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 Author Topic: Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre  (Read 15907 times) Description: Grassmann Mathematische Annalen (1877) Volume: 12, page 375-386 0 Members and 1 Guest are viewing this topic.
jehovajah
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 « Reply #15 on: July 05, 2014, 11:03:05 AM »

At any elementary level children are indoctrinated with the alphabet and a decimal numeral system of names to call out .

Both of these are cultural namespace systems, but they are taught or impressed on a child with little derivation?

They can only be derived for each individual from the space in and around them in which and of which they consist. These derivations are functionally divided into ordering space,(Let us specify ordering as arranging space relative to itself and the arranger) and ordering and constructing a model of our response to ordering space.

Ordering is a complex process involving sequencing by position, by evaluation, mental responses etc
At each process we move, count, are moved or are counted. In moving we translate, rotate, reverse reflect and many other aspects of what I shall call dance. In counting we respond sequentially rhythmically , numerically and vocatively in many aspects of what I shall call singing.

Thus in dancing and singing we account for, enjoy and respond to our experiences of all in and around us.

To focus then on 4 specific processes: dividing of wholes into ordered sequences of parts, which is factorisation, and the principal process in our nature, in increasing unit in this world.

Taking those units we have a goal for addition: to find the sum of the parts, to restore the whole.

This fundamental cycle of division or facto rising followed by addition that is summing the parts into a whole is the fundamental argument of Book 7 in Euclids Stoikeia. From this all classes of monads/ Metron or unit standards are assimilated and Arithmoi are defined and proto Arithmoi distinguished. It is here that standard Metrons are ordered into Arithmoi combinations or sums from which the pollaplasios or multiple form is distinguished and defined.
The division process naturally supports the subtraction of parts from the whole, again ordered into multiple forms.

The classic or ancient realisation was that all things can be given an alter ego, a formal representation. All things can be recorded in a mosaic form, a multiple form

And thus all multiple forms can be reduced to combinations of other multiple forms.

We can restate this, without changing its derived meaning, without giving ground to any modern interpretation as
Any product can be represented as the sum of other products.

Thus a product is the mosaic of some form derived by division. Such a product can be conceived as a sum of constituent products.

Grassmanns representational expression , for it is not an equating of equals, and neither are so many so called mathmatical equations:
AB  +  BC = AC

Is a powerful analogous expression. In it eventually Grassmann came to recognise that any product can be written as a combination of products. The fractal scale free implications are breath taking.

The rise of the consideration of ordering in in the roots of polynomial expressions that lead to group theory helped to identify ordering basics, but one fundamental arrangement process was so often ignored in the notation. It is rotation.

Cyclical rotation of letters terms etc were inconsistently apped to their spatial implications. So Cardano is left wondering what the square root of a negative expression might Be, and even what a negative quantity may be!

That reflection and rotation both seemed to be possible spatial ordering cognates, the descent from space to number made it hard to understand. In moving from ordinal or space numbers to cardinal or symbolic numbers a vital connection with space was lost.

Now I restore ordinal numerals to their preeminent place as denotations of the Arithmoi.

And the construction of products from other constituent products as fundamental to spatial ordering.

Here we shall see how Grasmann through constraints restablishes spatial ordering through the combination of constituent products of ordinal numerals

Through his monadic systems or combinations he records multiple spatial forms and how they are relatively arranged and counted, and what other multiple form may represent them either in count or effective spatial arrangement. To do so he restores rotation to its rightful position as a fundamental ordering process.

Reflection is usually added as a third process to translation and rotation. While it is useful it is a subjective intermediary. The third process that I allow is projection. This is a fundamental objective process generating shadows as objective forms.

The use of symbols and signs, in particular the contra sign must be carefully distinguished and interpreted so that what is objective and what is subjective is realised, and what is to be counted or discounted is clear.

The emotive use of terms like annihilated in physical phenomena equally must be sceptically received, for what is subjective nd counted and then discounted has never been physically present or physically removed!

While this is a head of its place, the designation of concepts like energy apart from their observables is fraught with difficulty if allowed to be considered as objective  the use and method of directed line segments provides a safeguard against such fauxes pas!

The product , one of many Grassmann constructs from his 3 product constraint groups he calls an averaged product of the 2 main products, or spatial orders, the outer product and the inner product. The inner product involves a projection process in addition to the cyclic rotation process in both products.

