In the early 1970's San Francisco pivoted between the spring garden that it was and the connection of urbanity that it became. There was a certain thinning of the air, as if a cloud of seagulls had flown into an updraft and had then been lifted towards the stratosphere. I, at 30, watched.

One day I was sitting by myself in a coffee shop in the center of the city and looked around at the other strangers. Each one of us was primarily engaged with our own self and some were talking with others from the surface of their being and some were writing and some were looking at the other strangers.

Being then inside an envelope of intent which linked mathematics, art and daily life, I conceived of a geometrical or diagrammatic structure for such encounters and nonencounters. I scratched out a few preliminary curves and then bent for the door. These relational structures became the diagrams which I later, for other reasons, called subpoints.

A text is like a road dynamited through mountains and cuts into the deep and varied strata of the historic rock. We who write often reside in our particular season, whether volcanic, organic or sedimentary. We should know that our depositions are also viewed through different lenses than our own, if they should ever be seen in the light of day or night. Whether we wear the logos as a warming cloak or expose ourselves to it as a cooling breeze, we may find ourselves writing in full or partial consciousness of multiplicity. And then it seems that we should be read with such an awareness of our awareness.

If there is an entity, then that entity is itself and this is a relation between the entity and itself. If that is too strong a statement then we can write that if we conceive of an entity then we can conceive of a relation between that entity and itself, which we might call identity, although that name and connotation of identity are not necessary for our purposes here. But when we have conceived of an entity and of a relation between that entity and itself, we have conceived of two things, and any two things in our conception have a relation between them, which we might call otherness, although that name is also not necessary for this development. Now we have three things and we can well conceive of pairwise relations among them or even of a triadic relation of all three, such as mutually belonging to the set of which they are the elements.

To use an old analogy we might call the above presentation of relations the warp of our woven text. Then what follows next might be called the woof. And we will consequently read how they mutually entwine each other in a fabric of understanding. Although the limit of our metaphor is shown by the occurrence of a third element, a mathematical sequencing which weaves the warp and woof together.

Consider a geometric point. According to the standard image, it has location but no extension. Very well. But no extension is usually implicitly assumed to mean also no structure. That does not necessarily follow. It is conceivable hat the point has an intrinsic structure without extension beyond its location. If so, it may also follow that the relations between points also involves, or might be extended to involve, some of this intrinsic structure which the points might possess. I have constructed the first forms of such ideas. They seem to hold some promise for new geometries.

What kind of structure can a dimensionless point have? There are at least two avenues to approach this question. We might approach it through the void, as in asking what kind of structure does a point not have. That is an interesting possibility but not the one that I have here followed. Instead I have taken the constructive route, and so we will ask the question. What kind of structure can a dimensionless point have?

A point is itself. There are two parts to the predicate of this statement. The existential 'is' and the self referent 'itself'. Suppose we combine them as a relation. A point is in relation to itself. We can name that relation as self-identity. A point is itself. We can also make a diagrammatic image of the relation.

In this diagram, the vertex represents the point and the curve represents the relation of the point being itself.

There exists a natural geometric representation of the relational subpoint structure.

We begin with the original relation of a point being itself and we can draw this relation as a loop with a marked point:

The marked point, distinguished as a dot and the vertex of the loop may, at least provisionally, be taken to represent the point itself or at least its location as a source of relations.

That the point has a relation of self identity does not prevent it from having that same form of relation repeated, which may be represented as another loop, and so on

Returning to our first self relation, we may also consider the relation, not itself a self relation, between the point designation and the original self relation:

We may also have a relation between the point designation and the newly drawn relation, and so on:

Returning again to our original self relation, that relation will additionally have an identity relation with itself, which may be drawn, along with its successors:

Returning to the case of two independent self relations of the designated point, we can also draw their relations:

Any of the forms of relation described above may be repeated and conjoined into complex forms:, e.g.

It is clear that not every set of relations can be drawn in the plane without intersection of curves, but it is also clear that all such diagrams could occur in three space without such intersection. As the essence is to represent the relational structure, rather than the diagrams themselves, we may use the simple bridging convention, familiar from knot theory, when necessary.

