Fractal Foundations of mathematics: Axioms notions and the set FS as a model

[jehovajah]

So the reference frame that the logos response utilises is a framework of ratios. It is not a framework that principally identifies position,but one that principally identifies relations, and ratios of those relations. It simultaneously connects every region to the whole by ratios of commensuration, comparison and distinction.

By focusing i have inherently available a whole network and nest of ratios related to each other in the region of focus, and related to the whole region of focus. I have to distinguish and chose those ratios that are most relevant to me and to my purpose. This is what relativity means and what relativistic reference frames are and how i work with them, and essentially what a tensor is, and is isolated from.

A regular pattern used as a reference frame is in fact confusing t the senses, if it is not spaciometrically dense enough to hold more than 1 structure repeated iteratively. Variation is helpful because it intimates a nest of ratios that differ without monotony.

[jehovajah]

The Logos Response i realise provides an array of ratios, a structure or an arrangement of ratios, a pattern and a patchwork relationship of ratios, a framework , a nest ,a nested arrangement or pattern of ratios.....

The adjectives and adjectival phrases can continue, to describe the notion of the Logos Response and what it provides to my experience as a fundamental output from my sensory system CNS processing structure.

The arrangement of ratios are spaciometrically nested, spaciometric rotations; with "boundary crossing" signaling differing spaciometric regional relationships; and boundarisation resulting from a processing outcome of a relational analysis: so that all ratios that form a relational group have a common boundary (providing a distinction of regions can be made). That is why monotonic regularity is so hard to boundarise if it occurs on a large scale.

All spaciometric forms have spaciometric rotations, and equally i can spaciometrically rotate around all forms either fully or partially. The cone of orientation  is a spaciometric form that can be used to orientate any spaciometric form relative to an observer; and  its apex can be positioned anywhere, but principally at the eyes for: line of sight observation, and spaciometric rotation measurement.

It becomes apparent that this rich ratio structure is a data structure best characterised as a relational database of ratios. Thus it is by earlier surmising, a tensor. The logos response provides me intuitively with a rich tensor response.

There is one aspect of the tensor which is still a mystery as i write and that is the rhythm response.

The rhythm response is ultimately related to every notion of motion speed and periodicity that i can experience, and its limiting factor is system processing "speed".

My system iteratively samples the surrounding and internal notFS part of my experience, and even if the signal input is assumed to be continuous, system output has to be sequential and discontinuous. Why?, because it is the result of a periodic iterative process.

Now that period is the defining output rhythm. This output rhythm is non other than a clock cycle, but where does it come from? How does it transform from a system of "molecular" motions to a periodic clock cycle?

If all motion curves/locii are spiral at all scales it is easy to transform this through any system into a clock cycle.

I could assume a standard helical motion at a quantum scale or even a conic helical motion. The difference would be the difference between a steady state sructure to the universe or an expanding, growing subatomic structure to the universe.

Nevertheless, this does not address the question of how a helical motion at that scale beats out its own periodic rhythm. How does a spiral "know" how to spiral? How does a spin "know" when it has spun its period length?
  
This hints at a basic form of "consciousness" in a spiral or a vortex. Which is not to say that a vortex is conscious, but rather that consciousness derives from the vortex, through this mysterious property of rhythm, that is the frequency of this spiral motion, that is the vibration of this vorticular rotational motion.

However i wish to describe it, periodicity is fundamental and mysterious, and intriguing. It exists, and it exists in the spaciometric form of a vortex at its most general, and it exists at a quantum level in its most mysterious manifestation.

 The quantum clock cycle has to be some form of vortex in space, and from these vortices all rhythm and vibration and frequency transform into all that there is including computational processing consciousness and the Logos Response.

Now when i started this post i was going to end up heading in the direction that i impose a reference framework on the logos response rich tensor by choice. We do not seem to exercise that choice as we lazily use Cartesian, Polar or Hamiltonian reference frameworks, so why would i need a spiral reference framework? What would it add?

