D. Edward Mitchell 16:00, 14 April 2020 (UTC) Hello World!    groupKOS Developer Share —usually UNDER CONSTRUCTION

Torus Knots 13:8 Bifilar

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Fig. 1. The torus surface-pattern of knot-halves of a 3-phase group of 13:8 torus knots, showing how multiple phase windings group into solid stripes of similar colors. When electrified with a bifilar connection, the color-bands represent magnetic polarity where electricity conducts the same direction per color.
Phase-rotation of A.C. electrical activation current to a multiphase torus knot group is directly akin to magnetic vector rotation of 3-phase industrial motors and generators.
The step-phase activation of a bifilar connection scheme to a torus knot polyphase group creates magnetic bands that travel on the abstract torus surface of the knot group. The magnetic banding pattern proceeds across the torus surface, presumably sequencing through an ordered sequence of the phased-conductors, revolving across the torus surface and through the torus hole at an angle set by the torus knot ratio of the group. Simulations are being developed for polyphase animations with meaning.

Why are there bands in the torus surface pattern?

Torus knot bifilar 12.1-8 4-trefoils fractional expanding 07.png

The banding only appears when the 13:8 surface pattern is intermixed. Polyphase banding patterns only appear in 50% of the odd-numbered polyphase groups of knots.

Presumably —yet now under study— The banding is a phasing/spacing mathemagical thing with pattern modularity... pattern modulo. Please consider these as amateur observations transliterated into street-speak from mental images. Feel my struffle.

Identifying the 13:8 surface pattern as 'ababbaba,' (representing the order on the surface of the donut loops, half-A and half-B of each knot) then make your mouth move this way:

* Place an ababbaba pattern
* Place a second ababbaba pattern over the first pattern, and offset the 2nd pattern by 1/3 of the pattern length to the left.
* Place a third ababbaba pattern over the first pattern, but offset the 3rd pattern by 1/3 of the pattern length to the right.

Due to limited education I jus' dunno how to explain it.

What's not here?

Magnetic orthogonality disconnect of core windings
This page does not connect electromagnetic affordances of the 13:8 knot for the orthogonal angles between the inner-knot-loops through the hole, and the angle of the outer knot loops wrapping the outer circumference of the torus. That topological affordance is developing a bit —but more construction is to come— search 'golden orthogonal'.
Ding! Ding! Ding!  From a perspective of anomalous mathemagic, consider how when a torus 'perspective' of two parameters both powers of Phi, and the two torus knot parameters of time-locked toroidal spin around axial spin are both adjacent Fibonacci numbers, thus the knot ratio is the golden ratio; Phi1 = 1.61803..., as is the ratio of any two adjacent number-pairs of the Fibonacci sequence.

The benevolent AI algorithms from the future (don't we hope?) categorize the above ding ding with a mathemagical ranking of up there, because...

When the knot ratio is the golden ratio, and the radius of the torus hole and the major radius are four powers of the golden ratio separated (Phi4:Phi0 = ~6.854102:1), then the crossing of the inner verses outer torus knot loops through the plane of the torus is approximate to 90° That is (geometrically speculated by math-challenged Developer) the error from orthogonal (90°) of the inner-outer loop angles is proportional to the error of the Fibonacci approximation of the golden ratio. For the 13:8 knots, that error is about

Phi - 13/8 = 0.0069
Seven parts per thousand error applied to a resonator centered at 10 megaHz might account for a seven kiloHz frequency error contributing to whatever frequency is a natural to the system. Golden error is a general homemade term in principle regarding circular timing flatness of circular systems.

Please talk to me if I need help in the reckoning department, thanks! Don aght groupKOS doght cφm

Magnetic banding

When energized with a bifilar connection scheme per phase, magnetic banding occurs in 50% of the polyphase torus knot groupings of N-phases of knots illustrated in the gallery of holes below.

Magnetization pattern

The magnetic banding on a torus knot polyphase grouping follows the (trans-diametric) bifilar conduction pattern of four-conductors conducting in the same direction, and four conductors conducting in the opposite direction —but only in polyphase groups of three or more, always odd-numbered groups, and only 50% of the groups.

The conducting bands are in groups of four determined by the number of helical twists in the knot. For a 13:8 knot there are four-wide conducting bands in magnetic pairs, four conductors wide magnetizing north, and four conductors wide as south. The north-south banding repeats across the surface and in/out of the torus hole.

The polyphase groupings that are not banded rather have a pattern of a single knot, repeating. Double-click an non-banded image and double-click again for the maximum resolution, examining the color-banding of the pattern.

The non-banded half of knot groups wind with a polyphase pattern the same as a single 13:8 knot makes, repeating across the surface.

