Fractal World when squaring the circle

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Fig. 1. Animated GIF images of the same scale-of-difference of square-to-spherical texture. This sequence of images is generated by a rotation of the sphere of latitude stripes, but at fewer magnitudes of order difference than the million-to-one difference of Fig. 2.

Fig. 2. 1×10-6 scale of latitude circles from a pole view.
This work done by Don Mitchell is original and of high quality.
There is new mathematics, when one studies any of the many kernels (the measures) in integrations of this type.
If Don or someone is capable of writing out various parts the mathematics underlying the Mitchell Sphere, this will be original work. It will have publishable value.
There is the possibility that this visualization might be linked deterministically with any processor or multi processor when a minimal instruction set is governing 100% of the process time.
Paul S. Prueitt, Ph.D
from private correspondence
06:45, 24 March 2014 (MDT)

Fractal seems a fair label…

…for the surface-texture dynamics that happen across-time, where time is considered to be a sequence of evolving frames. Practical time in this sense is then the time-line of a sequence animation, such as the animated GIF images of Fig. 1 and Fig. 3.
The dramatic ending of the animated sequence of Fig. 3 shows Δ–scale-difference between image frames over time. Each frame of the sequence shows the difference-image of the spherical Moire` pattern, but with the scale of the spherical stripes shrinking (of the apparent texture as a 3D sphere, as latitude lines growing finer, with more appearing from flashing dots at the poles).

Epoch of pulsing dots

After the sequence of shrinking spherical latitude lines shrinks past the one-to-one scale-difference between the lines and the pixel-grid of the computer screen, the two grid patterns begin to interfere with each other, creating interesting Moire patterns.

The Moire patterns shrink and grow more complex, until pulsing dots appear suddenly, springing from the periphery of the Moire patterns (the apparent spherical equator). The pulsing dots each flashing at a different rate. The dots continue a migration toward the middle to fill the whole Moire patterns with pulsing dots. While the scale difference continues to shrink, a new pattern emerges at the center, and seem to swell until it appears as a new flashing dot in the center.

As the size of the pulsing gridwork shrinks sufficiently in the epoch of pulsing dot gridwork, they begin to map various pattern tapestries that are morphing between circular and square.

The pulsing dot grid-work epoch finishes at the next sudden pattern change, which I call a time wave, as it modulates how fast the patterns change at any one local in the images.

Epoch of pulsing gridwork

After a sequence of shrinking scales passes the one-to-one line-to-pixel scale difference. Different dots appear toward the end of the initial epoch of pulsing pole dots. A new generation of pulsing dots suddenly spring up form the equator of the dot to end as a gridwork of pulsing dots.

The epoch of pulsing dot gridwork begins, at about around three orders of magnitude, scale difference on the Moire maps. And there's much more.

As the dots pulse, each with a different rhythm, any certain image in an individual sequence will be showing some of the pulsing dots white, and some of the pulsing dots black.

As the size of the pulsing gridwork shrinks sufficiently in the epoch of pulsing dot gridwork, they begin to map various pattern tapestries that are morphing between circular and square.

The pulsing dot gridwork epoch finishes at the next sudden pattern change, which I call a time wave, as it modulates how fast the patterns change between images.

Epoch of time wave expansion

The epoch of the pulsing dot gridwork ends as the time wave begins to make the center patterns swell larger. As if a magnifying glass was focusing on the patterns. As they magnify, a center pulsing dot at the pole of a sudden rapidly grows into a smaller copy of the original epoch of pulsing pole dots. Clear, crisp, and surprising. This closes the fractal regularity, divided into an aeon of three epochs of pattern morphology.

Time-wave evolution of the spherical Moire pattern change dynamics

The so called time-wave emerges within the 'time-line' of images of the black and white 2-D 'pattern-surface' as a pulsing center dot. The dot grows large then expels a ring during one 'pulse' of the pulsing dot. This expelled ring continues over-time to expand as a circular ring, like a ripple expanding from a pebble thrown into a still pond. The 'ring-wave' expands until it contacts the outer circular periphery.

The time-wave is so named because the patterns, over time, change more slowly as the wave approaches that locality, then speed up again as the wave passes. This modulation of the rate-of-pattern change-over-time suggest the origin of the term 'wave.'

The expanding, concentric wave-ring approaches the outer circumference of the circular 2-D Moire patterns where it is then reflected back toward the center-point. The 'wave' looses resolution while nearing the outer periphery, and after reflection it fades into the pattern details. However, the wave permanently compressed the patterns near the periphery of the circular patterns by a slight amount. This leaves a region of a diffuse image of a ring like the flotsam at the limit of the high tide. The compressed, darker image is accompanied by a slightly rarified ring just inside the compressed ring.

Pattern condensation through the 'time wave'

The 'time-wave' ring, expelled from a center dot over-time, begins as mostly sinusoidal, but morphs as the wave-ring expands into a wave composed of a summation of odd-harmonics, or a square wave, which wave-form profile has a flat top with steep rising and falling edges.