I do not fully know but i am seeing that it may add something.

I find it instructive to look at how we impose reference frameworks on the rich logos response tensor and i will go through some of that at another time.

[jehovajah]

A question i woke up with: is the Klein bottle a topogical surface for a vortex, thus enabling a vortex to self refer for its boundary conditions?

In other words can i draw a spiral vortex filament on a Klein bottle so that it retuns to the spiral filament beginning at or near where it started?

And, can stable combinations of klein bottles be formed? Or can a Klein bottle link to another Klein bottle through a "klein- moebius link" thus forming a more complex Klein  bottle structure; and would the simplest of these be a Lorenz surface? (A lorenz surface being the spaciometric surface formed by a Lorenz solution filament in space).

[jehovajah]

Feynman remarked to his wife that 1/243 had an amusing decimal expansion

1/243 = 0.00411522633744...

And there are many other anecdotes to the man's intuition and insight, and inspiration.

He inspired me from just listening to a few youtubes by him. Any way i found my number  and it is 2π-6.

I am going to call it a "spiral number" in the hopes it may turn up again when i do sort out a spiral reference frame to impose on the ratio tensor sructure from the Logos Response.

2*pi-6=0.2831853071795864769252867665590057683944

1/(2*pi-6)=3.531256652965522884896502576285279021366

10*(2*pi-6)-e*cos(√29)-e*sin(√7)=0.00009123888935780705434713174019675644282 (using degree angle measure)

and from i can derive this linking rule =

And from this i can mimic = where N is an integer, and F(x,y) is a general rule linking x and y parameters to a value parameter.

Why do mathematicians always use the same notation for 2 distinct things though related? this is what makes the brain ache! F(x,y) is the value parameters rule, the value parameter can be called anything, but its function(role) is to provide values for the working of the linking rule. It is these steps within steps that are so clearly iterative and linked that need to be celebrated as iteration, and connected to dance and rhythm and movement and music an colour if the bauty of them is to be appreciated, not balked at.

the rule is so sturdy that different linking rules can be put in each of its places and different integers. My favourite at the moment is sinx and siny.

z=x*y/(1-3/(3+x*y))

z=sin(x)/(1-3/(3+sin(y)))

z=e^x/(1-2/(2+e^x))

z=x/(1-2/(2+x))

z=x/(1-2/(3+x))

[jehovajah]

z=sin(1+1*pi)*y+cos(3+0*pi+1*pi^2)*x*y

This s trying out the aggregation based on mod(π^n). which has a natural home inside the brackets of the trig functions. This particular rule procduces a saddle rhombus, and this one produces a rhombus which when clipped is a perfect hexagon:

z=sin(1+1*pi)*y+cos(3+0*pi+1*pi^2)*x

[jehovajah]

So i guess that 2π-6 is a fantastic spiral number from the results i am getting when parametrising, using it. Its application is broad enough to include knots and weaves as well as spirals. So i am wondering if this number has anything to do with winding numbers in string theory.

From the relationships above it is no suprise that e is also a fantastic spiral number, but more used for knots and cones, than windings.

.

Graphed using free version of excellent Runiter 3d grapher.

[jehovajah]

There are a lot of things of interest in the current search for a grand unifying theory, and many theorists have conjectured from before Maxwell to now. The formulation of Coulomb's Law and Newton's law is so striking as to be deliberate. That Newton attempted to relate point masses and Coulomb point charges does not excuse the obvious abstraction in the formulae.

The formulae "speak" the same language, the linking rule for the parametric ingredients is the same, but the overall recipe is different, and one gives a gravity plum(b) cake and the other an electrostatic seed cake.

Nevertheless i have found some who unify the two and some who do not.

I suspect that the difference is in the ingredients that are left out like dimension, spin, charge distribution, permittivty (density and viscosity), and vorticular Coriolis effects, and the direction of gravity's operation.The bond energy Quanta for nuclear and chemical interaction and stability also need there place in the formalism.

