The occasional interaction along the way.

Originally posted by author:

I want to post a note about the iterative structure of the rod and cone sensors and how they are formed for their function and provide stepwise variation in signal output which is akin to digital sampling. Thus the notion of analogue and digital evaporates as a biological sensor can be shown to demonstrate digital sampling like all of the camera sensors which are in use nowadays. Plus how a drop of oil provides an interferometer /diffraction system that isolates "colour" frequencies and enables a colour signal output from a cone, while a rod provides a contrast ratio  signal output.

It's all iteration, its all biological and it's all there, man!

The rod cell
[img]  .The cone cell

Image from The Internet Encyclopedia of Science.

The rod and the cone cell develop  the rod and the cone from a  cilium. This hair like structure is associated with the mitochondrial cell elements and are usually spiral or helical in form and function. Thus a cylindrical helix and a conical helix form are the precise descriptions of these cell segments.

The helical form enwraps regions which are here called flat and parallel unit membranes but which in fact are helical "slices". These slices contain the distribution of the photosensitive pigment; in the cone they contain 3 pigment types and in the helix just one.

This spiral arrangement of pigment provides the analogue to digital sampler framework. The heliical forms both the cone and the helix act as collectors of light signals (photons) and the light enters into them in the human eye from the base of the image. Immediately the role of the oil drop in the cone becomes apparrent as does the cone shape. The oil diffracts the light into its constituent colours and refracts it to different  parts of the cone. Thus each part of the cone registers different colours and has a different photosensitive pigment concentration. It also digitises the analogue signal to that part of the cone.

So the helical sensor due to its size and shape is capable of measuring intensity of light to a very fine contrast ratio and the analogue signal is digitised by the amount of pigment that gets changed increasing the electron transfer to the helical wrap spaciometrically. Thus the duration for a helical element to transmit its charge to the mitochondrial elements in the rod cell is also measured via the "spiral length".

Intense lights therefore convert more pigment and take longer to fade the signal output. The mitochondrial capacitors are slower at discharging because there is more of them, but the ones in the conical helix structure discharge more quickly because there is less of them. Consequently the conical system is faster and able to deal with finer detail as well as colour but not as sensitive to intensity or contrast as the helical structure because the "raw light" is impinging on the helical unit layers while the diffracted and thereby diminished light is impinging on only a part of a conical unit layer.

The complex functioning that produces The Logos Response i have discussed elsewhere, but suffice it to say that despite the sophistication of the sensor it is down to the processing and wiring as to how the signal is managed; and at every stage we can identify a reduction in informaton content as the signal is transduced to make its way to the CNS.

Thus whether we liike it or not the setFS is only a poor model of the set notFS, and human beings who claim extra perception may in fact be able to demonstrate this in their wiring diagrams, processing algorithms and perception schemas, to name a few.

My joy however is to point out the presence of spiral and vorticular structures in the biological solution to signal processing and to suggest that these forms may be useful in modern electronics.

Originally posted by author:

"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"

This maneuver is not unique to the circle, but in fact relates the perimeter of a convex 2d shape to its area. By finding the geometric centre of a shape and measuring the perpendicular distance to one of it sides that gives the height of the rectangle measure the shape will rotate into along its perimeter. The area of the rectangle is twice the area of the shape.

Hello Hermann

I apologise frequently to you and others about the difficulty in following my thread, or particularly my train of thought. In part as i have explained before this is due to the intuitive nature of the writing, and the declarative style that i seem to have. I have also to point out that for me this is a journal, a noting of random thoughts and intuitions that drive me to write before the notion fades back into the vast computation that surrounds me, and of which i partake but a few morsels, crumbs of consolation for my tardiness and human frailties!

Also, as for a long time no one has been moved to interact with the thread i have not needed to clarify anything much except when i am moved to do further work on an idea. Sometimes i find myself inspired to write a full description of an idea, a truly satisfying or sometimes driven experience, only to find i have written the idea down somewhere else already! Some of the ideas i admit would be better spun off into their own threads, but i do not propose to impose on our host by multiplying topics endlessly.

