I have written before about the abstract notions of point and line, but not about side and face, or evem specifically about volume.
The motivator for this line of thought is the Moebius surface. We can treat it using Strecken as Audehnungsgroesse, but the most importamt place to start is what i am going to accept.
I will accept only those phenomenon for which i have a referrent, and which i can empirically explore.
Thus a point has a physical and a mental referent, and this is going to be the case for all these discussions: subjective and objective reference.
The subjective reference is the problem. Without it i may not proceed, but with it i must proceed with caution and rigour. Thus if i refer to a corner of some object in my subjectively derived experiential continuum as a point, i cannot them magnify it and say that it is not a point but a collection of what i mean by point!. If i refer to a corner as being an imprimatur an impression of a point, then my subjective designation of a point is based on an inductive comparative process, not on a definite object. Thus a point becomes a procefural process which has to return a coherence value as an output. Thus certain properties have to be assigned values of correctness, and thes values are assigned by some subsidiary process, and the fractal iterative loop goes on, unless truncated by some arbitrary truncation point. By the way we have to understand that arbitrary has no meaning unless it refers to some relative point of reference.
So subjective references have the quality of being fractal and subject to some arbitrary internal decions which may or may not be unconscious.
The strip of paer used to make a Moebius surface is not an ideal plane. As we have seen an ideal or abstract plane is bases on some com[arative procedure, not a specific object. Thus from the outset the moebius surface does not belong to any ideal world. After having made the Moebius band, some have attempted to idealize it as a surface, and to retain some characteristics of real surfaces.
However, real surfaces do not exist without some regional properties, and it is precisely these regional properties that are ignored when idealizing the Moebius band.
The solid strip from which the Moebius band is made has only 2 "sides" : inside and outside. Each side may have many "faces". Thus anyone traversing the faces of the solid strip will never get inside without "passing through the face. Joining 2 faces to make a band and twisting before the join, removes 2 faces but only relative to the old form. The new form des not have those faces and the inside and the outside of it has become more complex. What is meant by inside the new form?
This is precisely where the idea of a Moebius surface becomes sloppy in some treatments. Clearly i can still traverse the new surface, eventually end up where i started even if i have to retreace my steps backward, but never end up on the inside of the solid. However, if i remove the regionality of the solid ( thus i remove the inside) i can claim yhat the surface remains, (although it is patently obvious it does not!) and my journeys somehow take me into the "inside" of this new object.
I do not mean to ridicule this important set of relations, only the poor analysis that is being foisred on one as explaining what is happeneing, or what this surface is and does. These surfaces are very real and important in vortex forms, but not as these impossible abstract forms. Regionality is still needed to have such a form, we need an inside to have this moebius surface as an outside.
This inside of any form we usually symbolised by another abstract form called a "volume frame". This is made up of the edges of a standard form by which we define a standard volume. For the purpose of measuring volume the forms must Tessellate. Circles and spheres therefore provide a problem in terms of defining volumes which are not whole circles, and so procedurally there is a definite dijoint between circles and polygons etc. And yet Polygons clearly approximate circles.
This disjoint is the notion of infinite untruncated process which naturally seems to occur in real space. Or does it?,
At the end of the analysis i will never know whether i subjectively produce it as an output for measurements beyond the computing or measuring process, or whether it is a signal which i am able to respond to by a specialist subroutine for circles once identified, ie i have a special circular sense especially for circular objects , For example i know we have a special orthogonal sense system within the ear canal systems, thus orthogonality is a real "Sense" not a defined one.
Thus the volume frame is a concept subjectively imposed on spatial magnitude to define the notion of volume and to specify regionality by volume frames.
Do Volume frames have an inside and an outside?
The concept of inside and outside is important for the notion of curvature. In fact curvature defines inside and outside reltive to the subjective processing centre. Closed curvature is what i usually look for to decide on insideness and outsideness.
Closed curvature is not just associated with form, it is also associated with motion. There is another "medium" curvature is associated with, that of electrostatic and magnetic fields. Fields cannot be seen but the intensities can be measured by suitable measuring probes. Do these fields have an inside or an outside? Could such fields deliver vortices that mimic the klein surface?
Gravity is described as a curvature of space, but this curvature could be the combined curvature of the electromagnetic fields. Since all matter is electrically and magnetically active gravity may simply be an indirect effect of the curvature of these fields and not an independent force. Just as curves are in some sense independent of polygonal sides, the "continuous" curvature of the fields is independent of any directional force. I have discussed this fully in my Shunya Field conjecture, but the point here is that inside and outside do not preclude permittivity or reversal of orientation in our reality, so we do not need to invent some absurd 4th or 5th orthogonal direction etc. What we needed , and what Grassman supplied was a combinatorial technique for all the orientations in our own space. This is what we sometimes call Vector addition, but Grassmann's conception is much more natural and spaciometric and Heuristic.
I have remarked before that spider's Webs are n dimensional spaces, merely to highlight that any polygon or polyhedron is an n-dimensional space if it forms an open "spiral" space or "curve", and "quaternions" that comply with this constraint are a suitable vector model for 3d spaces of this form.
I cannot pass without commenting on the confusing use of "sign" that is + and – . These symbols have taken on a life of their own and mean many things. Their convenience is outweighed by their ability to obscure. The subject of mathematics may have gotten its shape through these 2 symbols alone, and as such, it places mathematics on an insecure footing. Many other symbols derive their power from these 2, and so things or topics which ought not to be thought of as related receive a spurious connectivity through the similarity of signs. Things that are analogous should be clearly analogous, and the contrasts emphasised over the comparison.
Analogies can be tiresome despite their power to engage, because they do not advance our knowledge, thay analogise it. Thus the Yi Ching and the Taijitu despite their power became moribund and in need of reform due to the encrustation of analogies and YI. Lai Zhide updated the model with the modern understanding of his time, and the Masters of the Yi should continue to do the same if the model is to remain incisive.
We have no real use for the sign value that is not cosmetic or for shorthand. The contra notion does not require it, and then one may see that this notion is indeed a complex notion which may serve us better if it is treated in its cases at all times. Thus a Klein surface is orientable because it is all about orientation, direction and rotation!