We have no other explanation for attraction other than force fields, but the form of these force fields is what i being investigated. My speculation is that the vortex torus form will be found to be the best fit and the best explanation of attraction and repulsion. Gravitational attraction will be seen as a spaciometric equilibrium of the two in a distributed regional field skewed by elemental and compound chemical reactions: the electromagnetohydrodynamic laws of order, which eventually  leads to a quantum value that repulses the bound mass according to the Coulomb law, which is generally an explosive decomposition of the regional order.

[jehovajah]

It looks to me like the "Sumerians" and the Harappians began the forms that would later become the subjec of polynomials, by devising tables of squares and cubes and even higher powers. And the Harappians were familir with the sphere the cube and the tetrahedron/cone.

All were intimately familiar with the spiral, but mythologically the magi appeared to hanker after the astronomical forms of the circle or ring while some Dravidian cultures revered the spiral or serpent, a more earthly form. It is only as one travels further eastward towards Burma and China that the spiral/snake is given a truly cosmological significance, and finally combined in the Yin Yang philosophy of chinese magi/sages.

The ring/circle as a magi-cal protection stems from this style of reasoning/analysis and heavily influenced western culture, whereas the spiral/snake concept informed eastern culture.

The fractal nature of polynomial bundling appears to have been appreciated more in southern regional cultures, particularly african where arrangements of campsites can show a self similarity in the design that is not only artistic but geopolitical: the status within the community is indicated by the position in the fractal and the complexity of the fractal design of ones dwelling.

This trait can be seen in all communities plant or animal, but the distinction between human animals is whether the spiral or circle is the fundamental paradigm. Those that adopt the circle adopt perfect abstract forms in their designs, those that adopt the spiral have more natural curvilinear forms.

The proto-polynomial forms of the Sumerians / Dravidians, bases around their base 60 (mod 60^n) aggregate system tended to represent regular arrangements of squares and cubes in a fractal array, with sectors of the circle typifying the perfect fractal arrangement into divisions of 60 (secs of arc, minutes of arc degrees of arc). These special polynomials were used to base their number naming system on, and to aggregate their value namespace, and to order their whole value and rank systems. By arranging their aggregates in this way they formed the first place value systems and the first power series systems of aggregation which are truly polynomial in design and use.

Thus their counting iteration was entirely consistent with the myths they had of the cosmos and truely magi-ical, but because it did not take account of the fundamental "spiral" form/ the foundational vortex-torus form as i now percieve it , their cosmological sytem always had to be corrected as it "spiraled" ( we now call it precession) out of sync.

[jehovajah]

I have tended to ignore anticyclonic vortices, under the mistaken idea that they were not common and shortlived. However though not as stable as the cyclonic vortices in our atmosphere they are common and reasonably long lived. In fact the red spot on jupiter is reputed to be an anticyclonic vortex.

When i was flying my kite it dawned on me that the air around me did not blow as in a jetstream but in eddies and vortices, somr vertical some horizontal and some at all "angles". And these eddies transport air from low to high pressure to low pressure eddies by swings and roundabouts. The motion is always spiral or coriolis and the strength of the wind gusts depends on how big the eddies are, what the density radiation rate is, the viscosity and elasticity of the air and the temperature and pressure differentials between regions at all levels above sea level.

Ed Lorenz in modeling these systems single handedly contributed to the  modern notion of chaos and non periodicity, but also established the overarching role of the vortex in making sense of all these motions and relationships.

Anticyclonic vortices spiral outwards, but i have more to learn about how they relate to cyclonic vortices and whether they too have the torus helix form.

[jehovajah]

Spiralman has constructed spiral reference frameworks based on orthogonal spirals which are damn nice . I will explore these in time, but i am slowly adjusting to the fact that the spiral is not in competition with the circle. It wins hands down anyway, but the circle is a special spiral form, even one of the perfect forms that some look for.

In trying to standardise the building blocks of a spiral tensor reference framework it slowly dawns that spiral forms cannot be standardised like circles , cubes etc because a spiral has no constant defining unit! everything varies in a spiral reference framework!