We can assign a sort of double sequence to the relational entities, the subpoints. If we consider each subpoint as drawn, or related, sequentially, then we may number the relations, or graphical connections, by the natural numbers:

(1),(2),(3),...,(n),...

and, to be explicit, we shall call the relational or graphical entity created at the (n)th step by the name 'n', which name it shall retain, in the sequence, for further reference.

There are four fundamental categories of relations. There are the primary loops which are the self relation of the designated point to itself; these may be expressed in the sequence merely by the relation number in parenthesis, thus

1 =

(1),(2) =

There are the relations between the designated point and a previously mentioned or drawn relation. These will be represented in the sequence by first giving the new relation number in parenthesis and then a dash followed by the relation number of the curve or relation to which the new one connects. For example

(1),(2)-1 =

Then there are the self relations of a previously established relation or curve. Since these connect the entity to itself we will designate the new self relation by, first its relation or curve number in parenthesis followed by the number name of the entity which is self connecting, then a dash, then that entity number again, as in

(1),(2)1-1 =

Finally there is the case of a relation between two previously established entities, which will be designated by giving first, as always, the new relation number in parenthesis and then the relation number of one of the relations to be connected and then a dash and finally the relation number of the other relation to be connected. In the case of nondirectional relations which we are presently examining, the order of the last two numbers may be reversed. For example

(1),(2),(3)1-2 = (1),(2),(3)2-1 =

Given our general idea, and when we perform these special provisos, we have created a sequential numbering of the subpoint connections, each such sequence defining a particular subpoint. Let us look at some examples; we begin with the simples series:

has sequence

and

and

has sequence

and

and

has sequence

We can also create the geometry from the series, for example

gives

Having the sequences and the geometry we can define various operations upon subpoints. Here we look at three forms of addition and an operation called fission.

1. Addition One

Semidirect addition. We can add two subpoint sequences S₁ and S₂ by concatenating the sequences, except that we increase the sequence numbers of S₂ by the sequence length of S₁.

2. Addition Two

We can also add two subpoint sequences as in Addition One except that each relation number of S₂is increased by the sequence length of S₁.

 

3. Addition Three

We can also add two or more subpoints by condensing the designation centers. Thus each subpoint retains its original relational structure but the subpoints are conjoined at the same location-designation into one subpoint. This can be called fusion.

 

4. Fission

In the simplest form, fission is the inverse of Addition Two. When there are separate sets of primary loops from the same designation center, having no relations established between the sets, then it is simple to separate the subpoint by splitting the designation center into two or more and assigning each resultant one of the sets of primary loops along with its intraset relational connections.

Example

fissions into

and

A more complex form of fission occurs when we do not separate the original subpoint into non related sets of primary loops but instead separate it into sets of loops which, although they end up occupying separate designation centers, have connections between these sets. We then keep those connections as connections between the separate subpoints.

Example

fissions as

We can establish various numerical measures of subpoints.

L = sequence length which is how many relations are in a subpoint.

N = loop number which is how many primary self relational loops are in a subpoint.

D = depth which is, for all (n) a-b in S, the maximal a+b in the sequence, except that if there are only loops and -1's, then the depth is (for geometrical reasons) 2.

We can find some simple formulas for the number of varieties of subpoints which can occur for a given sequence length.

If P(n) is the number of different subpoint structures which can occur of sequence length (n), then

P(1) = 1

P(2) = 3

P(3) = 10

and these may be diagrammatically established.

We have the general recursion relation

P(n+1) = P(n)*(P(n)+3)/2 +1

and we have

P(4) = 66

P(5) = 2278

P(6) = 2,588,060

We can also write some rough approximations for large m

P(m) ~ (P(m-k)/2^1/2)^2k

or the equivalent

P(m+k) ~ (P(m)/2^1/2)^2k

Suppose we consider the possibility of extending the construction of a subpoint towards all of the relations; that is, between every pair of relations we establish another relation and we may also consider extending the loop number unboundedly and the count of designated point to loop relations for each loop unboundedly. We could call this the complete subpoint.