Because this thread purports to be about Axioms i have had to spin off the axioms just to provide a more straightforward access to them, but as you will notice my housekeeping is not the best, and i need to tidy up some of the numbering system. I will get to it, just as i eventually get to my typos, which are frequent and occasionally interesting: for example Axiom! i have never changed it because it says something i cannot define!

So when i started it was not out of the blue but after years of self reflection and working through practical philosophical issues about the nature of knowing: Epistemology and religion and language and communication etc.

Man i had a lot of issues! But any way the epistemology had to start with me, because i had no idea if anything else existed independently of me. But of course this is not satisfactory because "me" is a construct developed over "time", etc and "space" is a construct i develop over time.

I had got as far as the concepts of infinite possibility space condensing into probability space instancing in a particular statistical reality space,from which and in which i have my experiential continuum. This was driven by a common religious concept of "ALL in ALL", and "in whom we have our being". Also the infinite pattern formations were inescapable, but the epistemology of pattern recognition or perception was not clear.

Then i came upon Fractals, and immediately found a resonance. Space then became infinite fractal possibility space, etc.. I literally woke as if from a dream and began to look at this thing called fractal. Who should pop up bur Benoit Mandelbrot and then Arthue C Clarke. Arthue i could understand having read his science fiction from a child. Benoit i had come across earlier in life, seen the clouds and the mountains , been put off by the mathematics and put it back in the library as crazy abstract mathematicians stuff! (it seemed arch and Facile and it still does, but at least i have the resources now to get at the basis of it- i intend to explore dimensionality along with parametrisation).

I had a long term beef about numbers and why "i" could not possibly be a number imaginary or otherwise. Thus i started my exploration into FractalSpace, and the foundations of maths, seeing that "Fractals" were fundamental to the order of all things.

I start off all Axiom this and corollary that, quite the mathematician , darling! A bit up myself? Not really, just Autistic and not sure where i was going who i would meet along the way, how out of date my ruminations may be etc. I had assumed that the artists here would all be jobbing mathematicians. How ironic and special to find that they were players and playful! Just what i needed.

The thoughts range over my areas of insight and interest, and some do not apparently appear mathematical. They probably "ain't" in the old sense, but since i discovered, uncovered The Logos Response all things are mathematical or strictly "ratios".

My current aims are to fully explore the Logos Response and its impact on Epistemology, to continue to review the basic "mathematical" structures of which Spaciometry is primary as a fundamental output of the Logos Response, and in conjunction with spaciometry algebraic thinking. Algebraic thinking is how i formally construct and apply and deploy axioms, definitions, transformations, products of transformations, theorems, notation and rules of computation or manipulation,mapping, iteration, boundarisation, enumeration and transformation rules.(Some of these overlap).

from these Algebras i derive specific arithmetics which are the fundamental computational tools i use in everyday calculations of one sort or another. Arithmetics can be very intense and algebraic at times, so i welcome the tools of computation now available, especially symbolic logic applications.

Calculus is usually marked out on its own but is a type of arithmetic for dealing with change, so modern computing platforms that can animate those changes are a fantastic aid.

I have found out a few things of great interest so far, and i need to note something in more detail about the change from ratios to Fractions, and thus the whole misleading conception of number leading to the complex numbers so called.

A lot of the "genius" of early mathematicians is due to the fact that they thought in and were fluent with ratios and proportions. For example i recently found that the greeks were not squaring the circle to find its area they were triangulising it (if such a word exist!). This lead me to observe a curious fact between rotating the perimeter of a convex 2d shape into a rectangular area, and the area of the shape. I am currently wondering if a similar relationship exists between surface area and volume, or is it just rotation around a given axis.

Hope you find this helpful Hermann. Essentially the whole of my thoughts are to explore as rigorously as i can the fundamental function of iteration in defining and developing our mathematical ideas, and as a consequence the inevitable fractal nature of everything, but this time not in Benoit"s sense but the modern sense of self referencing, self similar patterns at all scales.

Originally posted by author:

Hallo Jehovajah,

thanks for all the commends on my work here in this forum.
When I read this thread I get a lot of new Idears cause some things are very fundamental but I have a lot of Problems to followe your train of thoughts.