The constraints define what we must do and when to carry out this combined process of ordering space!
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 « Reply #16 on: July 05, 2014, 11:42:14 AM »

The inner product in the Vorrede has been a problem to understand , because the first ordinary line product was expressed dynamically as 2 adjacent and or thrusting together line segments. By parallel projection one may construct an intersection point which then defines the 4 points of a parallelogram.

The inner product is defined in terms of 2 line segments projecting vertically into each other to form a trig line segment in each of the ordinary line segments.  Now I have taken the inner product to be the product of these 2 adjacent trig line segments, as per the ordinary definition. Gone ver another product may be also implied. The trig line segment producted with the ordinary line segment it is lying in. This product in general would be
bcosøaa which is bcosøa2 = a2bcosøu2a

ua is the unit line segment in the a direction

I will wait and see if he clarifies this here or in the 1862 Ausdehnunglehre

There is an argument that one should define the projection of a line segment onto another in terms of its unit line segment. This is usually called normalisation . At this stage it is an option I have yet to find Grassmanns word on.
 « Last Edit: July 06, 2014, 02:12:07 AM by jehovajah » Logged

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 « Reply #17 on: July 06, 2014, 02:57:50 AM »

Combinatorics and Kombinationslehre were first introduced to me by Lancelot Hogben, in relation to ordering playing cards. I only really looked at group theory last year or so when Norman started a course in it, and I took opportunity to explore a subject I could not connect with at university.

So it is not surprising then that it has only just dawned on me that the cyclic groups are precisely the same as combinatorics!
http://www.millersville.edu/~bikenaga/abstract-algebra-1/cyclic/cyclic.html

It was as a result of looking into cyclic groups that I constructed Newtonian triples.

Now combinatorics always reminds me my beginnings with permutations. From permutations I can remember selections and from these the combinatorial choice formula, and finally the link to the binomial coefficient.

In my blog I did some thinking on sequences and spatial ordering, trying to grasp at some connection to direction . I found that ordering with repeating of options and without repetition of options from an option pool had 2 formulae to give a count, but basically these counts were only significant for limitations. The infinite case was not really fathombe, or even interesting!

I came up with a fractal " cuboid" spatial arrangement for permutations , taking it beyond the table or gridlock format, and finally found I could make a nested Circe arrangement  with spirals.
The basic tree diagram for choice was what I remembered and used and found it was precisely the same as the 2d array for sequence of lengt 2 nd the 3d array for sequences of length 3 . Thus I was stymied for sequences of length greater than 3 until I came up with the nested spherical array.

Well guess what, all meters that use dials or dial equivalents are using cyclic groups, even product cyclic groups. Our number system is based on product cyclic groups.

The connection here is simple and profound. Grassmann uses the " rotation" in cyclic groups to model rotation or gyre in space.

Thus for a parallelogram the four points are cyclically ordered ABCD . The points are rotated cycliclly BCDA etc..

However we can take any 3 points , order them cycliclly and rotate among them cyclically. We can do this also fo any 2 points.

If this is done systematically, ie as a meter dial system we can actually model sequences of rotations at any level And structure reference frames , basis line segments , rotation systems etc cvordingly.

When I realised that Grassmann was using cyclic interchange to control sign changes I had no idea he was actually using cyclic groups to define spatial rotation and  orientation.

Strictly speaking I glimpsed this from the work we did in hunting for the holy grail. It lead me back to create the Newtonian triples, but I struggled to get a clear picture , a memorable understanding of what I was doing, what Grassmann was doing and what the hell were we all doing symbolically!
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 « Reply #18 on: July 06, 2014, 10:52:05 AM »

The cyclic " meter concept" leads me to use the fundamental expression as follows

AB + BA = AA . This I am going to call a phase product.

Similarly

AB + BC + CA = AA

Any cyclic rotation around a set of poits at the vertices of a form will begin and end on a phase point. The thing to note is that a line segment has phase points!

For a triangle we may establish a 2 dial meter based on the triangle cycle and the line segment cycle.

Thus
Setting the meter readings horizontally , a letter ( point) from each dial we get
AA
AB
BA
BB
CA
CB
AA

We can clearly see the phase points marking the beginning of a phase and its End/beginning. Without going ino detail this is precisely the role and meaning of Shunya, the alpha and the omega!

