The aspect of a point left out of the discussion until now is how we kinaesthetically respond to a point as a seemeioon.
The main reaction is the kinesics of the upper limbs, the motor cortex generating a gross motor response that orient the body to make the point centre focus and in addition extends the arms and fingers in that direction as well as the tongue. We become poised to respond on multiple levels to what is out there t the distinguished point. This is the seemeioon response.
The seemeioon response is at the basis of more generalised motor responses from different levels of the system we call self or actualising self. The chief response I want to identify here is the manipulation response, the mandibular one, we respond with our fore limbs in a way that grasps and manhandles, and our poise is to be in the best position for a fight or flight response.
This fight response which leads to finger , hand arm extension can result in a motor response that is a " blow" against a potential or actual opponent. This strike itself is an attempt to leave a mark on the environmental factors. For some this is as crude as a punch in the air, for others as graceful as an extended fluid motion that tracks the point and for still some others a means of applying tools and mediums to a surface to represent the retinal image of the point and it's environs. All responses derive from this same poise to react to the perception of a point of interest.
I focus now on the artisanal response to a point, in that it induces a representation on some adjacent surface or medium. The point in space as a seemeioon experience is represented by a mark on the ground, paint on a wall or ceiling or s pebble positioned in some fashion, etc.
One of the fundamental tools that Astrologers came to formulate was what I denote by the word mosaic. The mosaic derives from many practices that use stones to mark important shadow or light ray positions. We can group monolithic structures which formed early astrological markers such as Stonehenge and other structure, to those observatories in caves or at the top of man made pyramidal mountains as well as at the base here shadows are cast in each case light and shadow was marked by poles or pebbles in the ground. These positions were actual points of view.
Not only were physical objects placed in the ground to form these mosaic patterns, but also lines were dragged into soil, or trampled into paths, or scratched on rock, or painted in media to represent paths of connections between important points . The art was to make the pattern memorable or to draw attention to similarities and congruence in the patterns of the points to everyday experiences.of bulls, gazelles, sheep fish etc.
The combination of these aesthetic mnemonics led to more figurative and thus mythological mosaics. The skill or style of the artisans was reflected in how smooth and how level these ground mosaics were to become.. The patterns and styles reflected philosophy as well as mythology. It was soon noticed that angular stones created smoother surfaces, and rectangular clay tiles created smoother paving still.
The development of these geometrical forms went hand in hand with the revision of the artistic and thus astrological notion of a point. These refinements however do not detract from the seemeioon response, although they bury it deep in cultural, mythological and aesthetic excrudescence .
The mosaics were to become representations of astrological points in the plane. But there were other representations, including solid spheroidal shapes, onto which astrological marks were made. Or inside darkened observatories through which light beams would stream creating pinhole camera phenomenon, detailed marks could be drawn on walls and floors as the sun and moon projected light into the rooms. The rooms, though not spherical nevertheless represented aspects of the sphere.
I now turn to the representation of the sphere , particularly in Appolonius work to understand the sophidtication of the notion of seemeioon or the point response.
Before we can even define or identify the sphere we need to identify the concept of a bounding or bounded surface..
The definition of a surface is experiential. It requires visual input, visual sensors. The distinction between the phenomenon of light and a lighted object needs to be apprehended.. Whatever the source of this attribute called light, it is important that objects are identifiable as being lighted or in light.
A surface is the first distinction. The surface is that on which light falls.
Some surfaces seem to " reflect" light, and so appear shiny. The light falls on another surface showing it is reflected, and that light is some separate property in the space environment, that a surface can reflect. Some other properties of surfaces and light can be distinguished including the Kinaesthetic and auditory properties.
The fundamental notion of a surface is Epiphaneia in Greek.
The spatial property of a surface is distinguishable by first noting its boundaries. Fortunately we have visual sensors and logarithms devoted specifically to this task. A boundary is a visual experience, again reinforced by auditory and kinaesthetic experiences.
Curvature of a surface is thoroughly dependent on kinaesthetic sensors, direct and peripheral. The proprioceptive sensory system uses focus and muscle sense and memory to develop a spatial model of the surrounding environment.. Curvature is a complex experience of form and boundary and surface reflection properties.. Shadows and other local light variations on environmental surfaces contribute to this complex notion of form, magnitude and curvature.
The type of curvature of a surface becomes distinguishable. We know when a surface is " flat" and when it is spheroidal, when it is disjoint and when it is jagged. Smoothness is a kinaesthetic synaesthesia with a visual in distinctness or reflectivity.
Because of the senses for boundaries we recognise edges and boundaries of surfaces. These edges are fundamental to surface, they are also fundamental to an artists response to a spatial scene. By " drawing" lines on a surface an artist can represent a spatial view as a skesis, or schematic, a pattern of lines. This pattern can be viewed by others and maybe they can recognise an actual spatial view, from the Sketch.
Skesis are very important bridges to representing the world.. Just as an artist can drag lines onto a surface, light can cast shapes as shadows onto a surface, and through a pinhole a camera image onto a wall. The properties of light and eyes are so similar that a good artist can spell bind his viewers. This is the basis of the philosophy of Pythagoras School, and the power of the mosaic.
One cannot sensibly progress much further without acknowledging the fundamental and crucial role of the artisan. It is through the work of the artist, sculptor, weaver and potter that detailed distinctions are woven into the languages of people's. Distinctions that are communal and cultural experiences, which allow an individual to take a subjective view.
