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Most of us have played with bricks.

When you set up an array of bricks say 2 by 4  we know we can set that array in any orientation on the ground.  I also know that you would immediately sy that there are 2 ways to construct this array.

Well then , imagine the bricks scattered across the room. To construct the array in the centre of the room you could collect each brick individually and return it to the centre and carefully place it in contact with the previous brick if there was one. Once you got to 4 you could then suddenly place the next brick on the top or quarter turn side of the first brick. The sequence is extended,by carefully returning and placing the bricks next to the previous one and in contact with the " row" of bricks below one brick to one brick. You woul stop at the end of the bottom or first row with your last brick in the second row on top of the last brick in the first row.

In so doing you have created and defined a row of 4 bricks and an array of 8 bricks consisting in 2 rows of 4

The combinatorial product of your effort can be defined as a 4 by 2 array, where the 4 signifies the count of bricks in a row and the 2 signifies how many rows., and  the word array means they are patterned not scattered, and indeed they are as contiguous as possible.

Well that was 2 products of a possible 3 using that particular method. The third variation is to start the second row below the first in contact with the ground. The second was to stack the second row on top of the first row.

Well of course there are more methods of forming this array. Now we know what it looks like we could save time by collecting 4 bricks in sequence before returning to the centre to place them carefully.. We save on the repeated foraging, but now at the centre we have to decide which sequence to place the bricks in, and of course we have 4! Choices now.

 There are many other scenarios that result in the same product but involve different choices in layout of the array. The "other" choice you may have considered traight away( why?) is where the trip involves collecting 2 bricks at a time and constructing the array 2 bricks at a time.

We still have several choices if we lay the bricks in pairs. We could for example lay the bricks in pairs in a row that totals four. But the one you are probably screaming out now is the laying of bricks in pairs, but carefully placing the pairs on top or to the topside on the ground or below the first pair on the ground so they are contiguous. The next pair is placed on top, above or below the previous pair until 4 pairs have been laid.

These are the pragmatic choices one is faced with when constructing or synthesising a mosaic array .

Now ,after all of this we are expected to note that the 4 by 2 array has become a 2 by 4 attay.. It should be clear that this is a rather meagre reduction of the combinatorial choices we have just touched upon. In addition we are supposed to assent to the term commutate as if it means nothing spaciometric ally. The fact is it is a symbolic , literal device referring only to marks or symbols on the page.

Giving our assent to it is all that is sought!  The methods and processes are distinct and distinguishable and do not say ab = ba!

To get to this statement we have to agree that row counts will precede column counts, thus 4 x2 means 4 in a row,2 rows.
2 x4 means 2 in a row 4 rows.

If I now apply this to the measurements of a single block say 3 cm laid out horizontally to me , left to right on the ground and 2 cm vertically on the ground, forward of me then I get
 
4(3) x2 for one method  and 2(3) x4 for the other arrangement

Equating these 2 formulae is equating the lengths of one side of the bricks! I have ignored the depths of the bricks.

Similarly I can write
4 x 2(2) and 2 x 4(2) to equate the other dimension.

However, now my notation has to be defined. The brackets are to signify the measurement of the fundamental brick in the direction of the row or column.

All this is usually finessed away. It is confusing to write it down in longhand and the student is not shown this instead they are expected to just pick this up from the demonstration and the exercises.

It is very tempting to ignore this detail, but it in fact is crucial to understanding multiple forms in symbolic notation. It was discovered or noticed by Hermann Grassmann, and it is the reason why logicians have fundamental difficulties with deriving multiplication

Let us use the full notation of this form now

4(3)x2(2)  can be symbolically equated to 4x2(3:2)

This format can now be defined to represent the foundation of multiplication of scalars.or ordinary arithmetical multiplication. The bracketed  pair are the dimensions of the brick and the notation relates them in a proportion or ratio format. The ratio format can now be defined as information about the single brick and representative of a single brick. The scalars now multiply to 8 the known numbers of bricks, and we have now captured the relevant information  for counting and identification purposes that should enable anyone to construct this mosaic array in any orientation in space.