While the time-wave is mostly flat-topped, the existing patterns it approaches expand further apart from each the other pattern element. As the leading 'edge' of the wave approaches very close to a normal pattern region, the relative scale of the pattern quickly increases, and much faster very near the edge. But the pattern scale expands isomorphically as a ring expands through the patterns. The expansion is much faster in the tangential direction around a concentric circle than the expansion in the radial direction away from center. The expanded scale within the flat-top portion of the wave-form profile remains unchanging. The pattern then morphs back to a normal scale as the falling-edge of the wave profile passes.

The isomorphic, circumferential expansion of the pattern scale within the flat-top region of a passing time-wave is sufficient to 'stretch' the pattern elements into a complete circle concentric with the spherical Moire pattern image. These new circle-patterns within the ring-wave assume a white color, or a black color, as various patterns of the whole image are traversed by the time-wave. These black or white circles create a black or white band in the color profile across the flat-topped time-wave. The narrow rising and falling edges of the color-band create an appearance of a raised-ring as a 3-D pop-up of the surface within the whole image.

Fractal regularity of a 'time-wave'

In edit:

Flotsam rings demarcate an 'epoch' period along the image-sequence time-line.

The anthropomorphic era

Almost spooky. Faces appear in the tapestry, ornately. Kings and maidens, gnomes and knights. A plethora of ink-blot opportunities of subjective impact as a totem of faces on the orthogonal and the diagonal axes from the center pole.

Algorithmic solution

After a kernel function of the effect at play is available, any of the 10^20 (thereabouts) patterns may be available for sudden generation of any certain range of scale-shrinkage. The generations of change in the epoch of pulsing patterns is of no certain sequential meaning, unless the image per frame has a shared recurrence at the scale steps of the pattern anomalies.

The kernel function will also afford selection of a sequence navigation on a certain delta-scale between frames that will essentially make any certain pattern seem to freeze over the sequence, with very little change save for the evolution of the epoch.

While these things are possible also with POVRay**, the generation time could stretch into days to prepare only fixed sequences, where the kernel function is anticipated to approach slow video frame rates (hoping for ten per second). ** POVRay ray-tracing software

I took the ray-tracer to its numeric limit, which was at a 10^23 scale ratio. Time waves are yet visible as they modulate the density apparent of the fractal dust.

Though, fractal dust yet has time waves.

Assistance for co-authorship sought

I'm seeking assistance with the calculus breakdown of the algorithmic solution, and the derivation toward an analytic function for the color of any one pixel at any one scale-rato betweem the pixel and the pattern stripe width.

I would prefer to co-author a paper with an astute mathematician for at least non-peer-review publishing. A third co-publisher is welcome with the comp sci skillset needed to render the kernel function to a flow chart level. I will code the solution.

Please contact me as for project enquirey or suggestions for a prudent approach to the project.

Thank you in advance.

A visual sequence —25 megaBytes— slow to load (but then repeats fastly)

Spherical moire 1000 shrinking anim 0.1004 to 0.0001 640x480.gif

Fig. 3. One visual epoch in the fractal regularity, as an long, slow pattern shift growing exponentially faster toward a dramatic ending of the epoch.

From the pole-view of extremely shrunken latitude circles, the pulsing dots shrink to a curved rectilinear pulsing grid…

Earth latitude longitude lines-JMalone320x435.png

Image © J. Malone.

©2010 DEM
13:49, 6 August 2012 (MDT) Narrative added 12:11, 1 January 2014 (MST)

when at the center, something organizes and swells larger, with a central growing, pulsing dot.

Spherical moire ga 0053.png

Like a an old tree that swells the crack in the side walk to an arboreal volcano over the decades of growth, the exploding, pulsing dot becomes an outwardly expanding ring. This ring expands faster than the patterns it meets, and these patterns disappear, to reappear as a condensate, per se, on the inside of the ring.

Spherical moire ga 0192.png

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As the outer pulsing dots are subdued by the expanding ring, they condense from the inner wall of the expanding time wave to a rectilinear pattern of flashing dots at the center. First sides are dripped, then the corners, and continuing alternately to reform an organized grid work of pulsing dots, themselves morphing over time.

Spherical moire ga 0364.png

The exploding dot, discovered after a slow compression over time of the grid work of pulsing dots, heralds the epoch of condensation, a quantum jumping of patterns across sudden, dynamic pattern growth that emerges as if by a magnifying glass.

Spherical moire ga 0653.png

Spherical moire ga 0850.png

Spherical moire ga 0861.png


Spherical moire ga 0865.png

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Timewaves ensue across fractally complexifying epochs…

…with the accumulation of compression bands of temporal-equivalence across the pattern changes on the surface-apparent, and which surface location lenses the evolution-rate of pattern changes, expressed as changes between images in a sequence of images, portrayed as a series on a time line of shrinking scale-difference of circular and rectangular interference patterns.