So let me ask you at the begining some simple Questions:
What does the abreviation FS mean? Fractal System?

For me a set is something like this:
{1, 2, 3, 4, 5, …}
But what is a universal set?

For me a fractal system is something like:
zn+1 = zn2 + c;

When I look on the set {1, 2, 3, 4, 5, …}
I have the Problem with the … which means Infinit, which is something I canot measure and what is not computable.
So I would prefere to have something like:
{1, 2, 3, 4, 5, … n}
From this kind of set it is easy to come to the set:
{1, 2, 3, 4, 4, … n+1}
So I will be possible to define a lot of Maths in a recursive way like a fractal.

Is this the direction in which your idears go?

Not really. Set theoretic descriptions are a language algebra with operators in the notation and rules of the set. The set you denote    (see how the language has to change to become rigorous?) is not a real set except by definition. You won't find it at the bottom of your garden for example. You won't even find it among communities of religious monks dedicated to chanting and counting! So what is it you are denoting? My contention is that this is not a set but a cultural iteration called by me +1. But now perception comes into it. What do you perceive you are looking at? I do not know, and any way is it the same as what i perceive or intended to communicate? Again i do not know.

Only yesterday i was struck by the enormous assumption we are taught to make through number bonds: 20 *20 = 400. Is it really? What the hell have i just written? some marks on your screen, some buttons i have just pushed, and all of us reading that agree that it is correct. How did i do that?  Did you do in your head what i did? i did 2*2 =4 put 00 on the end you get 400, but i could have done 2+2=4 ,10*10=100, these are 4 hundreds, that gives 400! i might even have done 2+2=4 put 00 on the end that gives 400. Why do these numbers do that? Why is one wrong and the other right, but they both give me the right answer?

So i never could understand Russels Problem, until i realised it stemmed from the flip flop of undecidability. Undecidability exists in the real world and we solve it by making a real decision. The real decision is a bit like Archimedes principle, we avoid these "unreal" ,not everyday abstractions and deal with what is quantifiable. This is what you have done. But hey when you make that decision you set up consequences that shape your world, a bit like Schroedinger's cat!

The only set of all things that can include itself is the set of all things. And as soon as you realise that you realise the iterative nature of all things and how one condition drops out a universe of consequences from that set which partition that set of all things. Thus as soon as consciousness or perception is allowed the set of all things is partitioned by that perception. This essentially and abstractly is the effect of the Logos Response: the set of all things is ratioed, that is partitioned, proportioned and related irreducibly in some way to a portion of the whole.

I find now that i am comfortable with iteration being foundational to every perception and notation, and thus the set with the ellipsis means an iterative process is being evoked to describe what goes in between the brackets. Your alternative definition is equally valid and in fact i would define away the issue by formally making them equivalent, much in the same way that 0.99999999999999……. is formally equivalent to 1.

My whole thesis if you like is that mathematics is a recursive system, that is all of mathematics is iterative, and finite systems are only stages in the overall scheme of things.The setFS was initially going to be a Universal set, but then i realised that it was really a model of another set that is universal: notFS. SetFS then has really become my mental model of reality if you like, and the place where all paradoxes live! Thus it represents discovered knowledge particularly overtly iterative.

There is a lot more that can be done to make this obvious, and a lot of overlap with computability. I am happy and grown up enough to accept maths as a subfield of computing, despite the obvious irony. In "truth" maths has always been at the behest of one patron after another, like music,so where is the problem?

Originally posted by author:

I can begin to address the issue of the concept number. At the heart of it must be spaciometry. I have abstractly refered to the tensor spaciometry as quantity and "number". The relational ratios in a tensor encapsulaing Quantity and the boundary of the tensor encpsulating unity or one.

Numbering is a simple naming activity, but the naming activity can be made complex or rhythmic or repetitious or systematic. It is these intuitions that cultures bring to their numbering that inform their concept of number. Value is also attributed to their cultural counting /naming iteration, and that value is proprtioned throughout the whole process of numbering so that each number may hold an ordinal value or a cardinal value or both, and a rank according to the proportion of value. The map between value and number is spaciometric and thus provides a circular or tautological basis to value. The source of value lies within our own neurology and is culturally maintained, defined and standardised and enforced, at least in weights and measures and SI units etc.

