From Sequences and Series to Grassmann Algebra

The so called vector sums and vector products are in fact combinatorial conjunctions of various sequence process results.

The generation of sequences is a fundamental process in space. That we can characterise some of these processes and the sequences they generate is perhaps the defining ability of a quality we may call intellect, or intelligence. If so it certainly is shared by all animates, not just homo sapiens.

By starting in space and quantifying it by form , that is its several magnitudes of experience , I will be able to chart the innovative, but otherwise largely previously mapped out course to Grassmann's Algebra. It was of course previously mapped out by Plato and Euclid and ancient Greek conceptual models(myths) and analogies.

On the way I will relate how the Muses inspired a particular style of art that is referenced in book 7 of Euclid as the notions of Arithmos. The arithmoi will also be associated very pointedly with a form of artwork known as a mosaic, a type of decoration found particularly at shrines to the muses. It will also be pointed out that such shrines we're the model for Plato's Academy, and other such Academies founded in Alexandria in Egypt. It may be that the true documented history of Pythagoras is associated with the spread of such shrines in Ancient Greece.

I am going to write in 2d about a 3d spatial process. I would like to show what I mean using blocks like Lego , but I would have to film that. So imagine I am filming and animating what I am discussing. In this way I hope to avoid the traps that lead away from the direct route to Grassmann Algebra.

Grassmann felt keenly that Geometry had been misapprehended and particularly 3d geometry, that of space. He felt Phorometry justified his opinion, and would soon demonstrate the geometry of space as being foundational to Mechanics and physics by exposing the real Kinematics of motions in space. Thus Grassmann thought and notated in 3d, that is in terms of ideas or forms not marks and squiggles. In 3d we do not have operations such as multiplying or adding. Instead we have construction and destruction, processes of combination and transformation, growth and decay. Also, and this is fundamentally important, we use real quantities of space, not thought ones. Thus if I talk about a line. I physically draw and cut that line out and use the cut outs, or bricks or whatever. This is about how empirical space works, not idealised, imagined symbolic space.

In order to study sequences in space I need to have a clear notion of dimensions. So as I have discussed in previous blogs dimension is the notion of orientation with motion along that orientation: that is what I have defined previously as direction. Thus dimension and direction relate directly to each other through orientation. Now the unit quantities in a given direction mark off the dimension in that direction. Thus the count of units in a given direction is the dimension in that orientation.

Choose an orientation. Move in that orientation. The units you move past in that orientation count off it's dimension in those units.

We do not have to have unit objects, as we can still count off objects as we pass them in that direction, but of course we do not then have a "metrical" measurement, just a count. A dimension is a metrical evaluation of a direction along an orientation.

For general sequencing we need a non metrical dimension! Now this I am going to call a node "unit".

What I have done here is set up a tautological definition of a node, this is so a node can be both a general or arbitrary object or a unit quantity in a given orientation.

So now I can define a node as a symbol of a unit quantity or an object;
A dimension as a direction containing nodes;
A direction as an orientation along which one is able to move which contains nodes, or in which I may assign nodes if none are specified..

In dealing with real space size matters, but units do not. So size is an arbitrary experience of a spatial quantity, which is distinguishable. Thus different experiences are related to different sizes, and comparisons, and metrons etc can be assigned on that basis.

Nodes allow me to discount size for the rhetorical purposes of this discussion, and therefore discounted processes have always to be borne in mind when any real application is derived from the discussion..

Now because I have 2 descriptors, dimension and node count, it is worth relating this to real space. Dimension can be in any orientation. Therefore I can describe space as any dimensional, in order to make the glide to n dimensional space.

In general space is n dimensional. How would I set up an n-dimensional reference frame?

Fortunately I have given some thought to this and proposed a dual spherical coordinate system. In fact it could be as many spherical coordinate centres as one could cope with, but I think 2 is sufficient for mental analysis. However, it becomes obvious to mechanical engineers and planetary explorers that more than 2 are necessary for real applications.
Again I will discount the reference frames until it becomes necessary to pick them up again.

In real sequence producing processes like the mRNA transcription produced my tRNA , or the protein translation produce by the ribosome acting on mRNA, or even the DNA replication, these sequences exit in a dimension relative to the transcription assemblies of enzymes.

Though this exit direction may be fixed relative to the enzyme complex, the enzyme complexes themselves move dynamically in any direction, thus facilitating sequence folding. However, packaging of sequences is a related process all of its own. The two reference frames enable at least a minimal account of such a process to be analysed.

There is a difference twist eukaryotic and prokaryotic cells, which enables sequence variation to be expressed separately from the protein translation phase. This separation is spatially organized and bounded within a part of the cell called the nucleus. Within this space the process of combinatorial variation takes place. The details are very interesting but need not concern me here. The resultant sequence is produced by a number of different sub processes that select the start, stop, length and combinatorial sequence of the end product. The main model is that one template programmes the whole process on a node by node basis along its dimension, while the product comes off in a different direction. The product is spliced and packaged as a selected sequence to be sent on for processing outside the nucleus. This combinatorial process is able to produce a variety of sequences from the same template.

I am going to model this nucleus by a fractal structure built of arrays.

The template will stand as the option sequence and the NSp will stand as the finished variety of product. The varieties of product are assigned their own dimension and of course they have nodes in each dimension.

Now I can factories each product based on the number of nodes. If I collect together all products with p nodes, then I have that many dimensions to account for. To account for them I have to use G{NSp} where N is the count of nodes in the template. Spatially I can lay this out dimensionally in a tree diagram. The template gives me the initial number of dimensions in the tree, that is in a spherical reference frame with branch options at each node generating new dimensions of smaller node count. Thus the dimensions increase factorially.

Now it turns out that if I have up to 3 branches at every node I can collapse that branch into an array either as a cuboid or a rectangular array. If there are more than 3 then the combinatorial chunk form becomes a mixture of all lower forms, which can be packaged as arrays, but very carefully following a formalism to ensure all dimensions are accounted for. The tree dimension account will therefore be the standard for all dimensions, but I mention the array packaging to highlight the connection to computer RAM storage solutions, and for any computational advantage that may be achieved for larger n dimensional product trees.

Already I hope you can see how Grassmann was thinking about the combination or synthesis of form from subforms. Now this is a fractal iterative process, and it is a natural process, both views Grassmann did not have the terminology to express, but he did have the insight to apprehend.

In a real sense Grassmann's analytical method synthesises algebraic templates. What we use those templates to build, ie what useful algebras we make of them, is entirely down to our translation of them into viable building blocks.. But thins activity takes place outside the nucleus of transcription . The king of products Grassmann envisaged were no less than new physics and mechanics as sciences, but I daresay we can allow mathematicians to make thier fields and rings and groups with them.

The other reason for relating them to arrays is so I can identify the various sequence products used by Grassmann as product processes in his algebras.


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