64 Segments of Squarings.
- It is noted that a squaring is not the same as a squire. Some say a squire is the attendant to the knight. And is known by the knight the pattern of its movement makes possible the completion of all moves of 64 upon the array of 64 when made eight square as a chessboard.
And this is done as unique movements involved with the proposition that the past cannot be revisited as a movement.
Now a sort of different proposition presents itself. In the knight's journey it returns first next to its origin as the completion of one double layer, where the path back follows the path in. It starts from one corner and folds back after reaching another corner.
Now when we introduce the analogy of this being the square windings of a square coil; and if the volts equals the no. of winds in the coil, each successive wind is separated by one volt by virtue of the length of the electrical path, which most importantly dictates the amount of electrical current conduction along the entire conduction pathway, established by the entire length of conduction path being the lowest path of conduction. Therefore by definition there is 1 volt per winding established by length of conduction path, but this does not include the actions of the associated electric field when those conduction pathways are in parallel as will be the case when they are fashioned into a multi-layered square array coil. For the first example of a laterally wound coil, each winding has one volt across it from the line connected source of current pathway, but it also has one volt at 90 degrees across that current from its adjacent winding's electric field returning to that conductor through the dielectric between windings; which in turn has the net result that two electric field vectors at right angles reduces the actual electrical impulse velocity to that below of light speed. Thus the question evolves; how low can low be practically made? How do you design a coil to have maximum sideways electric field deflection from the actual electron displacement current made from its line connections? Obviously this is done by intelligent pairings of highest and lowest voltage differences in a pattern; where having half of the adjacent voltages being only one volt obviously must be at least the lowest possible version of how to attain to this goal!
So here 64 square segments are given, each 1 ft long by ½ inch.
32 glass segments to fit between their halves of adjacent surface areas are also given, to form a glass plate capacitor of 32 inch by 12 in.
Now take only 36 segments, and 48 glass separators and accomplish the same result.
This puzzle is given to introduce the possibility of what we might call magic square technology. If the 36 segments are made into 9 square windings, the center winding can have all four surface areas for use in creation of a capacity in a 2D manner rather the the 1D plate capacity method, which only uses one side of four available for creating an "internal" capacity. In the first case here to make the 1D plate capacitor 64, or eight squared equal length segments are used. For the 2D construction method we have a vast amount of construction possibilities. Here we simply fashion the new amount of six squared or 36 equal one ft lengths into a single length. Now we decide to wind this long length into windings. Since it is the maximized use of surface area of the four sided windings that determines the rise of capacity vs no. of segments, we first see that all the external surface areas are not employed in the capacity, and only the internal ones, but not necessarily all of them will be used in the construction of this internal capacity by winding method. By creating a middle gap in the winding length, the capacity created in this 2D manner cross sectionally on the winding array depends on whether its internal surface areas are part of its own winding path or that of the opposing voltage's winding path. For this combination of 36 segments we can divide them into 9 windings of four segments per wind, but bear in mind if all 64 segments were used the larger square of 16 winds @ 4 segments/ wind could be constructed instead.
Now we can present the somewhat vast amount of winding possibilities present with the precondition that the center wind will be the middle wind, where the gap is then made between the middle winding length itself. The amount of winding options are 8*7*6*5*4*3*2 = 40,320 possible winding pathways. Apparently only four of these combinations will yield a magic square using 9 squares. With 36 segments the same capacity can be made in a 2D manner as with 64 in a 1D manner. However the magic square combination may not be the one containing the maximum internal capacity upon midpoint open circuit break.
Consider the simplest case of using four winds that cannot because of its smaller size be combined as a magic square. For four winds there are eight outside surface areas and eight internal ones, that can be made as four pairings internally. If the windings are made laterally only two of four of the internal capacities are used upon midpoint open circuit break, (MOCB). If the windings are made diagonally then all four inner surface areas are employed in MOCB.
But for the next highest square array of nine units, it should be impossible to divide the winding length in two on the middle wind and obtain a possibility where all the internal surface areas are employed in MOCB. This however could be done if the center wind was used as the fifth wind entirely on the diagonal movement side moving the MOCB point 1/18th off center. If the square array itself has an even amount of squares to begin with, the MOCB point then need not be moved from center for all the internally made capacities to be actively used by opposing diagonally based sequences based on patterns available by the white and black squares of a chessboard.
Here, considering that a coil is built by how many layers are employed, the minimal amount of layers that a coil employing adjacent windings can be is two, having two half sets of windings.
In the case of lateral windings on a nine square array, the winding layers are three. The simple reason that all the inner surface areas are not employed in a multilayered coil as MOCB capacities is that for almost all cases the winding path on either side of the center point will have an adjacent layering somewhere on its pathway. For the case when the number of winds on the square array is divisible by two, then if both winding paths were of adjacent diagonal patterns, as in the white and black squares of a chessboard, then all the internal surface areas of the winding array could be used in MOCB. In this extreme case because each winding set would have no (lateral) adjacent turns until the halfway point is reached, each half winding would have very little inductance compared to their sums in series.
It becomes useful here to employ Wheelers equations for the three cited cases of coil construction and predicting the inductance by the given parameters. Where the linear quantities of radius, length of windings in layering as either L in the case of a solenoid, or B in the case of a spiral layering, or all three of these parameters for the case of a multilayered coil; all of these three equations use R^2N^2 in the top term, which is then divided by the bottom term determined by the three parameters. Now if fewer winds are employed in the array, say for example 5 of 9 in the nine square array where the bottom term (6R + 9L + 10 B) will not change by omission of four of nine winds, but the top term does by differences of N^2 or 25/81ths or 30.8 % of its former 9 wind inductance.
(unfinished work, HDN)