External to the meter is the motion of principal orientations, principal directions and contra directions. These are usually encoded using the +\– Direction signs. Orientation is not encoded but usually the orientation is set by convention. However, we are not to be bound by convention, and may define orientations by 2 points of a line segment . In doing so we must distinguish principal orientation and relative orientations, and certainly understand that direction is not the same concept as orientation.

Now we have the meter and phase points we can understand that for a point a phase point is not defined, for a line segment 2 phase poits may be defined  AA and BB and for any n gon n phase poits may be defined.

However when written in meter form the meter reading shows phase points as extended alilterations, ie AAAA.. Depending on the cyclical structure being recorded. ABB for example is not a phase point for the 3 meter reading( that is 3 nested cycles) but it contains a phase point for one of the nested cycles. Similarly BBA is not a phase point but it contains a higher level cycle phase point "waiting" to complete the higher level cycle.

Putting it this way I hope makes clear the sequential ordering which is the topic of combinatorics, and how it naturally encodes spatial rotations of forms, as well as contra statuses which come from the line segMent rotations! We might call these reflections, but they are rotations through a point ( or round if you want to include 2d and 3d  degrees of freedom).

Psychologically a reflection is constructed from these rotations through or round a point. Physically they may be impossible to perform as rotations round any point but the point is definitely rotated contra as it "goes" through the point of reflection. Again this is subjective processing of information returning to the source by Reflection at the point of reflection. Light does not pass through this point.
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 « Reply #19 on: July 06, 2014, 11:48:17 PM »

There are some fundamental issues with my understnding of Hemanns system that I can now address.

Certainly the labelling of points is the underlying handle to correctly determine wat is going on or being represented. But Hermann also uses a line segment notation.
e1 represents AB . If written e1 = AB I think this is a common misuse of the = sign . This occurs in the fundamental expression and is why I refer to it as an expression.
e2 —> BC and e3 —> AC gives
e1  + e2 —> e3

This does not resonate like the point version. It contains less information and thus requires a diagram to convey that missing information. So why use it?
It is a mathematicians conceit that if the essence of a notion can be expressed in the sparest of notation, that that in itself is elegance! However for most of us it is obscurantism. However, I will allow that if a teacher employs such symbolism in rhetoric, explaining as he discourses both verbally and non verbally then it is a piece of dramatic art, and like an orchestration enhances the teachers performance in exposition.
To write such a thing in a book is however an inanity. Just as magic written down is not magic at all, but mere pedantry!

However Herman means to show the nature of the spatial ordering of products and the invariance in the form for a product of line segments as compared to a product of points. The summation is less significant, especially as it is not a sum in the arithmetical sense.

Several adjustments or constraints on use of this lineal notation are made, designed to give maximal mnemonic impact. Thus in a parallelogram ABCD AB is // to DC and has the same length coefficient so e1 represents both these line segment and any and all parallel to it!

This deconstruction of space into parallel lines with a single lineal symbol is often ignored. What it means is that the whole of space is regularly tiled by this set up, and that what applies in one parallelogram applies in each parallelogram cell throughout space!

While this is not necessarily true it does reflect our psychological perspective. This is how we conjugate space.

The product e1 e2  is the sum of products in the plane, if you replace a lie segment by a sum or combination of line segments.

The use of the term combination is more in keeping with the process that is going on.

The combination of products is misleadingly presented as = . Thinking in these terms obscures the representational nature of the expression is lost. What we are doing is representing one product by a combination of other products.

The product itself represents a projective process.depending on the spatial relationship of the line segments the parallel projection involves a projection in the principal directions or one in a principal direction while the other is in the contra direction.

Also the principal rotation in the figure affects the description of the lineal combinations of the sides( line segments)

e1  + e2  represents both a clockwise combination and an anticlockwise rotation around the perimeter of the parallelogram. AD + DC is the clockwise rotated combination.

All these differences make it easy to make some unphysical interpretations of the spatial behaviours encoded in he 2 levels of notation.

Commutativity of lineal combinations in a parallelogram context are in fact clockwise and counterclockwise combinations.

Distributiyity of the product process over lineal combinations also needs to take caeful notice of the principal rotations as well as the directions of line segments.

This is all tied together at the level of the combinatorics meters in the previous post
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 « Reply #20 on: July 07, 2014, 10:07:44 AM »

The trigonometric convention for measuring angles and computing the sine and cosine and tangent Ratios has the circle divided into 4 quadrants. The convention makes the principal direction on the horizontal orientation to the page. In 3d that page swivels as easily as any universal joint!