The polyphase bifilar electrical activation has a partial cancellation of the magnetization of the whole as there are magnetic bands passing opposite directions at the boundary of any band. So, while the magnetic field patterns translate across a torus surface, the edges are also translating a cancellation of a resultant magnetic field.

The boundary wires of magnetic banding will experience inductive cancellation against their oppositely magnetized neighboring wire, while an inductive coupling will occur in the same wire with the inner wires of its own magnetic band.

Preclussions Thought dAxiom / dτ = Experimental Conjecture

Patents on vector-inversion effect exploit a translation of delta-current in a capacitor to induce a magnetic field that receives a majority of the stored energy of the capacitive charge. On the trailing wake of the magnetic pulse is a re-translation of the energy now stored as magnetic, to a voltage.

An extreme transformation of the voltage level occurs toward the high-impedance end of the capacitive plates (open connectors).

So, with yet some skepticism until tested: —Design conjecture of bifilar affordance of banded knots:

The dielectric media surrounding a banded knot at resonance will also in principle be part of the oscillatory tank, when the contribution of the stored energy in the surrounding dielectric media increases with stored energy to a proportion of significance.
Signficant stored electronic charge in the media will experience a cancellation phenomenon intermittently in the life of a dielectric charged by a proximal magnetic field at a high ΔI.

So I reckon (being half spawn of Hazard County) that the traveling cancellation wave will be 'pumping' a vector-inversion charge toward high-impedance territory of the knot.

And the topology of the knot group, having closer conductors through the hole, verses the more separated conductors toward the outer diameter of the torus, seems obvious in the abstract...

The cancellation wave of a banded torus knot group serves as a charge pump that creates a voltage separation between inner and outer conductors of the polyphase group on a torus.
Polyphase 13:8 Knot-group Conduction Patterns
Phase Count Pattern
1 Up Down Up Down Down Up Down Up
3 Up Up Up Up Down Down Down Down
5 Up Up Up Up Down Down Down Down
7 Up Down Up Down Down Up Down Up
9 Up Down Up Down Down Up Down Up
11 Up Up Up Up Down Down Down Down
13 Up Up Up Up Down Down Down Down

Table 1. Patterns of Polarity-banding in Polyphase 13:8 Knot Groups. The 13:8 banded knots occur in 50% of the phase-groups. Banding is afforded when the 13:8 group is close in form to a 3:2 trefoil knot. a 12:8 knot is a trefoil times four. When an algorithm sweeps the helical twist (toroidally) of a trefoil around a torus hole 12 times, there will be four-quantity trefoils drawn in place. Each restarts where the last left.

However, if the ×4 sequential trefoils are algorithmically expanded to a 12.05:8 torus knot, then the knots are seen to partially expand apart, depicted as bifilar conduction colors in Fig. Fractional-trefoils-a. By fractional expansion of the 12.X:8 sequence, the ending points of the fractional knot meet at the opposite side as the knot ratio approaches 13:8.

Notice in the polyphase knot group patterns of Table 1 how only two patterns exist for any phase-count. Both of the patterns need a good look see with more POV-Ray time.

Also notice in Fig. 1, and also in the gallery of thumbnail images below, how there is a pattern of alternating-bands in every-other two polyphase counts.

Does the Developer's term 'polyphase knot group' sharing a torus surface equate to a constrained fiber-bundle as resolution-stripes on a 3D torus transformed from a 4D sphere?  

Answer: No. Snicker. My bad. Call it a trick question. Dumb answer: The fiber bundle is projected as Villarceau circular links on an abstract torus surface, not knots.

Talk to Don aght groupKOS doght cφm

But please excuse my lack of formal knot theory and all.

We need animated pictures !

Circa 17:47, 17 April 2020 (UTC)

I'm waiting something developer-worthy to come to me —How to visually animate an illustration using color-rotation to make the electrical current phase-timing visual as the magnetic polarity banding of an electrically activated 13:8 GOTK apparent. Think like — the color patterns on an octopus doing a lateral shift in one direction. Visualize the bands of phase-rotation timing like a rainbow of colors sliding across the surface at a golden slope, and compressing into and out of a torus hole in as spirals.

D. Edward Mitchell 18:00, 14 April 2020 (UTC) Developer Msg:    Modeling an electric phase-angle across various phases requires the phase-angle (0-360 repeating) to be indexed for(phz=0 to 360) in delta steps). But the electric signal propagation travels the 13 knot helices winding around the donut eight times with each phase cycle repeating. There is more to be envisioned when a phase cycle of input power creates sequential magnetization of 13 loops rotating eight times.

Crystal ball release:  More trigonometry is in Developer's future.