Each new time wave places another embedded band of normal time, where throughout this normal band all surface patterns evolve at the same rate across frames in shrinking-scale-time.

The temporal-normal-shells of pattern-evolution of interference patterns was observed locally by the author with rapid sequence-viewing on a set of 1000 spherical moire images, using Irfanview for Windows, viewing images in fast sequence from a folder created by clocked-sequences of POV-Ray programming code.

While viewing the fast 1000-frame sequence of shrinking-scale interference maps, the temporal-normal bands were all locked to the same, slow, pattern progression. Differently stated, there was a blur of images viewed too fast across the 1000-frame sequence. However, as bands of slow-motion, embedded circles of normal-time appear between the bands of blurring images, where an almost un-changing pattern-image lives across hundreds of image frames, which is on an order of magnitude of scale-change in the sequence.

The temporal-normal bands are the nodes of the temporal waves that blue shift as scale-difference increases into orders of magnitude limited only by machine bit-resolution and image resolution. Testing to 10-22

The temporal-normal aeon shells are the same pattern of the base pattern, a pulsing dot creating expanding rings.

Work to do…

…remains large. A kernel function of the calculus expression for the bounds-threshold sampling algorithm is pending until urgencies shift.
Reckon the conduit of self-similarity in the pattern that echos future sequence in present digital-step-shadows, and pattern half-tones, etc.
Seek discourse with experts on topic: digital-rounding-errors map texture into threshold-maps that cross digitization-steps of giant-difference measure. This exactly means that much of the image is therefore a texture in the residue of a huge modulus. So then, Fig. 2 is an image within the residue-map of a million-to-one modulus.
Note: a phase-lock on the modulus may afford a field-optical microscope for a precision phase lock on a very high frequency dynamic, with a stability invoked as a macroscopic instance, many orders of magnitude larger in scale than the source frequencies in the phase-lock. The direction of this glimmer of plausible project inclusion with the ongoing research and development of the author amounts to renditions of algorithms that operate within millimeters of ultra-switches controlling the wave-front of radio-frequency nuclear spin-waves operating in a 2D circular polarization against the self-described point-center of stochastical quantum mediation of the resonant fields.
Straining intuition to the limit, the present now affords some suspicions of future inclusions as a harmonic controller of the inner involute of a virtual particle that condenses in the coherent envelope of a golden-orthogonal torus knot, phases for ultra-switch modeling of the phase geometry used to probe for the texture of the inner involute.
The inner involute forms as a handed spiral from an enticement of molecular structures. This is the enticement for the quantum-consolidation of the mediated virtual particle, or macro-quantum-dot, maybe RF BEC, if you please.
The inner involute is sort-of-kind-of akin to the monopole charge at the center of a magnetic resonant vortex.
So as the bottom-scanning sonic-algorithms used by a dolphin to probe for shellfish under the mud of a seafloor, so might complex scanning-frequencies be able to produce categorical singularity in a sensor-grid sampling from a sampled-plane in the resonant wave form.
Complexity is presented to human visual acuity, such that the parameters of the keys of the signals are computed by the subconscious acuity of geometerized space, via a neurological port observed by the guy turning the knobs.
Yet another Oh wow! simplification this Winter, 2012, followed a design-focus on the digital version of a category engine as a spectral outlay on a computer display of the dynamics in the edge-jitter (phase noise) of a chaotic ring oscillator.
The dramatic ramifications that shock me too…
…As the modulus-residue-phase-lock of the high-order dynamic would in principle be an amplification of a lock on the texture of a measurement, any effect created by the involute center of a structured chimera within a toroidal resonant envelope is mapped directly as the vector-description of the phase-lock.

Interpreting the song bird melody

  • Some complex sequence of coherent signals over time will occasionally find an epochal-signature within a spectral envelop of the coherence.
  • Two of the epochal-signatures will afford a mathematical grip on determining the frequency of the source as a function of the second. That function is found by a test between adjacent images in the time line of animation sequences. The logic discovered in the animation dynamics entails that there will be rings in any complex pattern of spherical to rectilinear conversion. Comparison of two sequence images with an algorithmic examination of pattern-drift, will reveal —conjecture by ponderment— set of radii in the patterns with a much slower evolution rate than other parts of the image. The pattern in the patterns of the rate at which the patterns morph, is —hold your breath— wait... IT IS THE SAME PATTERN AS THE BASE FRACTAL PATTERN OF LATITUDE RINGS.
  • While searching for words and evolving the draft of this article, the simplicity of discovery of comparing two-frames for pattern-change will reveal a first estimate on the location of temporal-normal bands.
  • The number of pattern-locked-evolution-bands is the number of time-wave epochs, which yields the orders of magnitude of the scale differential of the source-signal to the sensor grid dimensions.

See also