The principal direction is then always to the right of the circles centre , on this horizontal diameter.
However this then divides the diameter into two rays, each starting at the centre and directed contra to each other. These are termed ray segments and are as such radii of the circle. This old term reflects the fixed value of the length of the diameter and thus the radius or displacement from the circles centre.

Euclid defined, or possibly Apollonius redefined this property of a circle in terms of seemeia, or indicators. These " points" lay on the intersection of every diameter with the circle form, such that every point of intersection was the same line segment( length) from a unique point called the centre.

It is hardly ever pointed out that this is a mechanical description of a drawn form. As Newton observed, Mechanicl principles lie at the base of geometry. That is to say empirical and metrical concerns precede and define the fundamentals of Geometry prior to any reasoning or ratioing.

For example the 4 quadrants are empirically determined; the 6 sectors are empirically discovered and determined, various figures are mechanically constructed.

It is because these constructions are so basic and repeatable that they are universally held to be true. The " perfection of them is hardly noticeable. However, who has not had the slightly disconcerting experience of segmenting a circular arc into 6 by the Radius displacement only to find it does not quite fit?
That is when you make the 6th segment mark it does not land on the hole made by the compass point!

As a young geometer it is easy to believe that your fumbling attempts are the problem. More skill , practice and care should eliminate the discrepancy, but it does not. Instead what we do is perfect the result. After all it is a pragmatic thing to do. However I find it highly suspicious that the perimeter length of the circle is a little over 6 1/4 radii. The curved arc naturally should be longer than the chord on which it sits, but only 1/24 of a radius longer seems too small!

We accept it because we have no other measure by which to compare. In fact the issue is neurological. Our sensors are set to enhance certain signals for definition and contrast. So the boundary of a form is enhanced to clarify it. The area or volume bounded by a perimeter is also enhanced, but when we use a Metron the surprising differences in the counts between similar figures is just one of those empirical spatial behaviours that defies simplistic pattern making and reveals the flaw in pure reason not grounded in empirical data.

This is how it is. This is what our fabulous unconscious processing delivers up to us to reason with; to compare and contrast and distinguish by symbol, name or some internal experience of memory.

It is the " perfecting" of our memories that blinds our inquiries. Those who have insights have not obscured the flaws or ignored our process of pragmatic perfectioning.

Our conventional system of assigning Values to the ratios of line segments in a right angled triangle has been augmented over the centuries. The mechanically derived tables were initially partial and limited to a few indexing arcs.. These arcs associated with the appropriate corners became the concept of a corner angle..

The perimeter of a circle was divided in the Sumerian times by the Akkadians, into 360 parts. Each part was a day in an annual cycle of seasonal observations of the stars, moons and planets. Thus 360 was empirically derived, by marking sighting lines on a great circle.

During the Napoleonic years the Ecole in Paris decided to make the quadrant part of the metrical system. It was divided into a thousand sectors! For each sector the sine value was calculated. Needless to say it took years of calculation to complete, but was never introduced into general use! Copies of these greate tomes are in the library in the French Acadrmy.

Thus the angle, the arc and the circle are indissolubly linked. Even if you, like Norman Wildberger consider the current angle measure flawed you cannot escape the relationship between the right angled triangle and the circle.Thales theorem only asks you to accept what is empirically demonstrable: all orthogonal angles are equal. This statement can never be deduced, it must be induced from the circle properties and empirical measurement.

Before directed line segments were introduced, in particular contra line segments were distinguished by these circle measurements. Thus a diameter was evaluated as 2 radii, but the radii were rays contra from the centre. They were said to be opposite or 180° to each other. Imposing a – sign onto this ancient angle or arc measure system of defining direction was bound to lead to confusion!

In the first quadrant all ratios were positive(+). This meant that the principle orientation had the direction drawn to the right of centre, or vertically above the centre for the relative vertical orientation.

The relative vertical orientation was introduced precisely because of the right angled triangle.. Once the principal orientation is fixed the relative vertical is also fixed.

However the quadrant is limited to a quarter arc. Pragmatically this was not a problem because you only needed a quarter arc to define the other quadrants! Similarly if we defined a sector of a 1/6th arc then we would have to apply it in each sector. This means we are pragmatically forced to repeat values in each sector , and thus confuse ourselves as to relative orientation!

