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.
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.
Originally posted by author:
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?
Originally posted by author:
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!
Originally posted by author:
"our lives do not spiral out of control! click to find out
" Rather, they spiral into
"an exhilirating rhythm of existence
"which is uniquely
Originally posted by author:
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!)
Originally posted by author:
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 [tex]S(zeta, eta, upsilon)[/tex] 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 [tex]A(zeta, eta, upsilon) , L(zeta, eta, upsilon), and F(zeta, eta, upsilon)[/tex].
Originally posted by author:
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.
Originally posted by author:
The Logos Response compares and distinguishes Ratios, but the ratios are all related to one another, so focus is necessary to concentrate the attention on a particular region. But this means that the region focused on is in a ratio with the whole ratio data set from the logos response. Thus the ratios i may divine from a region of attention are in fact ratios within a ratio itself.
The Golden ratio is perhaps a formal recognition of this fractal state of affairs.
Originally posted by author:
The Auricle has other structures which function in the logos response for vertebrates; resonance chambers, pressure amplifiers through lever actions,pressure equalisers and head tilt sensors,gyroscopic motion sensors, amplitude and frequency sensors.
Although i began my exploration of the sensory system with the eyes and discovered the logos response in that system, it i clear to me that the auricle system is fundamental in a way the visual is not. The sense of rotation lies in the auricle system and the sense of orientation lies in this system also at least the sense of head orientation vis a vis( or rather ratioed against) the proprioceptive sense of orientation. The auricle and kinesthetic senses of orientation are more fundamental than the visual reference of that orientation.
Thus in the auricle and propriceptive systems i find the fundamental notions of orientation and rotational movement relative to the organisms form and structure. If the visual sensors are combined i find that rotational movement and orientation can be referenced against a visual map which includes a map of the organisms form and structure.
Thus i note the systems fundamentally provide information /ratios about orientation and rotational movement Where then does Extensional movement reside and the notion of extensional direction (from which and upon which we base the notion of Axis) ?
This is of interest to me because despite seeming fundamental Axis is not at all represented in the sensory system, and therefore seems to arise from the processing algorithms that form the basis of extension.
As far as i can tell the notion of extension arises because of the interference patterns produced by the binaural, stereoscopic, dual gustatory and rich proprioceptive web-maps of the inter-reacting sensory systems. The point here is that extension is a computational output that is at a different computational level to rotation and orientation. Thus extension will have more computational artefacts than orientation and rotational movement.
What this all means is that "Perspective" (as a relevant example is a computational effect) and the vanishing point are a computational effect. Of course the structural form of the eye contributes to this, and the ratio of "interference pattern source" signal processing also contributes. Thus if i focus attention on the interference pattern at a particular region in the visual data/ratio streams the perspective and the detail and even the image size output changes to reflect that. The output to memory thus can significantly differ from the raw signal output from the sensors themselves. However it is unlikely that the raw signal output will be interpretable without the processing that occurs to "make sense" of it so to put my point another way: the data that is currently being processed in your processor to produce this screen you are currently looking at will be unintelligible without the visual data processing algorithms that re-translate them to a visual image.
The perspectives in this visual image are not reflected in the raw data that is being processed as it may take only a few bits of machine code to describe the colour and area of the screen but masses of data points to modulate each pixel to give the correct amplitude and relative colour and persistence, etc. And of that vast ocean of data only the selected part is output to screen, and only in the window assigned to it, at the resolution assigned to it . Thus these very words are the focus of your attention, but the processor and arrays have present on the whole screen more information than you are actually focused on right now. Your perspective of he screen has thus been altered, so that your experience highlights these words and their references and not the physical display.
Their is no vanishing point on the physical display so every selected thing that is currently being processed is flat in front of you, but by focusing you have made various parts of the screen vanish from attention ( this is now called attentional blindness). But what if the algorithms put a vanishing point on screen to reflect the computational ratio of the data being processed? Thus the processes with smaller cpu clock cycles would be represented on the screen with smaller windows, and the larger cpu clock cycles with larger windows. The resulting quilt map would mono-scopically reference a ratio field. If these windows were arranged on a spiral we would observe a natural spiral vanishing point that would immediately make visual sense to us.
The point then is that the notion of extension is as much a computational effect as a "real" spatial arrangement in notFS, and the notion of Axis derives from this computational output. The cone effect of the vanishing point is likely to be a computational artefact of the arrangement of processing loads on the CNS distributive parallel computational system.
Thus despite the seeming fundamental nature of axis it is not as important as relativity, and the relational arrangement of certain structures within the organism are sufficient for rotational movement and orientation to make sense.
Now orientation as a spaciometric attribute relies, as it must, on an individual organism defining it and demonstrating it. It is a real thing only in this sense, and makes no sense as a line on a piece of paper.
However, if the defining organism can spaciometrically reference these marks and lines they may have a significance as an orientation or set of orientations to another organism. You may have heard of the dance of the bumble bee? This is such a non human example of the general system i refer to, and exemplifies the basis underlying our graphical representation of orientation. When extension data/ ratios are combined with that orientation definition we have the basis for the notion of Axis.
It is worth noting that in none of this has it been necessary to define a surface called a plane. A plane therefore is even more of a computational artefact than an axisIf i spaciometrically rotate and keep the relative ratios within my organism the same i might visually reference a surface which i can define as a plane, but equally i might reference a surface which is a cone, or edges or surfaces of any form that encloses me. What i will not reference obviously is the spiral or vorticular nature of all my observations, in all sensory systems.
Originally posted by author:
This is a 2d notion of a 3d DNA wrap.
Hey Bib , Welcome!
I thought some evil genius had stuck a sign on my Thread saying "mathematicians only" ! No way man! It is a thread after all and not a blog
So i welcome everyone .
First let me apologise for my complexity. Although i accept the criticism/comment i cannot promise to change much as the thoughts come to me as is, literally. I am more interested in capturing the thought than in making it intelligible to the reader, including myself!
Anyway this is fun for me and a reference notepad of insights. Anybody can contribute , challenge correct, question (what happened to the alliteration?) , but i hope someone will want to collaborate. I do not have the answers and i do not intend to judge others contributions. You know the pedagogues of my day called teachers of math did that to so many that they have become timid and bereft of their mathematical heritage. Its a power trip man. Pure and simple! And while i am on a roll i have a beef with Gauss for his treatment of Riemann!
Anyway now i got that off my chest i feel a lot better!
I think the more i learn about Feynman and Dirac the more i see their Autism, but Feynman was able to use his Aspberger"s in a field where it was valued and he could mature in a positive framework. Dirac had to overcome much more difficult circumstances and a different level of Autism. So now what about Stephen Hawkings?
Mathematical physicists from Newton to Feynman are my favourite maths teachers, but of course they can be self righteous too, a trait i think to be avoided.
So welcome, Welcome Welcome ! All are Welcome.