Thus number becomes a name that identifies a stage in a cultural iteration onto which a culture encrusts many meanings, all of which reflect a spaciometric attribute of the many tensors in space.

Given this description i venture to add that the attempt to tie number mathematically to one abstract tensor, a linear fractal called a line distorted the concept of number and confused those who had cultural attachments to number. Newtons tutor John Wallis was principal in achieving this and despite the neatness of it the underlying fractal has come to the fore when mathematicians were not ready for them in general. Thus Cantor, Julia, sierpinski, Peano, all gutturally felt these fractals as monsters and horrors eating away at the basis of reality and of course mathematics.

The area of solving arithmetic problems using algorithms led to the development of Babylonian binomial equations to trinomial and quartic and eventually quintic. The increase in the number of terms in the equation reflected the effect of iteration on these algorithms as they described relational aspects in spaciometry,and the systematic relations that underlie manipulations. This "attacking" of a problem by "manipulations" is a very militaristic paradigm, and underlies all the notions f combination and permutation, issues that would very much concern the militaristic mind through the ages, but also the commercial or merchant mind would consider these aspects of the spaciometric tensors under its hand.

This rich appreciation of number and value is what the number line threatened, and that is why it was a tool for mathematicians per se. Fractions and the numberline are where mathematicians withdrew contact with the general culture and began to distinguish mathematics as a specialist field of study with certain enforced tools.

Fortunately for us the iterative nature of reality put a cold hand of dread on them and hopefully will prevent mathematicians from disappering up heir own anus!

So in the times of the great Taxonomists the subject of mathematics came under taxanomic scrutiny, and among other things the taxonomy of equations was updated to "polynomials". Mathematical reference for the body of knowledge to do with algorithmic solutions to quadratic, cubic, quartic and quintic equations became subsumed under the heading polynomial of rank or order 2,3,4,5 etc. The term binomial had existed prior to this for a while , so this represented a tidying up of the taxonomy for ontological purposes.

Early on in the development of the solution to the equations surds had been encountered as solutions. Surds are purely geometrical values, in that they naturally arise in euclidean geometry of the right angled triangle. The very name of the equations quadratic and cubic testify to the geometrical basis of these algorithms. Going beyond the cubic meant that no geometry informed the solution, Thus it made solution harder and less intuitive and relied much upon "abstract" relationships and symbolic manipulations,and analogy of form. Essentially try to view the quartic and quintic equation as some kind of quadratic or cubic one. That is simply to utilise the spaciometry of the day to intuit the solution.

Without formally recognising the difference mathematicians had come across a type of value in solving their equations which were geometrical, ratioed and measured, not counted. Thus they were not numbers, nor were they the ratio of any common or archimedian numbers. They were thus called surds and meant "geometrical measurements".

In the course of this feverish activity mathematicians came across a curious surd √-1. As mathematicians new general surds had a value they did not reject this as meaningless, but as some geometical measurement they did not yet understand. They were necessary for many solutions of quartic and cubic equations and so had an algorithmic value.

It was not until Argand that their geometric meaning was hinted at, and by then the number line had queered the pitch and the surds had become irrational Numbers, rather than geometrical measurements. it could not be seen for a long while that √-1 was a geometrical measurement of rotation. It is still not appreciated as that even today.

Due to abstract and symbolic manipulations some mathematicians had developed algorithms that gave solutions to the quadratic and cubics both as numbers and surds, particularly when the negative number rules had become well established. The negative numbers were another geometrical value, but because they were defined  in terms of a balance, and from that the commercial bookkeepers financial sheets/ tablets, their geometric meaning  was obscured. Their geometric meaning is in fact a rotation through π radians, if one accepts the number line.

The chinese and the indian mathematicians had a good understanding of them in the commercial context, and in the context of quadratic equations, but they were not easy to accept just as √-1 was not easy to accept.

It is of great importance to realise the spaciometric origin of these quantites and how they do not exist without the awareness of a mathematician and his/her paradigm. The concept of number is a cultural totem, based on identifiable tensors in space. The relational ratios in a tensor quantity are key to the distinction i am about to make:measurement and distinction .