To overcome this 2 quadrants are used to define the trig ratios. 2 scales were written on every protractor. It was crucial to state which scale one was using.

Eventually the mathematical board decreed that 4 quadrants be used, that contra rays be used and that the ratio signs be distributed in the quadrants by the CAST mnemonic. It also decreed that the conventional principal rotation from the principal orientation was counterclockwise!

As usual this was ignored at the elementary level. So we still refer to the interior angles of closed forms and the exterior angles of the same forms., or we defy the board by rotating the figure or paper into the " legal" orientation and then measuring!

These games we play have serious consequences. We become confused and use incorrect values and our calculations then do not match the mechanical reality! Which do we believe? Unfortunately most are taught to believe the calculation!

However is the decreed method practical? For example how do you measure an angle between 2 lines when both are not horizontal? That angle is the Difference between 2 decreed angles, and that forces us to accept that angle measures are fundamentally differences between decreed angles..

Using that notion our sine table values have to guide us to the correct ratio values  for angle differences up to 2$\pi$ radians.

The overlaying of the negative(–) contra sign onto the older rotated contra orientation is still a fundamental confusion in the notation. But mathematicians seem to love these quirks!

If I start with 2 contiguous rays, some call these parallel and touching, I always want to call them collinear, but perhaps I might just use collineal; then giving one the conventional principal rotation means it's relative orientation changes anti clockwise but it's principal direction remains positive (+). By this I mean moving away from the centre the arrow I attach in the initial position is fixed in that line. By this means I can perceive how my experience of its direction gradually changes from one direction on the principal orientation line to the contra direction. To see this clearly I have to do a vertical projection from a fixed point in the rotating ray.

Now I associate a decreed rotation with a projected direction on the principal orientation. Note that the ray projects a direction onto the principal orientation not just a length.

So an angle difference strictly projects a ray difference or a direction difference onto the principal orientation.

The simple minded approach often ignores these subtleties. To be fair, they arise only due to the need to be consistent, but that need arises because confusion reigns else!

Why do we make it so confusing? Or rather allow it to be so ambiguous? This ambiguity is confused with flexibility, or rather as we all know it allows a fudge factor to adjust our confused thinking to empirical evidence?,

These issues were all considered deeply by the Grassmanns in an attempt to promote rigour in these kind of discussions. Justus in particular showed a dogged determinism not to give in to the logical inconsistencies, but it is so hard to eliminate them in confused thought. Hermann by his system of notating everything at least made them obvious, even if he then was forced to define them away!
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 « Reply #21 on: July 07, 2014, 02:56:40 PM »

The issue of commutativity can now be tackled from this geometrical view. A line segment in a parallelogram as it rotates into the relative orientation of its adjacent line segment may have its principal direction changed to contra, but certainly as the adjacent line segment rotates into the relative orientation of the other side it has to become the contra of that orientation. .

This cyclical rotation is clearer in the point notation, but it is possible to see how the projections onto the principal orientation confirm this..

This behaviour applies solely to the parallelogram form. If we wish to use different forms we will have to accept a more complex relationship between sign change and line segment orientation change. Hermann in keeping with his scheme refers to these line segments as factors. As yet I have not found him referring to the cyclic interchange as commutativity. This is a denotation imposed by a later more robust form of group theiry, and it obscures cyclic interchange in the geometry of the parallelogram.

It is to be noted thst this interchange does not apply to triangles, because contra directions are never achieved..
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 « Reply #22 on: July 07, 2014, 03:10:27 PM »

Grassmann defines the inner and outer products in these cyclical terms. The outer product is where the line segments step apart from each other. He says the value gains worth as it does so. The inner product arises in the context of vertical projection, and this value gains worth as they step closer together.

This only makes sense if the arc of a circle is split into 2 by these line segments. As they step apart the arcs behave in a contra fashion, one increasing as the other decreases.

For the product to increase in both these directions the vertical projections must involve the sine and cosine ratios.  The outer product must involve the sine ratio and the inner product the cosine ratio.

As yet I have not clearly seen that in the 1844 passage in the Vorrede, but I am still looking.
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 « Reply #23 on: July 08, 2014, 06:35:48 AM »

It is heartening to find that even today traditional math topics are still being taught, but it is disappointing to watch how formulaic and rote and devoid of understanding much of the teaching is.

Watch these videos and try to understand how this relates to hermanns averaged product.