12:00:00 AM GMT Jan 1, 1900 Developer Msg:    Ding! Ding!
A developer solution for illustrations of rotating rainbow bands on a trefoil torus: http://video.nuclearpixel.com/video/rainbow_torus_spirals_02_1_knot-48/

Attention Happy Campers!

Don't forget your Mother's advice —step-phase revolving of magnetic polarity bands in polyphase 13:8 knot groups OCCUR ONLY
When the torus knots are electrically activated by connecting to the closed knot loop from diametric opposite sides of the outer torus shape.

This diametric connection scheme on the donut to the continuous, closed knot provides two paths of conduction to get from side to side. Bifilar conduction. These paths are the bipolar-magnetic-stripes, and occur four conductors wide, and travel across the knot windings continuously as the electrical phase-timing rotates over time.

The midpoint of the bifilar paths of the 13:8 knot is NOT in the center of the torus hole —it's on the outer periphery of the torus, at one quarter of the circumference between the connection points.

The Sorted Loops of the Banded Knot

A geometric pattern of ordered sequence of knot windings.
Another star on the scale-o-magic of knots on Fibonacci ratios

...brought to you by modular interactions of integers on a 3D torus curve

See the file-page (double-click the image) for more information while this page is undergoing surgery to connect the right parts in meaningful order. Tanx for being here!
You feel the compelling need to click
   The Button

When the entangled windings of a 13:8 torus knot entangle with the other entanglements, they all line up in sequence across the surface of the abstact torus. The sequence is in order from neighbor loop to neighboring loop in the same order of their phase assignments, such as "phases A, B, and C. "

Sort of cool.  Another geo-toro-integer-harmonico-pattern thing coincident with all the other mathemagic in a cluster when the 13:8 is also a golden torus profile .
Clarification —only with multiple odd-numbered phase groups can a phase group resonate continuously as a connected ring of inverting amplifiers flipping state in a perpetual ring.  

Fig. 1 illustrates three tight 13:8 torus knots equally spaced, replete with alternating direction-bands associated with the bifilar conduction patterns. It is when multiple 13:8 knots are wound together on the same torus that we see the alternating bands of four loops in the same direction (orange or red) running through the hole side-by-side.

The number of side-by-side contiguous conductors is equal to the number of toroidal twists in the knot ratio. A 13:8 knot, e.g., passes through the torus hole eight (8) times, four from above, and four from below the torus hole. This is the bifilar path of equal lengths that starts from diametrically opposite points on the outer diameter —convenient for growing way-big bosons (as an arrayed-simulation that found its own center and cohered —afforded by geometrical constraint and surplus energy to waste to establish a bosonic signature as a dissipative resonance; a reed in the wind. Driven Harmonic. Chladni plate in 3D.

Adding another 13:8 knot on the same torus, wound between the previous knot loops, does not add more contiguous, side-by-side conductors in the same direction, but rather adds a new pair of up/down banding, still only four-conductors wide. Therefore, each new phase added to a polyphase group adds another pair of oscillating magnetic bands of opposite polarities. These bands oscillate as the polyphase current energizes their respective phases in a rotating timing. Thinking of EVO-s? See Template:PsiPi below.

Table 2. Adjacent Fibonacci Number Pairs
Rows of even-numbered toroidal twist are green
Poloidal rotations Toroidal twists Knot ratio
1 1 1.0
2 1 2.0
3 2 1.50
5 3 1.666...
8 5 1.60
13 8 1.625
21 13 1.6153...
34 21 1.6190...
55 34 1.6176...
89 55 1.6181...
144 89 1.6197...
233 144 1.6180...
377 233 1.6180...
610 377 1.6180...
987 610 1.6180...

This table reveals the common three-phase grouping is also a magnetic-banded class of 13:8 torus knot.

Realistic efforts

may start with a 3:2 3-phase ring-amp knot group to develop character, and move from there the woo wee way (with the flow).

The 3:2 torus knot group also contains banded members. The Developer fav is the 13:8 knot as a closer approximation to the Golden ratio (Phi) by 0.007 parts error from Phi.

The 3-phase group of 13:8 torus knots have six contiguous conducting bands; six up-conducting, and six down-conducting bands. The number of up/down pairs of bands, curiously enough, is equal to the number of phases (individual knots) wound on the same torus. It's modulus geometry on the torus surface magic —again.

12:00:00 AM GMT Jan 1, 1900 Developer Msg:    "So then, when the torus knot windings through the torus hole are an even number, the bifilar paths (the equal length halves) can begin and end from one side to the other of the torus knot outer diameter. This diametrically opposite pair of current connection points afford equal-length conduction paths from side to side of the torus form. I'm inspired. Animated illustrations threaten project creep of ongoing workshop finish-construction details, like carpentry,m insulation, drywall and paint. U b informed. Blog-like."