The Logos Response provides me with measurements of ratios. These ratios are a field effect in my experiential continuum, and i respond to them by processes within my CNS and Peripheral NS with an action that boundarises regions in that field, based on comparison of the relativistic motion attributes within those regions. Thus the field of ratios from the Logos Response is a Motion Field. Although i cannot say much more about that yet i am working on it in the thread on the Axioms of setFS. Nevertheless the point is that Measurements of ratios not Counting is the fundamental response to the motion field in the set notFS.

The distinctions we make by boundarisation are the source of our language response. Thus our language response holds the bounded distinctions in and among  the tensors. One aspect of our language response is the identification of plurality, which means the recognition of more than one and the recognition of repetiton of bounded regions: identical, similar, or none similar. At the same time i recognise the relativistic relationships between these regions in this plurality. Thus the spatial arrangement is inherent within this notion of plurality. Thus to sum it up almost the first notion that arises through the logs response is a measurable spaciometry; the second notion is a languaged spaciometry and the third notion is a countable spaciometry, in that order.

All Founding mathematician exclusively engaged with the spaciometry in doing and thinking about their mathematics. Thus while a region is real when it is in front of one, it is also a real memory that can be in front of the mind at the same time , Recognising this the indian mathematician in particular were able to conceive of debt as an absent region, a re-balancing of scales or the filling in of a hole, or the removal or changing of a colour. The chinese used coloured rods to represent a region that was removed from the direct view of the mathematician, but was important to account for the regions in view. Spaciometrically the red rods were removed from the relationships under purview, but needed to be accounted for. Each red rod thus told a story, and the story might be one of debt, loss, investment, advance or retreat, whatever the mathematician wanted to account for over a sequence of events. a set of relativistic motions.

With these spaciometric tools and memory tools in mind Indian mathematicians were able to give rules of manipulation which became the rules we use for signs today. How they arrived at -*-=+ i have yet to uncover, but our mathematics is the way it is because of this rule. We now can explore different "sign" rules and see what mathematics they produce, but the one we have resonates in so many ways with the natural order that it is unlikely to be replaced.

Thus the geometric/ spaciometric underpinnings of number are clear but mathematicians began to confuse measurement and a number concept. The geometric measuring scalar fractal was and is different to the number line concept, although of course the number line concept is an analogous system that principally John Wallis used to great effect. However Wallis used it as a geometric measuring line, later mathematicians like Cauchy and Dedekind ripped it away from these geometrical roots and created the number concept we use today. This number concept is a aggregation of number, memory tools like number, surds and Fractions, and infinitesimal like limit values including continued fractions and e and π cognates.

It seemed crazy to extend numbers by fractions, even crazier by negative numbers, but to then attempt to add √-1 was a step too far. Mathematicians have resolved the conflict by the invention of vector mathematics, but few recognise the work of Bombelli 

1572 book-keeping was highly
developed in northern Italy,
but even "simple" negative numbers were just introduced
(and the + and – signs unknown). So the hydraulic ingenieur
Bombelli  wrote a poem about "piu" and "meno" to teach
calculating these.  In his book "L Algebra" he didn't try to
solve x²+1=0 any longer; instead he recognized the "necessarissimi"
existence of squareroots of negative numbers and introduced

Just insert numbers and you can calculate every combination.
So he introduces them as new members to the family of numbers,
or, more precise, of quantities, lets say of a different branch (we
express this by the word adjungate or adjoin). In modern words
he is calculating vectors. For him, it was not an abstract construction.
Solving equations were done with geometric constructions and
Bombelli used L-shaped rulers for this:

The aftermath of this age long construction has been a confusion in the concept of number, carried on today through the use of number when referring to geometric values and operations, and relationships. The concept of a tensor has the power to resolve this issue of geometrical measurements in a vector type relationship, or a matrix or indeed a relational database called a tensor. This allows quantities and measurements to be separated from number and number to be returned to its cultural role in naming stages in the counting iteration.

Tensors, by which i mean weights and measures and dimensional units are the geometrical heirs of the "number line" concept, modified now to a geometrical vector.