The second video gives the ancient circle theorem that explains how arcs, arc measures and angles as chord or secant intersects are related.

I have known and used this theorem since I was a lad , but the arc measure and the radian was never explained to me . Later I learned of the use of the radian as an angle measure. Even later I realised the circle arc measure itself was a universal measure of Ritation. Using parallel lines and translations I could show without angle measure that the sum of the angle measures  in a triangle, converted to arc measures was always the semi circular arc!

What I was looking for next was how to deal with arcs that were circular but not drawn from the vertex or intersection of the angle.  These videos show the sophistication of Geometrucal ideas in ancient times and how the circle was fundamental to that sophistication.

In relation to line segments and the inner and outer products it is clear to me that these products are not only related to the vertical projection of one Ono/ ino an adjacent one but also the circular arc that passes through the 2 end points of the adjacent line segments.

A vertical projection naturally establishes a right angled triangle. But a right angled triangle is a structure within a semi circle! We have two important circles both intersected by these line segments ( extended if necessary). The behaviours of these internal arc measures and external arc measures governs the apprehension of the outer and inner products..
While outer and inner products are constructions of parallelograms, these parallelograms are intimately connected to circular rotations.and arc measures. The averaged product that Grassmann constructs to describe quaternionic Ritation, or any general rotation( ie not round a circle centre) is based on this line segment intersect angle measure. The point of intersection can also be barycentrically determined as Shan by these secant formulae..

While I have more research to do to show this conclusively, it is intuitively a sound conjecture.
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 « Reply #24 on: July 08, 2014, 10:05:07 PM »

Quote
Now Thereupon I was going over to the  wider implications and motivations  of these things, that I was appointing for the second volume of this Work, wherein I have commited to refer to specifically everything of those things, which totally simlarly,presupposes  the label/ handle of the corner angles or of the Swivel

This implies a full trigonometric useage, even at this stage.

Quote
Yes, this second volume, which will complete the whole work should first be printed Later ; so it seemed to me  there was a need for an overview of the whole, which related Revelations arising here going forward point to something more exact.
To this end I have initially given to you  the Results which already had revealed themselves for  the melded together reworking. I have even pointed out how the product of 2 line segments can be apprehended as a parallelogram! Even if as happened everywhere here the line segments will be with the direction firmly attached.

The restriction or constraint of a fixed direction does not prevent one from constructing a parallelogram from  any 2 line segments. Clearly the properties of parallel lines are implied here.

Quote
But how this product thereby has been defined, how the Factors ( cyclic) interchange can only happen with the sign change; while at the same time the two like directed line segments are openly visible as 0

This definition of a product of 2 line segments implies that they rotate relative to each other, but it is not clear that this was the mechanism of the " cyclic" interchange . It took me a while to perceive that this was how he saw it in 18444 when he wrote these words.

These directed line segments or rather line segments with an arrow symbol firmly attached indicating direction , he thought of like the hands of a clock. When they point in the same direction there is clearly no parallelogram! However when they point in opposite directions this is not so clear, but he makes no comment about that here. At the beginning of the Vorrede in 1844 he does comment on oppositely directed line segments with the arrows firmly attached, but in this case the hands of this " clock" point in the same direction when at the semicircle arc rotation .
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 « Reply #25 on: July 09, 2014, 09:16:45 AM »

Quote
A Different label/ handle places itself besides this label/ handle( for ordinary line segment products and how they are calculated), which likewise relates to directed line segments, that is line segments with the direction arrow firmly attached.

Specifically, even if I project  the line segment vertically onto the other ( line segment), the arithmetic product of this Projection in  the line segment Which has been projected upon reveals itself  just so like  a product of those line segments  , provided that also  for this case the multiplicative Relationship applies to the addition .

Here Grassmann is not specifying a formula, but a well known projection process! Even the other line segment is not specified. The only specification is the projection is vertically onto. There are 2 descriptions of a vertical projection of a line segment! They both relate to the right triangle.

The construction line is called dropping a perpendicular or vertical line. The direction of the line segments is important to specify where we drop the perpendicular line.

Grassmann here makes no mention of the construction line just the arithmetic product. The vertical line is also an arithmetic projection product.  The reader seems to be conducted to the line segment that is being projected onto. Suppose the line segments radiate from a vertex then the projection would involve the cosine arithmetic product. A third line segment , the proection line has to be invoked to use the sine arithmetic product.

