On Fractal generation, generators and motion equations

So watching Richard Hammond's invisible world with interest and noticed the raindrops and milkdrops sequence. Surface tension and viscosity are related. The spherical shape maintaining its shape at freefall speed depending on droplet size and air density and viscosity is interesting,as was watching it disintegrate when equilibrium conditions were exceeded. The instantaneous shockwave curve seemed to initially be a bezier type curve of a high order before being overtaken by a radiating conic curve going from a bezier suface to a conic surface. By this i simply mean that the underlying curves for thr observed shapes moved froma bezier description toa conic sectional one.
I noted that the milk dropped vertically followed a curve that was conically either straight on the conic surface or direct from the apex of a conic surface through evry connectected entrained conic …


So watching Richard Hammond's invisible world with interest and noticed the raindrops and milkdrops sequence. Surface tension and viscosity are related. The spherical shape maintaining its shape at freefall speed depending on droplet size and air density and viscosity is interesting,as was watching it disintegrate when equilibrium conditions were exceeded. The instantaneous shockwave curve seemed to initially be a bezier type curve of a high order before being overtaken by a radiating conic curve going from a bezier suface to a conic surface. By this i simply mean that the underlying curves for thr observed shapes moved froma bezier description toa conic sectional one.
I noted that the milk dropped vertically followed a curve that was conically either straight on the conic surface or direct from the apex of a conic surface through evry connectected entrained conic apex within a given conic surface. therefore iwas eager to see if cylindrical and helical motion ensued on contact with the viscous surface. Further analysis is required but it looked consistent with expectations, with the elastic bounce effect being entirely consistent with anti vortex behaviour at the instantaneous centre point of impact.
The pressure shock wave was also evident in the water surface and quickly seemed to damp whether that was hypebolically or harmoniously requires further analysis..JPG]
I understand references to mass as a volume reference to a dense material and force as a pressure per unit area but wonder if pressure per unit volume might not be more in keeping? Although force is a derived unit as is pressure as is acceleration, and really is a quantification of motion and arises out of observations of motion, i feel that the conditions under which the observations took place are not fully reflected dimensionally in the definition of force if the definition of mass is changed as i advocate to a density per unit volume of some substance based either on Avagadros constant or a Planck unit.

Things in the set FS are ordered by the conicsectional ,helical and spherical curve behaviors entrained by the universal vortex . Anyone of these curves can destroy the order of a system if they move in a non equilibrium direction. In that sense order and with it life and computational consciousness is very fragile,and it is very reassuring to see a strong region of coherence in vortex behaviour.

i proposed somewhere on a physics site that the proper paradigmatic medium for light waves is light, just as ocean waves are in te ocean etc. So now the shock wave it seems to me represents a boundary condition for any advancing medium with idiosyncratic wave behaviour being either side of the boundary. The boundary in set FS is always a fractal structure and thus represents intensified energy conditions in this structure. I would expect light to produce a shockwave in its advancing edge and thus as is currently stated this advancing edge would travel at or above the speed of light with the following wave behaviour averaging out at the speed of light. The advancing shockwave is what would initialise all the properties of light such as reflection, refraction etc.

Of course this would be generalised for all electromagnetic radiation.

Now a question: are the magnetic field lines around a solenoid a helical torus structure? This is to say would a test electron hypothetically take a helical path around the solenoid passing through the open or vacuum core of it repeatedly as it traverses the toroidal surface ? And would the toroidal surface the electron travels on helically through the core, depend on the energy of the electron?.JPG]

Originally posted by author:

I want to redefine force as pressure per unit volume P with the actual definition being

          P= [tex]int p.dA over V[/tex]

Where p is the "gas" pressure and the [tex]V[/tex] is the measuring volume of the gas and A is the surface area of the measuring volume.

this then leads to

P =[tex]int frac{F}{dA} .dA over V[/tex]  (1)

For a constant pressure this has a solution

P=[tex]F.int 1 over V[/tex] =[tex]frac{F}{V} int 1 [/tex]= [tex]frac{m}{V}.a int 1 [/tex] = [tex]rho .int a.1[/tex]

Where [tex]rho [/tex] is the "gas" density and [tex]a[/tex] is the observed mean acceleration of expansion in the gas which integrated over the surface area of the measuring volume will be a constant that is non zero but small. This represents an inertial /equilibrium solution with non equilibrium or non constant solutions resulting in resolved component acceleration.

by Boyle's law we can link [tex]int p.dA [/tex]  to the mean gas temperature, T:[tex] p.V[/tex]=k.T

I think that the simplest sequential method of integrating over a surface is by means of a surface spiral. Any space filling curve could be used.
The [tex]int 1[/tex] is an operator in this instance as it is undefined for calculation. As soon as a product can be passed into it it can receive a defined calculation procedure, in this case sum spirally around the surface area. As the product admits no change on an infinitesimal basis if [tex]a[/tex] is constant the result should be [tex]a.1 [/tex] . However the mean acceleration is not a constant and so a small variation in the sum should be expected as the infinitesimal areas go to limit. However i would need to have a function for acceleration of expanding gas based on surface area expansion.