It is the creation of these commonalities that Create a consensus, and it is the consensus that enables symbiotic agreements that is sunthemata to be made and sum bola to be anchored in cultural experiences. The creation or founding of schools , the development of disciples are all parts of this rag tag movement towards a cultural identity. At the same time they engender deep philosophical examination in some Individuals suited to that task. The natural or even zoological structures that emerge in cultures and colonies self organise into organismsl specialisms. Thus philosophers and artisans have a social role beyond thir individual predilections.
The creation of the wisdom about spatial surfaces is not just a personal or private motive. In some sense any human culture or colony expects these skills to be developed and used for the benefit of the whole. Quite often, altruistic behaviours can be traced back to the notion of a quorum sense of what the whole colonic structure needs.. Thus the colony as an organism is a real source of religious ecstatic experiences as well as inspirations. When several individuals come up with solutions to the same problem, communication has taken place byRome means.
Usually it will be found that individuals are in communication by obvious means: letters, seminars, common networks. But occasionally the communication is more mysterious. We cannot rule ot chemical or electromagnetic or viral communications.
Thus we forge a common understnding of concepts like urface nd boundary lines, but it is among the Pythagoreans that these notions are powerfully organised in a systemic pattern.
Starting with the spherical surface, the Pythagoreans analysed space down to something they called seemeioon, an indicator.
It is an artist that distinguishes an indicator, for it is the artist that responds to that indicator manually. Such an artist being a sketcher has to draw from that indicator of something interesting, something that catches the eye. Or being a sculptor has to carve out that indicator in the form that he sees in the material, or being a potter works the clay into the form indicated by hs object of inspiration.
To go from that subjective experience to the mathematical notion of a point actually involves more developments by artisans such as architects, engineers and astrologers and surveyors. For these professional groups the seemeioon was often a small dot in the distance whose position was determined by 2 sightings . Other dots or positions could be indicated by 2 crossing lines or string cords. Crossing shadow lines also marked important points.
The role of mosaics in formalising a part of an image as a point in a jigsaw . Thus a seemeioon was not a mathematical point in general. It had a wide range of meanings. In mechanics and in philosophy, however it did come to signify where synthesis begins. Having no parts, parts have to be added to it to chives the full synthesis, and the nstural parts to add to it were the line and the surface.
Today we have pixels and color dots, processor chips which easily fulfill the notion of seemeioon and the related notion of point, but we have developed a style of rhetoric in which points are undefined things at the ends of other undefined lines which f straight, also undefined, lie in a structure called a flat figure of some sorts..
Norman Wildberger actually defines a point intuitively and then symbolically in an arithmeticall structure. While it is a fresh approach, it actually relies on the consensus understanding. I define points based on the cultural background discussed hitherto.
Firstly our tool is a rigid object with a sharp point either end. For example a pair of dividers. The pointed ends A and B will now be used to indicate certain other points. These points are defined as follows: with A fixed in some surface B can indicate all the points in the surface of part of some sphere. These are indicated points.
Now fixing B as one of A's indicated point I can indicate a set of points that lie in the surface of a sphere.
Precisely those indicated points which are the intersection of both spherical surfaces are defined as a circle of Dual points..
Fixing A or B to any indicated dual point C allows us to define 2 other circles of dual points and a point that is a triple dul point where these three circles meet .mthis triple dual is always part of a pair . Fixing A or B to this point we cn define yet more circles and yet more triple duals.
We find that the thre points ABC lie on a circle which is the intersection of the indicated spherical surface points of these 2 triple duals. The triple duals D and E are unique to the circle through ABC , and so it is true for pairs of triple duos for circles ABEL,ABD,ACD,ACE,BCD,BCE.
It is clear that there is a hierarchy of points by construction, some are indicated,some are fixed,some lie on circles of duals some are distinct triple duals,etc. in addition we have structures associated ith each type of point. Surfaces, circles and intersecting circles.
It is also clear that we cn expand this structure out to fill any region of space , and it provides us with an Arith,os, a netok that allows us yo count spatial regions, and this is the notion of a medical space with the sphere being the Metron, but the igid pointed tool being the fundamental unit of measure in any direction.
This is a MULTI compass vector Algebra or rather arithmetic of space and it is commonly called a sacred Geometry.
In addition the classification of the points is determined by how many points I chose on that original circle of dual points intersecting the mutual spheres fixed by A and B.. Thus we have an infinity of oriented arithmoieach distinct and together forming circles and triples of even higher " rank count".
These hierarchical points I call Schwerpunkt to highlight Grassmanns method of Analysis and Synthesis.
We apparently have no planes or straight lines. This however is soon re tidied when we recognise the intensive Metrons needed to define just these concepts. This is when we find the significance of the circle and the pairs of triple duals uniquely associated to each. We also come to see the essential flexibility adopted by Apollonius is the best approach to a general arithmetic of space whether written in symbols or not. Algebra, the umbilical arithmetic also derives it's proper significance relative to the Arithmoi.
Algebras that are not based on the metrication of space, while possible, are nt Mathematikos. While a Mathmatikos may philosophise in this manner, the results are in the encompassing subject of Astrological Philsophy and decoding of the position and alignments of stars and planets to advise on Kairos, the opportune moment for any action or inaction.
While many reject Astrology many more guide their lives by it. It is therefore a great responsibility to be an Astologer, and requires great pragmatic and philosophical wisdom to correlate the position of noise on space with events in cultural life.
Today's astrologers, using satellites and other measurements, are bringing more accurate space weather forecasts and earthquake warnings, based on physical models of interactios. The more esoteric predictions are still hocus pics at the moment, but when we sort out the effect of electromagnetic storms on our Psyche we can expect forecasts in that more personal arena.
In a direct way Google may be leading the way in making these kinds of trends analysable and correlateable.