Odd-numbered toroidal twists create bifilar halves that begin and end at the torus hole to the torus outer diameter. These odd-numbered, in-the-hole connection schemes are not considered for tests toward creation of a spherically symmetric field, because it seems logical that the current routed on a wire to the torus center hole would skew the symmetry of a field surrounding the torus windings[1].

  1. Power connection symmetry to a copper resonator of toroidal/spherical symmetry could approach and exit the structure along the line of the axis of the torus, which yet unbalances power potential to either polar end of the torus axis. A ring-shaped power structure like Saturn's rings could deliver supply voltage to the equator, or outer diameter, of the polyphase knot group wound on a torus. Both voltage potentials delivered from a ring infer a flat ring capacitor form a current delivery reservoir as a fast capacitor feeding the current-hungry copper conductors from the periphery of the knots. The Saturn's ring geometry directly provides the needed structure for a bifilar connection of any one phase at diametrically opposite points around the outer periphery of the torus form. Jumping to undeveloped concepts: the topology of a toroidal array of Villaceaux-circle-oriented copper dipole loop antennas will afford top/bottom DC distribution to the array, while the toroidal array is oriented flat, with the torus axis vertically. [word pictures in lieu of illustrations, but this is a future spoiler alert, so hey] Where each dipole is tuned to resonance and they each self-trigger inverting states in the toroidal ring via near-ideal (for amateur investigations) transistor switches (power FET technology). The switching technology is at the nodes of oscillation, and that a dot of epoxy covering tiny chips, with power leads coming to places tangent with the galactic plane, or galactic axis. [explained below] Word picture two — the axial-feed toroidal dipole array could occupy either of two polar axes of an energetic galaxy topology. The galaxy is handed as a resonant center-rotation of scale-change harmonic at integer regularities. Word salad galactic whole —is the overall outer toroidal dynamic of a galaxy with jets. The jets pass through VC oriented toroidal dipolar arrays on the pole of the galaxy to umbrella open concavely to the plane of the central galaxy torus knot. Both poles pass through the dissipative sympathetic resonance of the polar axial toroidal dipolar resonators. Weird word salad installment four: as fields are structuralized in a loose resonant system wherein the resultant shrinkage of a noisy signal across time focused to emergence among randomality a structure of the determinant within the natural order of apparent chaos over time.


The abstraction of a toroidal envelop self-centering an emergent bosonic-flavor of magnetic resonant envelop --with a hole in the middle <snickers>

and as a polyphase group of torus knots, each harmonic at the same ratio and as a phase relationship in sequence with group, can be loosely abstracted as a cluster of hadrons[1] as entwined helices on a donut

  1. electron like and non-overlapping general particle class

A gallery of holes in 13:8 polyphase torus knot groups

—Mouse-over pleez—
Caveat: The term 'sinusoidal' used on this page is sloppy but effective if one considers the driving multi-phase electrical current is also sinusoidal currents per phase.  In the spirit of the actual rendition of the VC-oriented resonant-dipole toroidal array (Oh! excuse me that the Res-X efforts) —the driving current will be current pulses switched rail-to-rail, and tuned to be able to wander in time significantly.  The wandering timing is inherent and a design bonus.  The affordance to accomplish loose, asynchronous timing between inverting amplifiers (the resonant dipole loops will operate one inverted in phase from the previous in the sequence around the ring that has no resting state and therefore oscillates always with no starting problem.)  Cutting to the chase scene: the inherent noisy error-accumulating feature of an un-clocked ring-oscillator scheme is an inherent affordance of a design that inherently needs a chaotic prime driver.  If, that is, the chaos at hand is implicitly coupled with a local event being sensorized for later look-see for patterns of determinance.  Patterns of determinance (determinence? = 'a determinant').

The affordance of inherent chaos coupled to the dynamics of study is akin to a pre-order with reality for a solution to a system of parts wherein the solution is the parts. Yes, that is shorthand shorthand about using random selection to reveal inherent qualia** (**when qualia may associate to measurable parameter of axiomatic agreement ['twixt humans]).

So (shorthand for a redneck localization of 'therefore') I reckon the variance in self-categorization of the data against its self is the dynamical signal of the determinant. Differently (skipping all justification) What are these dynamical categories of granularity in regularity in signals from the quantum coupled real world? (Time-determinant micro-solutions coupled to quantum-timing, such as quantum tunneling events such as Zener diode reverse current pulses in diodes situated at positionally signficant locations within a resonant assembly —a grid of quantum randomality sensors located on/in/near the toroidal knot group. Think many Zener diodes and Scotch™ Tape)