Originally posted by author:

I want to redefine mass as density per unit volume D which is written in this way:

D=[tex]moles of substance over V[/tex] = [tex]{Avogadros constant times V over V_s} over V[/tex] =[tex]A times V over V_s.V[/tex]=[tex]Aover V_s[/tex]

Where [tex]A[/tex] is Avogadros constant [tex]V_s[/tex] is the volume of the substance containing Avogadros number of elemental or compound particles as a molecular weight in grammes and [tex]V [/tex] is the measuring volume for the standard and molecular weight includes atomic weight for elements.
This is an iterative redefinition of mass so that the initial definition of mass for calculating the atomic weight is the current one.

The cultural iteration +1 is an example of specialism, in that it defines counting as addition. A similar structure defines multiplication as counting in fixed value groups; +2,+5,…. being the 2 times and 5 times table numerals. General addition then is characterised by adding variable value groups and s called aggregation.

Similarly the iteration -1 defines counting in reverse as subtraction and -2, -5, …. as division which is subtracting fixed value groups and noting the remainder and how many subtractions took place. `in multiplication we note the final value and how many additons took place. General subtraction involves subtracting variable value groups and is called disaggregation.

Now mathematicians have modeled a form of aggregation which is variable but in a specialist way : the variable groups are formed by "multiplying" each variable group by the "group size", so these are termed power law additions and condense the notion of addition into a new process called multiplication by a base.

Distinguishing multiplication in this way tends to obscure the essential iterative base of it so that it seems to be a binary operation when in fact it is a unary operation on a binary base operation of counting.

Binary operators require 2 elements to define there operation. A unary operator requires only "one" to act as a modiier. So the operator 1*, 7*,… makes no sense without the base binary operation + Which instructs me or another to combine 2 elements by adding group size to a previous named element and to continue to do so until told to stop, calling out the names as i go.

Aggregation is specified by the 2 element names and a counting iteration starting at one numeral and continuing +1 the second element numeral of times. In the base operation the 2 elements define an iteration instruction or procedure one being the starting numeral the other being the iteration control numeral. This says stop after this numeral has been called out in the counting iteration while you have been continuing the initial counting iteration from the name of the first numeral. Thus 2 iteration counts are established the second controlling the first. This can be reduced to one counting iteration mod( the size of the aggregation of the 2 elements) . So 2+7 is not only 9 but also +1 mod(9) which is count or iterate forever mod 9 .

Multiplication operates on this binary base not on another element of a set, so in this strict sense multiplication is a unary operation on a binary addition iteration, groupsize . The elements of the set being operated on are usually 0 and .

Multiplication by a base is discrete as far as counting iterations go, as each numeral in the value name space has to be arrived at by its own independent counting iteration. There is no counting iteration that gives the sequence of numerals in a power law addition sequence. Because of this power law sequences are useful as linearly independent basis vector forms or polynomial forms. This type of sequence of aggregation products is a fractal iteration arrangement allowing independent counting to take place in each part of the sequence such that the power numerals record the cycles of the itereations in the preceding power numerals, ie 2 based power laws can record normal counting as cycles of 2 iterations.

The reverse of this leads us to look at disaggregation to find this special form of power law subtraction and in fact we find the basis for understanding this as a new process called division or better inversion. For inversion to make sense i have to have a constructed set of counting names that deal with the enumeration of splitting or breaking a boundarised object into pieces either by a natural destructive action or as a biological division process. This started to be attempted under the name of fractions but continues under the name rational numbers. The construction of this type of numeral space is a power law tour de force.

t seems clear to me,but then i have thought about it for a while, that Newtons laws are of motion but that motion is de facto Euclidian which is why it is not as accurate as Einsteins which appears to be Riemannian. The more general Riemann geometry is not hard as it is the geometry of plants and trees,and bulbous fruits etc. However we now have fractal geometry which is more general still and may lead to a more accurate description of motion.

