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

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All Lines Are not Equal: Strecken

The Law of 3 Strecken is about producing a form. When that form has a metric applied an area will be calculable in 3 regions. Those three regions are related by a sum..

Now it is very important not to think in terms of addition and multiplication: these are forms of combinations and that is what we must realise. In fact multiplication is a misnomer for the binary notation. The binary notation depicts the factorisation of a form.

So Grassmann was looking at a form produced by the dynamic extension of its edges with a parallel line pair restricting the type of form. The third Strecken provided the opportunity for variation, but the parallel lines appropriate to that form were implicit in his notion of the form. Grassmann called it a product in the normal sense of multiplication, but the factorisation was odd ab = -ba!

Two jostling Strecken can be seen as producing a parallelogram. In this case instead of putting the sum of 2 Strecken line draw a third strecken in order to "multiply" in the way just described. Any two jostling Strecken should produce a parallelogram, thus the third Strecken is used to produce parallelograms with each individual jostling Strecken. Then we can see that one parallelogram is equal to the sum of the other 2 parallelograms

Mr Newton in his method of moments woks specifically with the rectangle. The two variable factors result in 2 parallelograms as the increment. The underlying multiplication is thus a parallelogram production the same as Grassmann.

The combination of Strecken fhat Grassmann initially identifies have 2 forms: one produces a line segment the other produces a parallelogram. When the Strecken are in line the parallelogram becomes a line . Essentially the area is 0, but if the 2 sides are metricated can we define an in line multiplication?

To be consistent we should be able to define a Schwerstrecken made up of the product of the 2 metrics within a length which is the sum of the 2 metrics.

This is an odd region in a particular direction with an "intensity" equal to the product of the 2 sides over that half perimeter distance.

Does this type of region exist in real vector situations?
http://www.fractalforums.com/complex-numbers/polynomial-rotations/msg37986/#msg37986

I need to look at the actual process of building or combining a multiple form in sequence and thus over time. Then a special form of this will be factorisation.

Hopefully i am free of the indoctrination of "multiplication", and can remain in touch with the fundamental processes of division and combination.

If i start with a jigsaw puzzle and separate it carefully one piece at a time, i am dividing the original form into ji pieces, where N is the number of pieces and j is an identifier, and ji is also its sequence position identifier. Each piece is distinguished in this way and each piece has a relative orientation to all the other pieces which i will denote by a set {jijh} where i is a fixed value and h roams from 1 to N.

The orientation set is arrived at by drawing straigt lines between each piece from a given piece. The lines may coincide but must not cross. This is an orientation web for each piece.

Now i also need to record the rotational motion of each piece, or relativistically how the orientation web of a given piece rotates around that piece, to do that i will use a circle in each piece, the centre of the circle willbe where the orientation web lines originate from and the points where these lines cut the circle will be recorded. Now each web is unique and therefore it will also identify the unique circle within each piece . If i add another coordinate ch to identify the point where jh cuts this uniqe local circle, i have enough notation, and to show that it is local to that set i can set up a coordinate bracket (jijh,ch)
and the orientation web becomes {(jijh,ch)} i fixed and h runnining from 1 to N

The system is what i have elsewhere called a compass vector metwork and clearly is a version of polar coordinates. I can later identify jijh with the notion of position vectors, if that becomes necessary.

Now i have not specified any metric for the circle or the lines in the orientation web, but i do specify that the lines connect circle centre to circle centre. In keeping wih relativity, once a goal jigsaw piece is identified then the two points at the centres and the unique line that joins them provide the basis for metrication both in the circle and in every other connecting line.

Now if i have this information, or record this information as i dismember the jiggsaw, i have a sequence for reconstruction and a reference for orientation as i construct, and internal confirmation of the correctness of each piece both in its sequence position, its orientation and it actual relative position. What i do not have is a duration for the construction, but i hope you can see that in this situation this is independent of the construction, and in most cases without this information is likel to vary between the correct placing of each piece.

For a fast and consistent rate of combination each piece in the sequence would have to be stacked or arranged in that sequence however it is disposed in space. Thus i may now dispose these pieces in any sequenced pattern in preparation for efficient reconstruction, recombination.

What if that sequence is broken? That is when w need methods of assessing the sequence, identifying what is out of sequence or missing and then locating and restoring the missing pieces to the correct sequence position. The time taken to do this prior to synthesis factorisation may be the same as if performed during the process of synthesis factorisation, but the potential for more disruption is lessened by preparation beforehand.