Grassmanns method involves considering a third line segment in the plane and in this special case all 3 Strecken form a right triangle.
However he does not discuss the right triangle here, instead he says " likewise" the line segments product as this arithmetic product in the line segment.
This is the unclear part that I struggle on.
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 « Reply #26 on: July 09, 2014, 07:41:41 PM »

Ok, I think that I have been misunderstanding the verb darstellen in this passage! I was reading or seeing only Stellen.

It seems clearer that Hermann was comparing his ordinary line segment product with a projection of that product into one of the line segments.

That is if ab is the product of two Strecken then abcosø is the projection of that product into one of the Strecken! Apparently the representation of the shadow product is similar to the product of the ordinary line segments.

I still do not get this sentence structure here or this comparison. If I have got it wrong I will rewrite sections on trig line segments which I derived from this section.
 « Last Edit: July 10, 2014, 10:53:38 AM by jehovajah » Logged

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 « Reply #27 on: July 10, 2014, 12:23:53 PM »

I went back to basics.
I recognised that I had a gullibility inbuilt by almost mythical accounts of Grassmann and Clifford algebras.mbut my research showed me that Grassmnn was a man or child of his times. He was well read and educated himself through life. His great Förderung was not a new mathematics but rather a new approach to leaning memorising and manipulating mathematical ideas..

Using his approach did not mean you could skip learning mathematics, but you could learn it quicker, memorise it more securely and apply it more efficiently. In addition extended use of his system developed a creative intuition about how we interacted most effectively with space and formed formal mathmatical models.

But it's power lay in his childhood observations, which he developed refined, modified and extensively reworked throughout his lifetime. The important words are Zusammnhängend and Bearbeitung. Hermann melded, reedited, redacted, reworked everything he learned guided by one simple set of ideas. His ideas were based on his fathers presentations of " logical" geometry constructed by human action and thought, but his observation was that of a child, he was enthral led by the way the Notation behaved!

I have described these observations elsewhere, but the fundamental concept that drove him was the way the notation behaved for the geometric products. The geometric products were the areas of geometric forms.

When I first heard of exterior and interior products I immediately reflected on interior and exterior angles and their use in area formulae. I was trained in traditional mathematics, but knew nothing but a bunch of tricks and formulae. It was a disconnected mess. Consequently I did not know that much of what I was taught at elementary level was in fact based on the ideas of the Grassmanns.

The parts I have struggled with in translation have been those elementary observations and definitions of area of form! Thus when Hermann says that the product of a parallelogram is the same as the product of a rectangle I was confused, and thought he meant something else! I mean everybody except a child knows that!

I could not accept or expect that a book on higher or deeper level Maths would be looking at areas! Well yes he was and is!

The area formulae are all arithmetic products. They are also geometric products. What is the difference? The difference is what subject area they are explained in! In other words they are the same thing, but in Gometry it is required that the formulae are justified by geometric proofs. These proofs can be a bit complicated to explain.

However using Hermanns notation and system these proofs can be given quickly, providing you apply his main idea which is that a line segment should be considered as a ray segmnt( in modern geometrical terms). In addition the sign of the product must reflect the underlying ray geometry. Thus if a ray is defined, you cannot just swivel it round without paying attention to its sign! But the sign only changes every $\pi$ radians of rotation.

In Addition he noted a relationship in the rays of a triangle. This he captured in a fundmental congruence. Unfortunately he used = signs to represent a congruence relation. However this " bad" geometry has so many resonances with Arithmetic that it has bern hard to alter without losing he creative interplay.

For example Möbius used some similar ideas, but by being strict he lost some of the immediacy of his method and the creative flexibility. He certainly lost the connection to the geometric products of points!

I think Grassmann in this section is saying that dropping a vertical to a base in a parallelogram gives you the arithmetic formula for area of a parallelogram. This formula is still in the form of his general concept: product 2 line segments  to construct a parallelogram.

However this statement requires the following section to be clarified. But I have retranslated this section above to be clear. As it stands it has no relation to cosø or sinø, yet . The basic formula is base x height!

Each of these line segments or ray segments can be viewed as pairing with others to construct parallelograms. Using that idea it is easy to show that parallelograms are the sum of products that are rectangles!

