Justus Grassmann looked at the combination of things. Things as "objects and things as ideas or subjective experiences. His guiding principle became find the fundamentals and then study how they combine to make the usual "forms". Only then will you truly understand the ususal forms.
His son Robert , who lived with him, while his son Hermann he gave to his chidless brother to raise, was particularly interested with the role of Form. Form is such a general idea, so to properly deal wth its implications requires a full and lifelong theory! Hermann drew on these ideas in his conception of the idea of Groesse.
Justus principle was not his own, it was heavily influenced by Schmid and Schliermacher and others; particularly those who formulated the scientic method as a heuristic one,depending on alternate cycles of analysis and synthesis. These ideas can be trace back to the pre Socratic Philosophers and to ancient cultures, particularly the Indian scientific tradition of prolonged and insistent observation. Thus how one interacts ubjectively with space is the elephant in the room. It takes a while to acknowledge it is always there.
For example point A is a singleton. What can be said of it? Only this: the presuppositions are more powerful than the point A; they attribute the "point" A to some output space. The output space is my output processing space assumed to be real space. Assuming the space is real, makes the point A real. But the point A is an idea i just "thought up". So now i have reeified/ objectified a subjective idea.
How i deal with this process is important to my processing of space: I either believe i am discovering truths about the "real" universe, or i am discovering truths about my own subjective processing. Of course it could be all three, the universe, the subjective universe or the combination of both.
To reduce things to there fundamentals, Euclid proposes the notions of semeion and gramme as fundamental deconstructions of Epiphaneia, the surface at which he is inclined to believe "form" starts. Of course a more fundamental idea than form is magnitude, which apparently disappears at semeion, or rather appears mysteriously through the semeion.
Thus Grassmann brings to the table ancient Pythagorean thought in modern algebraic symbolic logic form,itself derived from ancient Greek logic.
The notion of magnitude thus starts ith the mysterious semeion. Given the concept of a point mass, a Newtonian et al conception, it follows naturally that a line is a different sort of magnitude.
Combinatorially a collection of points cannot be shown to be a line or a gramme. The "drawing" process can be seen as distributing points in a sequence and in a relativity. This relativity is not inherent in the point but in the subjective processing centre. Thus the difference between magnitudes is imposed by the processing constraints of the subjective processing centre. Thus magnitude is attributed, relativity is attributed,spatial properties are all attributed relative to the processing centre's functionality. Focus, perspective, parallax are all functions of the processing centre.
So the proprioceptive networks,and their combinatorics fundamentally underlie the spatial combinatorics possible or distinguishable. To distinguish there are possibly four states: wholly within, partially within(joined), wholly without but contiguous and wholly without and not touching, Now in addition there are rotational relativities based on dynamic spin. The rotational distinctions are relative to a second and third point. Thus rotation is fundamentally distinguished by the distribution of 3 points.
The fundamental importance of 3 is recognised by all cultures, but 3 semeia are needed to distinguish rotation. This rotation is symmetrical to each point, thus all 3 points "spin". The distinction inheres reference frames in every point relative to one exterior point.
When an object is distinguished, its own internal fram is inhered. To rotate the whole object an exterior, additional point is required against which to distinguish rotation. In each case, the additional point completes a 3 some of points by which rotation may be determined.
Two points/semeia enable a distinction to be drawn, in fact they evoke the word or cry "logos" or comparative idea for which comparative words are inescapably necessary. Along with 2 points comes the notion of "translineal motion". Any line may be drawn or established between 2 points. Motion along such lines is what i term translineal motion. Translineal motion is contra statused. Thus one can move along the line with 2 motions that are contra to each other. Direction and orientation only have a meaning in the context of 3 points.
In the context of 3 points rotation and centre of rotation is naturally established. With this notion the notion of orientation is established as a a part rotation, and direction can be defined as translineal motion in the orientation specified for the rotated point.
Now i can specify a straight line as a line which is directed to a rotating point which fits "evenly" with itself when rotated half a rotation about any point between the beginning and end point of the line.
Although this sounds like the first idea of symmetry, it is by no means the first notion of summetria the kind of group identity which arises with 2 points and continues there on out.
All of these notions are brought to space by the subjective processing centre, and the next notion of spatiality is added by focus information. From focus information a notion of space depth is supported, and audiometry supports this notion. Thus a fourth point rotated relative to the 3 others, provides perspective and depyh notions, which may be confirmed by other sense data such as arm extension from the body which gives orientation, direction and extension information.
The proprioceptive map of space is therefore derived from an interactive history with space for which a memory store is necessary and a sequence parser is integral to the subjective processing.
The parsing of this information requires only a few markers to distinguish the processing requirements. Thus a reference frame consisting of an indication of spatial direction is all that is necessary to process the majority of spatial relations. It is important to realise that processing inheres this light reference frame into all objects as a part of the processing sequence. Thus the subjective e processing imposes a structure on space interactively. This imposed structure is the source of concepts of magnitude and dimension.
The processing cues which determine the output of magnitude also determine the combinatorial freedoms, the combinatorial degrees of freedom. These relate not to axes but to rotational symmetries and translineal symmetries.
The important statuses also determine where the combinations situate. The nature of the status can be described as meet or join. Join covers interior and overlapping continuous object spaces while meet deals with contiguous object space. The status where the objects are not in contact are combinatorially not significant. However, if a relationship exists between distant spaces, especially where there is no translineal link, then a spatial force field is indicated. The structure of any forcefield reflects the combinatorial freedoms of any space because it is the subjective processing centre which inheres the attributes.
The forcefield is structured around closed or open triangular vector forms, the firsr exhibiting rotation the second translation and spiral motions.
Having identified the processing cues for combination in space and for determining magnitude, the remaining sensory networks provide information on spatial intensity. The intensity associated with a magnitude or form completes the characteristics of a space. Meditate on this video
Two spaces which join are continuous. But the spatial intensity of each may vary even when joined. However the join is perceived as a unit, and the spatial intensity is thus spread or averaged over a larger volume. However a contiguous combination as in a meet means the spatial intensity is spread over a varying volume as the spaces rotate relative to each other as in the case of a vector sum. The spatial cues remain the same, but spaces that only meet exhibit their dull rotational attribute which interact precisely as in a vector sum. It id the spatial intensity variation that governs the vector sum, and thus trochoidal convoluted spatial intensities exists around such spatial combinations.
Thus we have a sufficient structure in the Aussdehnungslehre to fully describe the combinatorial synthesis of space, especially when rotating relations are included. The additional spatial intensity factor is also treatable as ana Ausdehnungsgroesse, and falls within the scope of the Ausdehnunglehre.
The combinatorial basis to the synthesis in space leads naturally to the chemical notation and terminology just as it leads to the mathematical and physical notation and terminology. It is only the Ausdehnungslehre that straddles all scientific fields consistently and coherently .