to two continued fractions. We get this effect by choosing N large enough and causing compression from the Markov N to the given N.

Knot theory is an equivalent formulation of partitions across independent sets. Holes resulting from multiple curvatures in closing surfaces. The curaures represent set matching resolution.

Log for example, is the sum of all set resolutions available from one set of x things taken one at a time, two at a time, up until x at a time. It is Euclidean and lets x take irrational values, equivalent to letting dimensionality to to infinity.   The -iLog2(i)  find the deviation count needed to make the fraction i go around the snake once and count to the nearest integer. This compute the relative size and frequency of the yardstick mark on the geodesic.   The marks on the yard stick should correspond to point in my snake where colors overlap. Spots where a more accurate deviation count is encountered.