Now i know that a similar synthesis factoring occurs in the protein molecule building in the cell, and the locating of the correct pieces in te sequence is solved there in a more general disposition of the factors and element of a factor. We also know that the duration of this process can be in terms of hours, but also its rate is phenomenal per factor!

This relates to the product of "vectors" directly, because the notation must describe the process. The multiple form notation describes the process of counting a finished static form, not a dynamic one which has not yet completed. multiplication is therefore a dangerous idea to associate with dynamic situations willy nilly, or because it looks elegant. The actual process has to be elegantly described, and the notation fitted to that in the most helpful way.This has implications for the freeze frame notion, as it sets some kind of limit or guide on when a frame is to be taken in a sequence, and whether it is possible to get ideal conditions for that.

Thus the notion of producting in line vectors is suspect until an analysis of the actual process is done thoroughly. In this light i will also need to look at the notion of contemporanaeity in the factorisation process given a sequential process, and thus what it means for the factors to increase"at the same time" when a sequence is involved in the synthesis factorisation.
http://www.fractalforums.com/complex-numbers/polynomial-rotations/msg37986/#msg37986

Finally i hope to look at differential lines which occur in line to reexamine the notion of in line combination.

Venusvenusvenus1 Friday, May 4, 2012 8:25:31 PM

contact me at

venusvenus1@myopera.com

I was reading and some think grassmann was eating some kind of grass, maybe a hallucinogen.

I am serious.

TH I S work affords a sad illustration of the spirit of
lawlessness which has invaded one of our ancient
Universities since the time when she rashly began to
tamper with her Tripos Regulations. In the good old
times two and two were four, and two straight lines in a
plane would meet if produced, or, if not, they were
parallel; but it would seem that we have changed all
that. Here is a large treatise, issued with the approval
of the Cambridge authorities, which appears to set every
rule and principle of algebra and geometry at defiance.
Sometimes ba is the same thing as ab, sometimes it isn't;
a + a may be 2a or a according to circumstances;
straight lines in a plane may be produced to an infinite
distance without meeting, yet not be parallel; and the
sum of the angles of a triangle appears to be capable of
assuming any value that suits the author's convenience.
It is a pity that we have not had an opportunity of showing
the book to some country rector who graduated with
mathematical honours, say, forty years ago; it is easy
to imagine his feelings of surprise, bewilderment, possibly
of indignation, as he turned over the pages and encountered
such a variety of paradoxical statements and
unfamiliar forrnulre.
Seriously, Mr. Whitehead's work ought to be full of
interest, not only to specialists, but to the considerable
number of people who, with a fair knowledge of mathematics,
have never dreamt of the existence of any algebra
save one, or any geometry that is not Euclidean. Its
title, perhaps, hardly conveys a precise idea of its contents.
It is, in fact, a comparative study of special
algebras, exclusive of ordinary algebra, the results of
which are taken for granted throughout. Such an undertaking
has necessarily involved a very great deal of time
and labour; for, in order to carry it out with any degree
of success, it is needful, not only to master each separate
algebra in detail, but also to adopt some general point of
view, so as to avoid the imminent risk of composing, not
one work, but a bundle of isolated treatises. Mr. Whitehead
has, happily, overcome this difficulty by viewing the
different algebras, in the main, in their relation to the
general abstract conception of space. Whether this plan
can be consistently followed throughout may be open
to question: it certainly works very well in this first
volume, the keynote of which is Grassmann's Extensive
Calculus.

jehovajah Saturday, May 5, 2012 3:14:10 AM

Thankyou for the comment.

Grassmann's Health was a great concern to him, as he seemed to suffer greatly with phlegm. Beyond that i have not studied any remedies he took. However cocaine and many other stimulants were readily available and used widely without censure.

Whitehead indeed was very impressed by Grassmann's notions. I am reading Grassmann in German and can understand why.Certainly his ideas were very novel because he did not rely on number, but on process and subjective processing of information. However, he did no conceive of his work in terms of vectors or algebra, but as the Theory of Thought Forms, with an analytical method which he constantly modified to make it more complete and general. In so doing he returns to and exposits the Euclidean Teaching material in a revealing way, that lays bare its essential dynamic notions. Thus he did not enhance Euclid, he revealed Euclid's deeper intentions, something that had got lost in the turmoil of the 18th century.

However he did retain some misconceptions of his time, which he would gladly have refined had he had the support of such as Whitehead earlier in his life, or even the critique of his contemporary Hamilton, who also retained misconceptions in his unfinished work.

The reach of Grassmann's ideas is extensive and perhaps unrealised by many Algebraists. Even so, some of his ideas it seems were wrongly understood and misapplied.

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