2x4(3:2), and 4x2(2:3) and 2x4(2:3) are all different constructions, but the scalar commute in this precise sense. Not in the undefined sense hitherto taught.

Of course this is still a meagre representation of all the possible construction algorithms, and this is where Grassmann gradually realised that the stretch of these dimensions by an extensive method was very important to describing position in space. Of course he was not to know the Pythagoreans had already developed this idea.

<a href="http://www.youtube.com/v/ScTJRrg8ZfU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ScTJRrg8ZfU&rel=1&fs=1&hd=1</a>
There are 2 important things to note here: by removing Arabic numerals the discussion becomes more geometric and a direct link to the Arithmoi is almost obvious. The arrays of 1 are geometric Arithmoi.

Secondly, because Norman has defined the arrays to be. Row arrays he misses the Law of transitive distributiyity in the associative law. He in fact illustrates it with the circles, but fails to name it the wedge product.

It's real significance is in crossing the operator imposed boundary. The so called operations are combinatorial arrangements so
2.(3.6) = (2.3).6 = 3.(2.6) but
In vector group algebra  the wedge (2+3) can be scaled  6(2+3) and this is the same wedge formation as (6,2 +3) or (2 +6.3) where the combinatorial+ joins the Strecken of different " lengths" into wedge forms which are rotated relative to each other.

The wedge form is a modern notation for the transitive distribution just pointed out. Because it is a proper multiplication it is confusing why modern notation writes it as a +

This is because of abelian group notation.

The Wikipedia article has improved somewhat since I started my research . It could be I understand it more, who knows, but I can recommend it as an overview.
http://en.wikipedia.org/wiki/Hermann_Grassmann

I still Have issues with some concepts  said to be Herrmnns, particularly the "linear " or vector references. The purest concept of Hermanns thinking is given in his Vorrede, as we shall see, but lineal means just that, a line! Ausdehnung is to reach or stretch out especially an arm as in pointing or exercising. This is therefore a theory based on stretching out lines.

Mathematicians often like to talk about extending a method in a certain way to make it more general or applicable, and I am positive Hermann was aware of this meaning, which is why he specifically puts lineale as the adjective.

Another way to think of it is as a tailor stretches out his tape measure to get the dimensions of a suit , or an architect does the same to measure out the positions of his footings , they perform a lineal Ausdehnung. They measure Ausdehnungs Größe and record them in a way that enables them to construct or synthesise some product in space. Hermann simply recognised this everyday procedure as a mathematical method of analysis and synthesis. The next thing he did which was down to his fathers interest was define the underlying group and ring structure for this method. He therefore intuitively recognised points and lines as fundamental elements. By trial and error he isolated the begränzte Linie as his fundamental mathematical object, which allowed him to see lines and pairs of points as duals of each other.

The third thing he always kept in mind was the dynamic nature of these entities. None of his terminology was ever meant to be spatially fixed, everything was meant to be variable and dynamic.

When Hermann rejected 3d space, he did not mean he was aware of some other higher alien esoteric spaces. He was saying that in der Raume the 3 dimensions have never been significant for proper construction. Every person knows that measurements have to be taken in any number of orientations to construct a form . Thus a form determines the dimensions most useful.

At one time he considered calling his method Formenlehre, because of the importance of form. The spanning of a space by a basis actually goes against his general idea. The basis was to be revealed as the kernel or the least structure that accounted for every point and which every vector could be reduced to, but this often involved using coefficients that were not convenient. If one insisted on unit coefficients or integer coefficients then there were a lot more bases to play with!

The diophantine constraints meant that certain bases could be used to solve or represent those types of mathematics, and other constraints led to other types of "spaces". The space where the coefficients are continuous or smooth is usually what is meant by a "vector" space, and this space reduces to a basis of 3 dimensions, but these dimensions are orthogonal by convention. It is possible to use more generalised reference frames.