For sure, Hermann, 2 heads are better than one they say

Welcome aboard!

Originally posted by author:

In the light of further research i appreciate that the rotational and extensional aspect of z^2+c is in fact codefying spiral orbits, as a rotation while extending is precisely a form of Spiral. Thus the mandelbrot set is what remains after vectors c  trace spiral paths through the vector field [tex]R otimes R [/tex] with a sculpting effect whenever |c|≥2.

This i think is what Vector is exploring, but the spiral orbits exist in z^n+c where n>1 regardless of changing power or logarithm.

Julia is a spiral orbit traced by the z vector, which is the stepped vector added to the same c vector every time after squaring, thus spiraling and translating.

So the geometrical /spaciometric concept of measure in the process of measurement is my foundation for aggregating "values" of various measures and forming a measure-line concept which includes the natural number names plus many more for example π, e. The point here is the measure line concept is a convenient organisation of spaciometric measures which uses numerals in a namespace to identify the measures specifically.

Remembering the fundamental process involves ratios was brought home to me by the following project.

I want o construct a protractor that measures in radians. A simple enough idea as most mathematicians measure in radians not degrees. To do so i came up with the notion of rolling a circle of circumference R around a circle of Radius R and marking off the cardioid points. Clearly this will lead to procession around the circle radius R, but i only want a simple radian measure at this stage.

The construction cannot be done by compass by the way, and the ratio is r : R = 1 : 2π. So i can use some euclidean properties to construct the measurement r given R. I need r to draw the circle to roll around the bigger one.

Firstly i choose an r and draw a circle. I roll the disc carefully to avoid slippage to mark out R. Job done. I construct the circle using this new measured portion.

But what is this measurement 2π ? By constructing a equilateral triangle base R and marking of the distance r we have a ratio comparison equivalent to division ( it is the geometrical basis of division). From this i can see that 2π is slightly more than 6*r but a good approximation is given by 3 : 19. I obtain this ratio by carefully "dividing" the small difference, and marking it off carefully on the other side of the triangle.

This process of approximation of the measure for the ratio can be repeated for further accuracy, thus highlighting the measure as the value of the division process, not a number!

Of course the numeral namespace has been constructed to be able to name these measurements.

Thus i think the measure line concept along with ratios provides a foundational basis for ordering and evaluating the world and with a namespace we can describe that evaluation.

In addition the measure line is a flexible bur inelastic measure and so can measure curves, depending on the degrees of freedom given between the fractal regional boundaries as the iterations are increased. Thus a tailor's tape is an adequate measure line for most discussions.

Originally posted by author:

Originally posted by author:

When I read this thread a lot of idears come to my minds. So reading and understanding nothing makes me very creative.
To think about the fundations of mathematics a first question comes to my mind:

Are this to numbers equal?
1 = 1

This should be less a question than a starting point of an essay I have in mind and would like to write down.
I think I can not do this in a short time. But I can come back later and can change this post.
For this I have to write down my understanding of a matched filter in digital signal processing.
Then go to artifical neural systems and then one has to understand the behavior of real neural systems and our knowledge of the human brain.

Big a program. I will surly fail cause of the lack of time and energy.

But I can post some reverences to books I like:

The first one I used at University. It gave me a deep view in digital signal processing.
For me this book was a starting point to develop my idears in a scientific context.
Digital Filtering and Signal Processing
from Donald Childers and Allen Durling

the book is from 1975 an I think it is no longer available from the book shops.
It is more my personal begining of digital signal processing and not a starting point for learning digital signal processing in the year 2010.

A great inspiration for me was the reading of the Chapter 3 the design of digital filters.
I was very impressed by a recursive digital filter.
Where the output of the filter was again feed back to the input.
Building an infinit loop that can produce infinit patterns.
(May be equivalent to Stephen Wolframs rule 30 when setting the parameters right. (So I have discovered it first!))

The next book is:
Introduction to "Artificial Neural Systems" by
Jacek M.Zurada
This book lead my idears further in the direction of how information processing and system control can be done in biological systems.