This does not fall out from the fundamental product sum equation, but it almost does! The mathematician/ geometer has to guide the interpretation of the symbols. Once that role or uty is accepted, Grassmanns notation makes" proofs" less onerous and quicker.
In ab is the product of two line segments and I can equate that to cd + ed then I have formulaically or algebraically shown that a parallelogram is equateable to a rectangle.

The use of the terms = or congruent are problematic, but we know what we are demonstrating: how a form can be transformed into another, preserving certain invariant properties like area!
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 « Reply #28 on: July 11, 2014, 07:01:09 AM »

Quote
Specifically, even if I project  the line segment vertically onto the other ( line segment), the arithmetic product of this Projection in  the line segment Which has been projected upon reveals itself  just so like  a product of those line segments  , provided that also  for this case the multiplicative Relationship applies to the addition(Distributive rule). However, the product was from a wholly different Style, the how  of that earlier Line segment product, in the case of the factors: the same ones were interchangeable without sign change and of 2  perpendicular adjacent line segments the product seemed as null!.

I named that product of the first line segment( type) " the outer type line segment" product and the last line segment ( type) product as the " inner type line segment" product, because that product only by outwardly stepping directions and this product only by the same directions approaching (, that means  being partially interlocked ) had a  potable value.

Ok, so what I have deciphered here is simply that the two products are the product of the parllelogramme sides for one and the product of the vertical height and the base or the other.
The former is called the outer product, and the latter the inner product.

The former is called the outer product because the irections of the line segments or rays spread apart to form the vertex angles . The latter is called the inner product because those sae line segments that form the rays of the sides of the pRLlelogram have to be approaching each other rotationally to enable a perpendicular to be dropped to the base.

Both these conditions must be met for the products to have a value. That is they can not be evaluated if one of these conditions fails. In this case either the outer product is null or the inner product is null . In the case where the outer product is null, the inner product is alo null or not defined. In the case where the inner product is null or not defined but the outer product is not null the line segments will be perpendicular. One might say the inner product and outer product become idntical, and can be evaluated as the arithmetic product.

The arithmetic product is the area of the rectangle that is equateable to the parallelogram. This is only 0 when the line segments are in the same direction or collineal or parallel. We could also say it is 0 when they are contra to one another.

If I am right in interpreting this section it means that the inner product is
Base x Height= line segment x line segment x sin of angle between.

Also this does not affect the designation of types of line segments I have posited in an earlier thread.

What about the dot product? That would appear to be the quotient operator in the following section.

If the outer product and the inner product evaluate to the same arithmetic quantity why remark on the differences?
The geometric value of noting these differences gives us a flexibility of approach in describing space and its behaviours. But how Grassmann uses these product patterns or constraints is at the heart of his method of analyis and synthesis .

If I have interpreted it correctly then this paper on Quaternions will demonstrate that or correct me, because it is all about these 2 product patterns conjoined to form an average product pattern.
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 « Reply #29 on: July 12, 2014, 10:49:29 AM »

I have now quickly surveyed the 1844 contents, addendums and combined index. The inner product as an important part of Hermanns system does not appear. The outer product and various other products and reduction combinations are the main point of the system.

The Abschatten product( the shadow cast product or cosine projection) the eingewandtes product( one relation product) and mixed products are all described in detail as types of outer products..

It is also clear that in developing his system the vertical projection and the intimate connection to the right triangle and thus through Thales with the circular disc and plane, and ultimately to the sphere and vortex was crucial to evaluation or metrical methods.  This means that the outer product could be as " abstract" or analogous as the mind could make it, but there was always a way to calculate a " numerical" answer, or really a count. What is this count?

The importance of counting is not the naming. The importance is in the ordering of space. The ordering of space is our most complex behaviour, from which we derive and induce and deduce many things.  The count became a check sum. This idea of a calculation checksum was taken up by the computer programming field in designing the format of a programme main function. In a main function a behaviour or process is laid out in terms of aggregation operations, comparison operations memory storage and movement operations, register shifts etc. the way to check this was being performed correctly was by checking bit sums.

The system evaluates in several ways. The main way is as a geometrical pattern in an orthogonal grid. This is essentially what the inner product is: an orthogonal evaluation of a general parallelogram.  The other evaluation is the calculation or count result.a final evaluation of importance is the null evaluations! These evaluations give orientation information.

The question is; why use the inner product in describing Quaternions, when it is rarely used in the 1844 version? Does it have a more significant role in the 1862 version.
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