Try to imagine the complete subpoint, the point with all of its relational connections. On the one hand the point will contain every possible set of relational entities; this means that every relational entity will be connected in relation with every other entity. We will denote the complete subpoint by K.

We can also introduce a larger entity. To the complete point we adjoin by fusion all of its subpoints. We call this the whole subpoint. The whole subpoint will include unrelated entities, such as unrelated primary loops, plus all finite subpoints and all nonfinite subpoints. We will denote the whole subpoint by W.

We will introduce a further distinction to nonfinite subpoints. If a nonfinite subpoint is considered to be already there, with all of its relation curves existent, then we will call this a Parmenidean subpoint. We might alternatively have called it Platonic or Cantorian. If the nonfinite subpoint is as a process of becoming, we will call it a Heraclitian subpoint.

In a Heraclitian subpoint we can imagine a continued growth of connections forming between connections; in a Parmenidean subpoint we have all of the connections already made. A whole subpoint includes both the Parmenidean and the Heraclitian subpoints but also all of the finite subpoints; it may also be imaged that the finite and Heraclitian subpoint subsets of the whole subpoint are always growing and, as they grow, are always being replaced by the growth of a smaller subpoint as it continues to connect itself. In the whole subpoint there are always new primary loops arising, which add to and in a sense replace the previously arisen primary loops as those are involved in connectivity relations. Thus the whole subpoint is not complete for it includes also finite subpoints. A Heraclitian subpoint , even though it is still growing, is complete in that if you ask of any connection, does it exist?, then the answer is yes, any connection that can be mentioned is already there, even while further connections are still being engaged. A Parmenidean subpoint is of course complete.

If a sequence tracks the step by step construction of a subpoint diagram then we will call it a normal sequence. There are certain properties common to all normal sequences. Each step (n) in the sequence will be followed by step (n+1). No integer in the relational structure of the sequence will be larger than the largest previously created subpoint number. Since we have introduced the concept of normal sequences we have thereby implied the possibility of abnormal sequences.

Begin with K, the complete subpoint. Remove one connection, a. Then every connection which is a connection of a with itself or with any other connection, or the designated point itself (say that terminates on a) will also disappear. This disappearing will propagate ad infinitum. Will it change K?

A relation is said to be hanging if, as a curve, it only connects once with one entity, with the other end ending in empty space and, as a relation, it relates one entity with the void.

Now, imagine K and remove an entity and remove all hanging entities thus created, and so on. This is a sequential or temporal procedure; it is never completed in Heraclitian space and thus the remainder of K always has hanging curve-relations like threads connecting it to the void.

Imagine now renaming all primary loops from K and then sequentially removing all hanging relations. We have left a structure with constantly disappearing centers, a growing void, which never consumes all of K, but always leaves an infinite set of relational entities outside of the empty center.

Imagine a complete or very large finite subpoint as a ball of cotton, where the individual strands represent the relation curves. And we imagine a cotton seed in the center of the ball. Suppose you pull off a handful of the cotton, perhaps separating it from the seed and at least leaving many hanging connections. This handful of cotton may represent a relational entity, a piece of a subpoint, we might call it a.

Subpoints and pieces together provide a complete model of all possible relational structures. And to the extent that reality is relational, they thereby provide a model of reality.

Now look at what we have imagined. We have an unbounded floating relational structure always disappearing from the center outward and always recreated and infinitely continued. There is always an inner disappearing edge of relations that were momentarily ago terminated and then their terminations, at least at one end, disappear and they are left hanging in the void from which they evaporate.

This is one of the logical, mathematical structures which we have conceived, all of it is solid, although mostly unexplored, mathematics to which we add only the temporality of sequentiation.

But if the disappearing interior is temporal, then it is only natural to post the exterior a temporally growing. Thus we have a sort of shell of relation curves, disappearing on the inside and growing on the outside. It can be imagined infinite and also represented in finite mode.

We can give a reason for the beginning of the interior evaporation in the arbitrariness of the original point designation. We needed something to make our point, to begin the relational structure; that something was called the designated point. But after the relational structures accumulates we are able to let go of that designated point, which then makes the primary logos of self relation non terminated and so they disappear.