The last book is:
The Priniples of Neural Science
Eric R.Kandel
James H.Schwartz
Thomas M.Jessell

I bought this book because I was very much impressed by the lecture that Prof. Kandel gave in the Iconic Turn lecture serial at the LMU in Munic.

I have not read the book complete but I am always impressed how recursive processes are implemented in biological systems when only looking at the pictures.

Cool! May the muse inspire you! :music:

Originally posted by author:

So i fell to wondering how the proto-Akkadians as early Sumerians and Babylonians were able to divide the circle into 360.

I remember Euclidean constructions and mistakenly thought that this was the sum of their technique back then. Thus the issue was trisecting and quntisecting an angle. This i thought must have been easy. In fact it is not that hard for a practical person. This is when i found out about Galois theory and the greek "game" or agonia of trying to do this with an unmarked ruler and a pair of compasses!

so i fell under that spell for a while and diverted much of my thought and time to impossible pursuits! Why would you?

This is precisely where Geometry and thus a whole swathe of mathematics becomes obscure and agonising! The greek love of the Agonia!

Like all things the game is a diversion that produces amazing and interesting responses, but ultimately is not a model of "reality", by which i mean many of us have had our world view twisted by this type of practice and constraint. It is also why i decided to coin the term spaciometry to reveal in myself these imposed twists by adopting a different viewpoint.

So enjoy the pursuit as much as anything else but the fundamental connection does not require one to undergo "agonia", or any other "initiation" rite into the secret society of the pythagoreans etc.

Now this is out in the open, my appreciation of the vortex and its surface manifestation as a spiral has been enhanced, and the contention that the circle is a special spiral has an increased confidence value. The fundamental motions in the spaciometric motion field are spaciometric rotation and spaciometric extension. These mean that the fundamental form is the vortex/spiral (torus-vortex in 3d). If i have a condition that fixes the extension to a constant then the special form of a spiral circle or sphere is the result. Thus the extension quantity and its ratio to the rotation quantity determine all forms from the so called singularity to the infinite "straight" edge.

The proto-Akkadians are linked by trade and via the Dravidians to the Chinese ancestors. There is also a land bridge between them admittedly over difficult terrain. The Chinese have a construction of the i Ching based on 24 divisions of the 6 concentric circles. This tool enables the practical division of any circle into 360 parts, by qunitisection, and bisection and trisection.

Originally posted by author:

Lai Zhide is credited with introducing the Taijitsu called the Yin Yang, and in particular contributing to Chinese "analogical thinking" or philosophy . One must not get hung up on the apparent absurdity of Chinese astrology, as it is in fact an ancient and well respected body of knowledge which is common in the west and indeed Astrology  gave birth to western Astronomy and Cosmology!

It is a thing of note that the Babylonian magi began to demonstrate philosophical thinking, dealing with abstractions and principles in their cuneiform tablet records early in our common history. Chinese Astrology is nothing if it is not an exact example of this ancient "scientific" analysis of the cosmos.

Analogical thinking requires a form to make the analogy with. In chinese this is called the Yi. Thus the five elements are just 5 Yi by which chinese philosophy assembled the chemical and physical properties of the cosmos. They represent in one analogy a kind of periodic table, and in another a type of electromagneto hydrodynamic spectrum. The simplicity of analogical thinking is that it can be ratioed to just about anything.

In this current western absolutist, abstract phase we find it hard to conceive of the usefulness of this conception in terms of dealing with information overload. In terms of creativity, it is not at all restrictive or stifling, nor does it preclude innovation. Chinese scientists are as objective or subjective as i and i require an organising principle or set of principles to store knowledge!

Lai Zhide i think would have been at home with Carl Sagan, Richard Feynman even Einstein and Newton in his philosophical reduction of the cosmos to the i ching and his consumate Yi-ology.

There is a recurring relationship between spirals vortices and the measure 3. this in fact could be π 0r e but this certainly has a spaciometic form which is irregular angular and cone or pyramid-like. This has a bearing on close packing in the sense that a spiral based around a 3 measure centre may provide the best bundling spaciometrically. Certainly i notice the measure 3 in a lot of natural plant forms and structures.