I understand why cultural norms have been based on these standard units that do not vary, but this has denied us the tools to reference the experiential continuum easily.

From spaciometry i know that extension and rotation are the fundamental motions, and to reference spaciometric rotation a form is needed, which at the barest minimum is to standard rod extensions,joined at one end. The rods are thus relative and free to move the other end in any motion. If i relax the rod to an elastic material then i introduce a chaotic order of another rank!

So it is clear that even at this bare minimum  conditions of dynamic / mechanical consistency are to be taken into account.

However it became a point of interest as to how the cartesian tensor was formed, and the ubiquity of the triangle becomes apparent, but nowhere to be seen. One has to survey crystal lattices to find anything close to a triangle in nature .

 Clearly the discovery of the right angle and from there the triangle and from that the right angled triangle, lead over time to the general notion of an angle and then eventually to a notion of rotation around a circle. Angles and triangles appear naturally only in shadow casting when measuring the astronomy of the sun , and the massive assumptions in deriving angle and triangle from these practices pervade all our thinking about the environment i live in.

In a shadowcasting exploration the stick remains constant the shadow of the stick varies, almost elastically! But we agree that the ratios are in proportion, so the variation is not chaotic.

Thus spaciometrically the two rods is not even a natural occurrence! One rod and one piece of elastic seems to be the basis of all reference framework tensors! The triangle form is the only tool that keeps the elastic under strict proportional control!

With that in mind it would appear that the triangle and the spiral are two necessary forms for a spiral reference framework. This , because of the angle measure, implies a special spiral form called the circle is also required and of course one side of the triangle will need to be elastic.

Can i really reference all things by this set of tools and constraints?

[jehovajah]

Archimedes spiral. I like his thinking!


Although not at all rigorous  this happens when one converts a circle to a supposed right angled triangle with a height = radius and a base= circumference

Area = 1/2*c*r=π*r^2.

Geometrically the deformation to transform a circle into a right triangle is fascinating requiring the centre to unravel into a line in some way that is not obvious. No wonder Archimedes saw potential in the spiral!

[jehovajah]

Surds

geometry

primes and more prime spirals

spiral numbers and crop circles

tensor structures for data


Remember......
  Remember what?

         Remember this:
               "our lives do not spiral out of control!  click to find out
           " Rather, they spiral into  
   "an exhilirating rhythm of existence
"which is uniquely
 "our
0wn
"
'

[jehovajah]

http://upload.wikimedia.org/wikipedia/commons/9/9f/Spiral_of_Theodorus.svg

Not much survives of Theodorus' work but this is his legacy. My speculation is that Theodorus advanced the greek conception of π.

I think it was Theodorus who turned the quadrature of the circle into the right angled triangle transformation of the circle, and he did it to understand these ratios 1:√2.

These are called surds, measurements that have no archimedian proportions as they are now called. These values cannot be ratioed by finite archimedian values. However as a student of Theodorus Archimeddes appreciated the spiral form of his teachers "proof" that these values are measurable but not rational in the common usage. It is ascribed to Archimedes that quantities must be "mensurable", that is not infinitely large or infinitely small, but i think he drew on the work of Theodorus.

Theodorus and many greek mathematicians were enamoured of the triangle and its ability to decompose other forms into these basic forms. Thus it did not take long for Theodorus to realise that he could decompose the circle into a right angled triangle of the same area. If you roll the disc a full turn on its circumference you can see each bit of the area of a circle enter into a rectangle of height the radius and length the circumference.

Imagine as a the disc starts with 1/4 of the area of the circle in the box; a 1/4 turn brings in 1/2 the circle area, a 1/2 turn brings the area up to 3/4 and a 3/4 turn brings in the whole area and the final turn brings in another 1/4 area. This manoeuvre actually covers twice the area of a circle as Theodorus realised, thus the area of a circle was  1/2*r*c

What was and is more complex is to demonstrate this geometrically by triangle decomposition, this involved the ratios 1:√2 and others like 1/√2:1, hence his "surd calculator".