It became possible to generalise this method outside of geometry into node or graph theory and category theory. The methodology has proved to be so robust that many computer programmers have been able to use it to structure their programming languages and concepts

The law of 4 points occurred to Hermann when he was studying some ideas of his dad about multiplication. As far as I know, Justus had difficulty placing multiplication in his logical and constructive " map"'of mathematics. In the end he resorted to a geometrical  definition of multiplication. Basically he said if you stare at the rectangle you will intuit that multiplication is soundly based!

I have explained the difficulty he had and why in the second post.  The difficulty arises from thinking numbers are real objects.  They are our response to a dynamic world and they are imaginary. The process of that response is called Logos, and it is a significant analytical concept discoverd by the Greeks , in particular the Pythagoreans
 
The Pythagoreans went further and compared logos with analogos, the basis of Eudoxus teachings in books 5 and 6 of the Stoikeia. However Justus had Legendres redaction of Euclid's Stoikeia and he felt therefore that Euclid was about geometry and his geometry was faulty logically .

Euclid's Stoikeia is not about geometry, but about philosophy, and it is an introductory course to Pythagorean philosophy. The real source of geometry was another subject called mechanics. Newton expressed this in his opening statements to the Principia.
Mthe artisan develops or discovers mechanical relations, tools that produce these mechanical objects. The skilled artisan refines these mechanical relations and objects in a meditative reflection called geometry .  This reflection abstracts and perfects these relations and records these ideas in symbolic notation.

The Pythagoreans went beyond this and looked at all ideas and forms in this abstract reflective way. This was and is the essence of philosophy. They uncovered invariant relationships at this level of abstraction and it is this discovery that intrigued Socrates. He playfully introduced the concept through the questions and discourses he had with his students amongs whom was Plato. Which one is reality? The sensory experiences or the meditative reflective experiences that reveal these invariant forms?

There is no answer. The listener must decide for themselves, but their decision will affect the whole course of their lives!

So Justus and many of his contemporaries who studied geometry through Legendres redaction found many logical inconsistencies not found in the Stoikeia, but introduced by Legendre 's redaction, which combined mechanics with the philosophical abstractions.

The problem with modern multiplication is that it was decided to base it on number bonds at some stage in the late 17th early 18th century. The renaissance men also failed to grasp multiplication from the quadrivium or Trivium in the seats of higher learning. The meaning of the Arithmoi was lost in multiple translations, and the consequence was the make do make piece called number.

So it was as if n a dream that Grassmann realised that the dynamic construction of a rectangle and more importantly the parallelogram  was the foundation of multiplication. He imagined the two required sides to be active and dynamic and they pushed forward past each other to create a parallelogram .,they also rotated relative to each other to generate different parallelograms. The crucial role of parallel lines in creating these multiple forms was clearly apparent.

 By constructing a parallelogram Hermann saw he could construct multiple copies that tessellated space!  That was a vision of an extensive measure that was fractal . The pattern of multiplication spreads out into space a one side by the other side.  It also resides in the single constructed parallelogram.

From a point A the two segments fan out to B and C. Through B a parallel line to AC is constructed, the through C a parallel line to AB is constructed, they meet at D and a form is made ABDC

AB x AC  = ABCD

Not very revealing until a third Strecke is introduced into the plane . For simplicity sake let us call it AD

Now the behaviour that makes this work is the proposition that says triangles on the same base and between the same parallel lines are DUAL( isos) . It was also generally known that parallelograms with the same constraints were dual. Hermann realised that if he made the combinatorial process of multiplication represent the construction of parallelograms then he could use this dual property with different looking parallelograms on the same Strecke !

So now considering say AB as dynamically extending along the same parallel line as he first constructed, he knew that knew that if AC was fixed then all these different forms were dual! And similarly for AC  . The point D might move further or closer so that the parallelogram at a went from acute to rectangular to obtuse, but ABCD would still be the same or dual.

It is important not to jump to the conclusion that we are talking about area. The area is not defined. This is the shape we will utilise as a Metron to count space. It is the notion of counting space that we have lost by losing contact with the Arithmoi.  The word area is derived from the notion of counting monads in an Arithmos.mthe original Monas is this constructed shape.

