I just thought for a moment that a manifold should be understandable, not arch or arcane. I never did understand engine parts anyway, but it is clear an engine manifold has got a lot of odd spaces to cover and seal.
When it comes to the very roots of reality, or the grounds on which one builds all of physics, it to has a lot to do. The fractal paradigm gives a structural pathway to the manifold. A fractal zoom into the Mandelbrot set, or the Mandelbulb is a wonderful experience. It seems you can go deeper forever, but we hit a technical barrier. That technical barrier is The Manifold. This is not so much a formal experience as a realisation we cannot resolve any deeper just yet. Thus on a screen we begin to see the squares of the manifold.
Where we see squares or pixels we are looking at a single inherent attribute. To every part of the pixel manifold we assignjan inherent attribute, be it colour, depth, intensity, motion etc, the attributes are distributed spatially on the pixel manifold.
But that is not good enough to help give a sense of a dynamic, vibrating manifold. Imagine the pixel screen was also a speaker, then the screen would visibly vibrate as the sounds pass through it, and Chladini wave pattern would dance around on the screen as thr pixels are undulated in a complex dance of convoluted movement. Even this does not compare with the variations in reality, from the depths of space to the intensity of ripe flavours.
But think for a minute: what if the combined movement of the Chladini screen, with the variation in the disposition of pixel light intensity, and the pixel block intensity variations that transmit color are phased or timed just right for a bifurcation of the light signal and the sounds signal? And what if my eyes and ears are evolved at just the right distances, the phorometrically perfect measurements to distinguish the two or bifurcated signal, and the stereophonically perfect distance to apprehend the manifold as a 3d experience?
It's an amazing thought, but sadly I cannot claim it as my own. In 1844 Grassmann wrote of this idea in his Ableitung der Begriff, when discussing the true Kinematics of Die Raume. He felt that Phorometry would provide the true laws of motion in space alongside the Mechanical Laws. Today the nearest expression of the same idea is the Holographic universe.
But just inhering visual and auditory signals to the manifold is clearly not enough. It is not rich enough to explain our experiences. We have to inhere all our attributes to the manifold of continuously changing and dynamically varying spatial interaction Experience.
Think now about how I might inhere all these attributes. Firstly they would need to be oriented. My processing centre orients all information using the compas vector networks associated with any arbitrary "point". Thus my first attribute to any manifold is a Local Reference Frame. That means I have my subjective reference frame, but I am able to attribute a reference frame to any " other" in my experiential continuum. The compass multivector networks are in fact spherical coordinate systems which look like rolled up hedgehogs!
Now to each radial I could assign a different attribute, dis positioning attributes radially. Now it could be that simple, except the space to which I am assigning a local reference frame is dynamic, thus not only translating, but also shrinking and expanding, and above all else definitely rotating. The manifold is dynamic
Thus it makes sense to mix the attribute disposition up a bit. Thus a unit attribute may be associated with one local radial, and or it may be associated with a group of radials. This clearly means that we are in to Combinatorial computations! Thus I will need a Process that select any group of radials from say n radials and disposes the attribute that way across the spherical network of radials. Depending on n that is a binomial disposition that could be very large.
But we are not done yet. What if the unit attributes actually need to "produce" a signal by some functional combination. That is to say , what if certain groups of attributes say r from any n attributes have to be positioned to produce a 3d object, just like Grassmann's parallelogram or tetrahedral pyramid? Again that would be a binomially distributed disposition of "products" over the spherical dynamically rotating space.
Somewhat does this mean about the manifold? We should anticipate it being a combinatorial experience, an entity which is dynamically combining all the attributes we can distinguish and inhere to it and bifurcating the output signal just enough to give us a phorometrie experience of its Kinematics.
Now in physics and mathematics we deal with dynamics using the differential notation. And to mix it up a bit we throw in the integral to kinda imply continuous combination. It looks off putting to say the least. Mathematicians will rave over the elegance of the script, the calligraphy of their important ideas, but no one will give you a handle on it. Hey they had to learn it the hard way, so why shouldn't you!
Any way here is the handle. The combinatorial tree diagrams can be written as 2 or 3 dimensional boxes. Higher dimensions are harder only cause there is more of them and they are more spread out. The 2d box is are defining standard. This box will enable us to see all the combinations of any 2 things from n.
To do this we have to make the box a square box and explicitly write down across the top and down the left hand side the n things. We then have to agree a formalism with ourself: say left always combines with the top, so any combination is always written in that sequence.
The formalism is crucial, because when it comes to comparing or combining results, the formalism must be the same. Apparently Hamilton used a different formalism to Justus Grassmann, and so Hermann got sign differences to Hamilton's quaternions, which he corrected for, but Bill Clifford demonstrated it was down to a formalism difference.