Archimedes immediately appreciates the basket weavers craft and the area of a circle and hoped to use the decomposition into a spiral to solve the problem.

This unfortunately was more awkward than he anticipated and so he never pursued it more, but his fascination with the spiral is well attested.

It is possible using Theodorus surd calculator to approximate the area of a unit  circle using √39 or√40.

√39+(√40-√39)/2-2pi=0.0015913521879919579970533383436099958616 =√40-(√40-√39)/2-2pi (pretty damn close!)

[jehovajah]

i have to work out how to post some spiral numbers i have done on spacetime.us as the app is awesome. The spiral numbers and the yin yang popped up unexpectedly in one this morning.

I have also just realised that a tailor's tape would be ideal for measuring lengths along a spiral reference frame. Spiralmans frameworks are where i will begin to explore,but already only two orthogonal spirals are needed to reference every point in space. The third orthogonal spiral i would use to measure orientation of the other two. thus a reference framework  is entirely possible where the parameters are displacements along orthogonal spirals. Spiralman has frameworks for the Archimedian, logarithmic, and golden ratio, fermat spiral which i will denote by .

[jehovajah]

The ear.



The Logos Response is based on the activity and functioning of the sensors, has a major contribution from the ear, and the structures in it are really interesting.

The two main structures that stand out are the semi circular canals and the cochlea. These arise from 2 "chambers" the uticle and the saccule. These join to the one auditory nerve conduit, but clearly two bundles of nerve fibres use this conduit. There probably is preprocesing that occurs along the length of this conduit before they arrive at the main brain centres for processing.

So we are already considering a ratio between the uticle and the saccule inputs from our auditory system.

The two spatial and special forms you will notice are spirals. The cochlea is a spiral acoustic chamber which is so neat! it is breath taking to think that a sea shell informs us of how our own cochlea works. This is crazy, because in one form we have a "speaker"/ sampling system that analogically seperates the pitches and amplitudes of sound to give us that crystal clear quality of pitch ratio distinction: we can hear the high notes and the low notes and those of "Mr in between". The note blends and chords and note boundaries are all distinguishable by this system. What a great natural design for a speaker system!

The spiral form orders these possibly chaotic harmonies into a ratioed musical system. A damaged cochlea, then, may mean that harmonic sound distinctions are confusing or non existent. WE normally have 2 of these. Interference phenomena among ratios seems to be a crucial processing strategy for apprehending the set notFS in a richer more informative way, and certainly stereophonic, sonar detection would not be possible without it.

That brings us on to the next spiral forms the logarithmic loop canals (so called semi circular) which are arranged orthogonally from the uticle or nearly orthogonally. How crazy is that! That is so neat and mind blowing, to think that orthogonality or near orthogonality in reality is the basis of our spatial awareness and sense of rotation and orientation in this fundamental way.

One cannot hlp but be struck at how our biological functioning imposes itself on our mathematical constructions,tools vis a vis the Cartesian tensor, the polar coordinate spherical coordinate tensors. This is to be expected as the logos response deals in comparison and distinction in ratios, thus the inate form of the ratios will always come through, as ratios like similar triangles for example are applied at all scales infinitely.

This does mean that our notions of notFS are exactly dependent on the ratios in our sensory systems, no necessarily on the ratios in notFS.  

So these logarithmic loops as orientation and rotation measures remind me of the spiral frameworks Spiralman has been able to construct, and their function in the CNS will inform how i explore these frameworks and their use in relativistic motion.

Again we have 2 of these sensors and it makes me think of 2 gyroscopes and the interference pattern from their ratios what are they detecting?

One use suggests itself in the hunt for gravity waves, That not only should the current methods use laser coherence to detect stretching and shrinking of a long pipe under possible gravity wave influence but also the interference of gyroscopes along those lengths. As i understand it lasers and gyroscopes are being amalgamated more and more in research institutions.