Originally posted by author:

I thought it might be worth exploring the notion of extension spaciometrically, as the notion of straight and right are cultural paradigms, and somewhat idealised. I will consider this more fully after i have completed a survey of kinesis and kinesthesia with its subfield proprioception.

The Logos Response is informed and grounded in my sensor systems and processing, and thus extension is not as simple as it is made out to be.

I rather suspect that spiral motion will be he outcome and that orthogonality will be important along with orientation and feedforward feedback cybernetic systems.

It is of interest that orthogonality is a distinct sensor system of itself, irrespective of any spaciometric definition. Orthogonality founds nd contributes to spaciometry, and in a very real sense spaciometry could not exist without it.

By the way i feel a clear distinction between humans and other animates is spaciometry as an axiomatic system. Other animates i am sure have spaciometries and even codify and pass them on to the next generation, but it seems only humans seek principles on the basis of analogy, and reconstruct there experiential continuum accordingly.

That is not to say that other animates do not develop erroneous concepts of the environment, cos clearly they do, but there information and processing is empirical not axiomatic. I guess in just about everything else we are similar including inductive and deductive reasoning!

The notion of relative ratioed radial expansion suggests itself as the fundamental motion with relative ratioed radial extension being the specific or abstracted orientation. Thus extension and orientation are the exact same notion. The ratio is specific to the form and or region of focus, but the relativity relates to the quantifying of expansion by ratioed extension in a standard orientation.

If the radius is special, that is inelastic it produces a circular expansion and the notion of perfect or universal applicability of the measure: a universal metric. However if the radius is "elastic" or deformable in some way then there is no universal metric and all measurement of extension is local and dependent on orientation. This would be the case if a spiral reference framework is utilised.

Vorticular motion fields may indicate that the underlying metric is based on a ratioed radial expansion that is "elastic" relative to orientation.
Dilative rotation

Originally posted by author:

Polar coordinates, spherical coordinates are the most natural relativistic reference framework currently in use, but it is still made out to be special, whereas it is due to historical primacy that Cartesian coordinates are still presented first. The weight of mathematical description is still presented as Cartesian, thus Special Relativity seems inaccessible and strange, and the "complex plane" can pass as a number system rather than the coordinate system which it is.

Spiral coordinates will be a truely strange system, but one i am laying the groundwork to explore.

Originally posted by author:

Plato was one of the first to discuss the problems of perspective. "Thus (through perspective) every sort of confusion is revealed within us; and this is that weakness of the human mind on which the art of conjuring and of deceiving by light and shadow and other ingenious devices imposes, having an effect upon us like magic… And the arts of measuring and numbering and weighing come to the rescue of the human understanding—there is the beauty of them—and the apparent greater or less, or more or heavier, no longer have the mastery over us, but give way before calculation and measure and weight?"[15]

from Perspective (graphical)
From Wikipedia, the free encyclopedia

To which i add the art of languaging, without which the full context of the other three is missed and the logos response- the calculation-takes abnormal preeminence.

By the way only the ascetics among humans describe the human mind in these pejorative terms: a confused "mind" is no weaker or stronger than a seemingly understanding "mind" it is merely confused. The actions proceeding from a confused "mind" may appear weak and ineffectual but appearances can be deceiving, as outomes determine the relevant value of states of mind. And in any case many confused 'minds" have taken decisive and bold action without understanding, just as understanding "minds" have taken weak and indecisive action.

Originally posted by author:

We have no real sense of "time" that is not spiral, thus our modern abstract notions of "Time" are simply deceptive, hiding the creeping advance of precessional motion and indeed all vorticular motion in the void. Instead of shadows we replace our connection to our universe with something more insubstantial: the notion of time.

Originally posted by author:

The eyes have it for more go here

The important point is  it is not the technology it is the cybernetic system, based on iterative processing called here convolution.

It seems that projective geometry, in particular mapping projections like mercator et al may be a way to enscribe vorticular spiral wraps onto any object, thus providing a transform from its usual geometrical representation to a sonic wrap. This spaciometric transform means thqat in essence i would be able to represent all objects by these wraps and deal with the motion laws for these wraps for a theory of everything.

Originally posted by author:

Somewhere nice and spaciometric to play


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