Now we are restricted if we think Arithmoi were just rectangles. They could be any pattern that tessellated, and even those that did not tessellated perfectly were used. The best concept I have found is to think of mosaics.

The third Strecke allowed Grassmann further insight he decided to use this to define the rules of addition and multiplication. In the parallelogram the Strecken were symbolic of every Strecken in the constructed tessellated space. Therefore a Strecke which was located at a point H say could be considered as brought into or projected into the same parallel Strecke in the defining parallelogram. Similarly what was demonstrated in one parallelogram was applicable to all .

A parallelogram construction must have two Strecken fanning out from one point. They do not converge to one point. That signifies construction so that AB. AC means construct a parallelogram whereas AB . CA is undefined.

The introduction of the contra concept allows this second one to be defined as well as BA.CA the converging case.

If a Strecken follows another through a common point then this is the law of three points situation, so the parallelogram contains two different law of three points that result in AD as the identity.

AC + CD does not = AB + BD, but they both have the same identifying Strecke . Grassmann overlooked this and thought he had commutativity. He clearly does not but at this stage I cannot quite explain what it is. It is certainly some kind of projective or reflective duality. I even think that Klein named it a glide reflection but I not certain

Emboldened Grassmann used the third Strecke to demonstrate distributivity of the new product and this is when he received the greatest shock?

AB = –BA
AC = AB + BC and CA = – AC = –( AB + BC)* = –(BC + AB) = –BC + –AB = CB + BA

Note the * on the bracket. Something else besides normal negation is going on and it seems to involve this anti commutativity through and through.

Finally AB can identify AD + DB and AC can identify AD + DC

The product AB.AC = (AD + DB).(AD + DC)

Assuming commutativity naively

AC.AB  = (AD + DC).(AD + DB)

These should all cancel but they do not!

AC.AB –AB.AC  = AD.DB – DB.AD + DC.AD–AD.DC

The first pair are legal constructions so they cancel, but the second 2 pairs are not legal constructions.
Converting them to legal constructions we get
0 = –DA.DB – DB.–DA + DC.–DA– –DA.DC

0 = –DA.DB + DB.DA – DC.DA +DA.DC
Carrying the signs through transitively creates the problem the right hand side reduces to 0 , but the constructions were changed to achieve this so
AD.DB= DB.AD = – DB.DA = BD.DA
AD.DB= –AD.BD = – BD.DA =DB.DA
AD.DB= –AD.BD = DA.BD = –DA.DB
The switching of symbols is confusing and misleading when mixed with the rules for signs. But he was determined to make it work so he eventually concluded that
AB.AC had to be made identical to –AC.AB

This ensures that everything coomutes to give zero but obscures what is really happening, and it is Cayley who pins it down in his matrix algebra and multiplication. Transposition is key to the difficulty, but by grasping the nettle and defining ab= –ba Grassmann captured an important part of the solution. The anti commutativity and the sign change crops up again and again.

Grassmanns product was therefore confused and that allowed him to make connections he might otherwise have steered away from.

It occurred to me, while trying to step out auseinander treten and ineinander treten with my feet, grasp what  Grassmann was saying  about the Senkrecht Strecke in the next important stage of his arithmetic, that the stossender Strecken may not just be fanning out from one corner as I imagined. These parallel lines would project out of every point of the defining Strecken , principally the three points that define the two Strecken A,B,C.

From A would project AB and AC . From C would project CD, and from B would project BD. these Strecken would thus be stossender, crashing into one another at the points of intersection, the 4 points A,B,C,D.

In addition we would have
AB//CD,AB=CD
AC//BD, AC = BD
And we could write AB.AC = CD.BD
And  AB.BD = CD.AC

Thus in the parallelogram direction and quantity of magnitude are inhered into the formal labels.
Going back to the law of 3 points I note Grassmann makes a point of apprehending the quantity of magnitude and the direction of each Strecke.

The equality sign is the misleading sign. The multiplication concept is not being extended it is being nalysed by Grassmann prior to extension.

If the forms above identify construction of a parallelogram, then the = should be replaced by —> to make a construction method.
AB.AC —>CD.BD

would then indicate that the sides are projected onto the following sides in that order by the parallel line projection.