As obvious as it may be i make he point that a 3d graphing programmw such as runiters focusses on the linking aspect of mapping while a 3d fractal generator focuses on the motion part of mapping. So the "surfaces" produced tell me a different story for each programme . Iteration is about motion and motion surfaces or orbit traps i think the correct term might be, and the fractal generators sculpt this motion. The simple rules for sculpting highlight the complexity of the sculpted motion. More complex sculpting rules may be possible to describe growth motions.
.JPG]

If i think of the fractal generator as a video camera then it produces images of motion stopped at a certain iteration. The surfaces and edges are frozen in various states of motion. I have hitherto thought of these products as fractal sculptures,but am starting to see them as still images of a dynamic process sequence. At one time z^2+c was a polynomial in z a so called "complex number" but i term it a polynomial numeral. Now it is a predicate of an iteration statement z=z^2+c. And this statement i now see as a description of how quad numerals are to move under the iteration! and the quad numerals are polynomial numerals in 4 variables with a basis of 4 linearly independent power-like operators which behave like power nomials in a polynomial. By using unary operator analysis i see the relationships between the constructed basis and hope to further devise tools to explore this further, but it feeds back into the construction of mathematical operators in general, in particular the fundamental ones of addition, subtraction and division and multiplication , and base-logarithmic multiplication/division on which we algebraically construct arbitrary numbering /counting systems. These numbering systems are turning out to be part of the language we need to describe classes of iterations in the observed experiential continuum. So splitting in cell division can be described, but the fracturing of a plate or a rock crystal- well not quite yet, but almost.

What drives this innovation and development of mathematical thinking? I think and believe it is iteration. The development of infinitesimal math at a time when infinitesimal thinking in science was derided as absurd by certain religious logicians(those that study persuasive arguments, and forms of confounding a proposition!) is a case in point. Derivatives and differentiation is constructed on the basis of iteration. The structure of these iterations are not difficult but are confounding if one is not used to iterations. I have heard that indian mathematicians relished infinite fractions, and through this understood one fractal: the limit boundary. The fractal limit boundary became crucial for Newton and Leibniz to progress with their mathematical description of motion by iterative mathematical products such as the limit boundary/value.

Of course summation and limits have always been associated with infinitesimal addition and this has developed into the integral side of calculus,using iterations to arrive at a sum. It is a wonder that so many of these operators are linked by an inverse relationship that can be demonstrated but not entirely surprising.

So the humblr fractal generator is in my opinion a fundamental mathematical tool for looking at processes of growth and destruction and condensation and radiation and a whole lot of other stuff.

[img]http://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA20&img=1&zoom=3&hl=en&sig=ACfU3U0DhX4KkHf9myeT-HbteaIHFNIY4Q&ci=257%2C467%2C677%2C907&edge=0

Looking upon newtons third law i am impressed that the notion of "opposite" is not so firmly expressed, nor explained as being linearly opposite or "right" as Newton earlier expressed the motion arising from the impression of a force.Newton i have observed to be every bit as tentative and qualifying as any careful observer should.

Thus i have already redefined the laws in terms of vorticular motion. But it strikes me afresh to observe that the yin and yang symbol describing and designating two fundamentally opposing "forces" in chinese cosmology both small "scale" and large does not indicate a right action but rather a spiral one. This is of interest because the western notion of right action has been influenced much by the cultural transference of information and it would lead me to think that newton in proposing these laws already had infinitesimals in mind. Thus his notion of right was as an approximation to the "true" motion which newton attempted to divine from studies of circular motion.

Newton was well aware of relative motion and relativity but not able to advance his studies to Einsteins level publically having no cultural need to or relevant data to inform a direction. .JPG]

The chinese however for some reason not clear to me yet had observed this relationship between opposing principles at all levels and divined a spiral or vorticular relationship. I do know that chinese philosophers did not give credence to absolutes in the western philosophical sense and this may be why they shunned any absolute opposition. In any case the evidence for spirals and vortices in opposing interactions is manifold.

just a note for further development. Mathematics is part of language as a category and as an epistemology is a function of the general faculty of thinking which in its raw state is information processing.This requires the notion of information which inturn requires referents informer and informee. The informer transfers the information to the informee who receives the information. Now the referent for information can be deined as the particular state that the informer holds, inheres, maintains and demonstrates etc . The informee is the recipient of this state in that its state is modified by the receipt of this state. The informee necessarily has a state and is thus an informer also and may in receiving information transmit its information to the informer.These state changes may be as simple as confirmation changes in molecular structures for example.

With this definition of information a more precise explanation of transmission and receipt can be arrived at iteratively.

The utilising of this information as in information processing is several stages down the line of this structural concept, but represents an epiphenomenal product of these initial constructs. At this later stage parsing and syntax will become core concepts and fundamental to any iterative definition of perception. I also by default acknowledge the importance of the turing machine concept which will be central to the development of the structure of information processing. By way of these notions i hope to arrive at general language concepts and in particular mathematical language concepts as a fundamental language model of our perceptions of iteration in the set FS. By extension or discovery i hope to find more structure to notFS

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