Formalism is crucial. Maxwell could not get the signs right in his equations using quaternions, and he did not know why, so he ended up replacing his quaternion notation with Gibbs vectors. He lamented bitterly the slippery nature of quaternions! This was all due to sloppy formalism on his part, or possibly on Hamilton's. The equations were fiendishly difficult to write out and fearsome to look on when written out! Who was going to notice a procedural error due to assuming commutativity?
Anyway, the box table when completed, lays out before you the dot product, the scalar product, and the cross product depending on n. it reveals the determinant rules and a whole host more as yet undefined combinatorial products.
This is the handle on the complex convoluted combinatorics of the dynamic manifold .
Oh, by the way, do not get caught up in kindergarten multiplication bonds. Multiplication unfortunately is wrongly used and taught. Factorisation is the overriding process in combinatorics. The box table helps us to factories all the factors for any 2 from n. factorisation is in fact a division process not a multiplication process. It is better to replace the idea of multiplication with the actual Euclidean process of creating a multiple form.
Have you ever wondered how things like plants and animates grow by multiplication, when they actually keep dividing the cells? The formalism we are exposed to at kindergarten disconnects us from the actual formalism in spatial objects.
So it is also worth pointing out that combinatorics works within the overall factorisation process called the binomial factorisation. Now the binomial factorisation of a group of things is a process of finding how many distinguishable forms we can make from n things if we take them r at a time. It is one of the most ridiculous processes to fathom, and yet it is absolutely crucial to do so. What is ridiculous is that in the long development of the polynomial and the theory of polynomials, the working of this binomial process is fully explored!. It of course was Newton who took it to the next illogical level of an infinite series expansion, but that required Napier's Logarithms to show the way of exponents, and the ages long calculation of the table of sines to promote that in Napier's mind. Add to that Cardano's gambling addiction and we have the beginnings of the theory of games specifically defining combinatorial processes and formulae for the same. Then along comes poor De Moivre,Newton' student, one of the greatest multinomialists in Newton's day ( before we started to call them polynomials) who knew like Newton the intimate secrets of the unit sphere. All measures are harmonised there! Cotes was going to use it for some extraordinary purposes, but he died in the process of formulating his plan. De Moivre wanted to use it to make his mark in a developing fied of game theory called probability, and he did. All this and the procedural call for the _/ -1 were all at their fingertips in the unit sphere, and this was due to the binomial factorisation process.
Always bear in mind that a probability measure is a binomial factorisation process inverted or as it is called normed to define it as a rational fraction twixt [0,1].
The probability measure threrfore naturally applies to the combinatorial disposition of attributes within the manifold of ceaseless activity.
These are a few of the major ways we have constructed a disposition model of the attributes on the manifold.one other fundamental design aspect is to describe the manifold in Lagrangian Mechanical ways. At the time Lagrange developed generalised coordinate systems, that is none Cartesian, but the most developed coordinate frames for rotation were in fact Hamilton's Quaternions, so an operator called the Hamiltonian was identified for just the rotational constraints on the Lagrangian formulation.
And what about the wiggly bits? Riemannfrom the start wanted the manifold to be an Undulatory substrate. This was because of the manifest wave behaviour of light. Thus the aether in which all supposed a manifold woul exist must undulate and so should the manifold .
Mannigfach is the same notion as Vielenfach. Fach is a department or team or kind and it's combination with number adjectives or adverbs is used to convey multiple. Grassmann was intuitively using the Euclidean multipleform factorisation in his fractal form theory to describe what he called ausdehnungsgroesse, but already he had an intuitive connection to periodic or Shwenken dynamics in his work on the ebb and flow of tides. And in his ambitious planning he saw a way to include a dynamic rotating process in his toolbox, so I have no doubt that he would be aware of the wave notion of the aether. To have proposed a Mannigfaeltigkeit may have been his next step, if he had had the frenzied collaboration he hoped for. So, that Gauss and Riemann express it in these terms is not just because falten means to fold or to ply in a periodic motion, but because the notion was current at the time, but worked through rigorously only by Grassmann.
Euclid introduces two powerful bits of terminology in book 2 . The first is rectilinear proportion model of the gnomon, the second is lineal factorisation. Euclid took a rectangular form and factored it into a joined line. The significance of this joining is that the rectangle is not made up of 2 independent line segments. The line segments are dependent. Thus whenever we factor any form we have to consider if the factors are independent or not. If not I must show the condition or constraints, as per Lagrange. When I try to factor whiteness of space, if the colour cannot be separated from the space it has to be factored in as a dependent factor, and that surprisingly still means Euclidean lineal factorisation notation.
Grassmann's Vielenfach are ideal for capturing these dependent attributes even the Undulatory ones. What Bill Clifford hoped to do was to solve the Undulatory manifold with a combination of Hamilton's technology with Grassmann's general overarching framework ,which just happens to be a fractal iteration structure. Unfortunately he died before he could bring his ideas to fruition.