Similarly
AB.BD —>CD.AC

Is a construction by parallel line projection. It is clear that the direction of the second projection has been reversed relative to the first construction method.

Commuting the 2 sides  should produce the same construction. We are not changing the orientation of the Strecke or their relationship in space. We are simply changing the order of notation

 
AC.AB —>BD.CD

It is clear that this projection maps divergent Strecken onto convergent Strecken

And

BD.AB —>AC.CD
This is glide deflection  mapped onto another glide deflection although the first structure is fractured by the commutation. But the non commuted structure mapped a glde deflection onto a fractured structural operation.

There is a sense in which the commutation has reversed some aspects of the construction process, but to define this Grassmann placed a third Strecke in the plane of the parallelogram set as structurally defined by these multiplication definitions..

Then he introduced the summation definition in the law of 3 points using the sides of the parallelograms to identify Strecken relationships.
To keep it simple  let AD be the third Strecke.

Now instead of construction forms being disallowed we can interpret them.

AC.AB –AB.AC  = AD.DB – DB.AD + DC.AD–AD.DC

The constructions give a pair of parallel lines, a parallelogram ith in the original and 4 half completed constructions, whichever commutation of the Strecken is chosen

This shows that in the parallelogram the combinatorial form of multiplication is commutative as before, but when we relate to the triangle we get a anticommutative result!

AD = AB + BC

AD.AD = (AB + BC)(AB + BC) = AB^2 + BC^2 + AB.BC  + BC.AB

Here we get 3 sets of parallel lines squared and 2 parallelograms.

Grassmann deliberately chose to make the parallelograms cancel! This was the beautiful result he was after, Pythagoras theorem. To get it he had to define anti commutativity,

I do not think he did it lightly , but knowingly and with trepidation. But as I have found it does not affect the commutativity of the parallelogram! By making this definition he had both the Pythagoras theorem, and the commutativity in the parallelogram where he needed it for calculation.

We can now distinguish the " fan" product from the " wedge" product. The fan product produces a closed construction of a parallelogram, the wedge product an open construction of a parallelogram , and these extend outside the original parallelogram. By adopting Grassmans definition we can swing the open ones inside the frame of the original so they are completed.

Flicking through an online copy I actually found some figures for the first time!
https://archive.org/stream/dielinealeausde00grasgoog#page/n315/mode/2up

<a href="http://www.youtube.com/v/132amJvoLpU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/132amJvoLpU&rel=1&fs=1&hd=1</a>
Now the word vector actually was coined by Hamilton in relation to his Quaternions. Grassmann kept the reality of what he was dealing with clear in his mind:Segmented lines in parallel formation. But the common meaning of stretch also he retained, and the dynamism of the stretch, however it is done, the difference between a point and a stretch from one point to another has to be kept clear.

The broader context of affine goemetry and projective geometry really need to be explained clearly, and Norman does that .

<a href="http://www.youtube.com/v/bmEM75lTLes&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/bmEM75lTLes&rel=1&fs=1&hd=1</a>

In this context Norman asks what is a circle. Turns out to be a proportion of 6 terms. This is twice the number of terms in a similar triangle proportion, because 2 specify the centre and the third the radius  in terms of quadrance. The first three terms are quadratic, that is quadrance forms.

To understand the full context you need to get book 11 of the Stoikeia.,then the definition of the meet and the join of points and then finally the quadrature of the circle..
<a href="http://www.youtube.com/v/ucczpxEySno&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ucczpxEySno&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/8rkrymhphMQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/8rkrymhphMQ&rel=1&fs=1&hd=1</a>
http://youtu.be/B-5EVtEOSAU
http://youtu.be/huR1yKwx-DM
The discussion about the radians is a side issue. The quadrance and spread have  major applications in all geometries and that is its significance.

Contrasting the proportion for a line with that of a curve is highlighted by this video
http://youtu.be/eMt09NN1WOE
In which it is noted that circularity simplifies the descriptions, and emerges as the fundamental property of convex polygons.