But our space of models is richer than just the disappearing shell. Suppose we allow hanging relations, as in

These might be called the relations between the entity to which they are connected at the closed end and the void at the open end. We could also allow completely hanging connections, which are graph trees

How could such forms be interpreted? They can lend wings. Suppose the disappearing center represents time past, going past, the past side of the present. Suppose the accumulating exterior represents time flowing in from the future. And then we might hold the shell of connected relations, those here for now and neither presently disappearing nor presently coming into being, as the thick present, that center of the present which in large, the complete present, is always disappearing into the past and arriving as if from the future, into the void, from the void, but here now. And in temporal sequencing this present space of connections is also always growing internally. Every pair of connections are related in the next generation of connection.

In terms of sequences, we use n- for a hanging connection which begins at n, as in

(1), (2) 1-1, (3) 2- =

Although we have introduced structures geometrically and referred to them as subpoints, we have very largely been working with logical relations of self identity and of relations with other entities. Our geometrical model is just one possible interpretation of what we call subpoint theory, and that theory holds independence as a purely logical relational system; in this sense it may be of much broader application than geometry.

Is there a physical correlation to these forms? There may be, but if so, then this is a more abstract physics than has presently been formulated, to my knowledge. Or at least a more subtle and svelte physics, or something like physics.

Although we began with a designated point all of our subsequent subpoint constructions involved relational entities only. The original designated point was, in a sense, fictitious, seeming to give us just enough traction to begin our relational constructions. Far from necessarily designating a geometric point, which we see now that it does not, our origin was simply a place of beginning, a possibility of identity. It could equally be eliminated, let us say allowed to evaporate, from the relational structure which maintains itself, as it were, floating by the bouyancy of its own relational structure. Thus what we have created is a relational structure, and possibly the set of all relational structures, which can be mapped and viewed according to what we call subpoints. We note again that the relational structure is free except for an imaginary beginning; it floats above the void.

Geometry as unfoldment of a point.

You may already have seen from the fission operation where we are going.

As you recall we found the original point designation superfluous after constructing the set of relations so, as it is a somewhat ambiguous concept to begin with, we dismissed it and so we are left with the richly dense relational structure for the point.

Geometric forms were long viewed from a reductionist viewpoint; A solid could be constructed from a moving plane which could be constructed from a moving line which could be constructed from a moving point. This is not a necessary point of view; each form can be considered as itself rather than as a set of points.

Consider yet another point of view, which we might call enfoldment. the complete point, and we need only one, especially as we have dismissed the designation, fissions into a set of subpoints which provide the geometry of forms. The idea is that the complete point fissions into geometric forms such as curves, planes, solids, etc.

We begin with a curve. What property most particularly distinguishes points in a curve? There is a linear order between points in a curve. You can only get from one point to another along one path (with opposite directionality in a closed curve).

Suppose the point P has fission products p₁, p₂, etc. then in a curve we can begin to show the relational structure with three points: p₁ - p₂ - p₃

Of course such a finite and even small number of points (or subpoints) is hardly a curve. We usually consider a curve to consist of an uncountable or continuum set of points and a constructable approximation to a curve would be expected to have at least a countable infinite set of points everywhere dense except at the endpoints. So the finite sets of subpoints are new things; let us call them pointlets. Although they are not themselves classical entities such as curves, surfaces and so on, they may, in their relational structure, point the way towards such entities, as well as toward things entirely novel.

Thus our three subpoint diagram above shows the beginning of that linear ordering between points where a curve begins.

We may now introduce a classification scheme for various categories of subpoints. We will make a primary division between the finite and nonfinite subpoints.

1. Subpoints which contain a primary loop are said to have property a.
2. Subpoints which have a relation from the designated point to a primary loop are said to have property b.
3. Subpoints in which entities may have a self relation are said to have property c.
4. Subpoints in which relations may occur between two entities are said to have property d.
5. Subpoints which may contain a relation between three entities are said to have property e.
6. Subpoints which may contain a relation between four or more entities are said to have property f.
7. Subpoints which may contain mulitplicities of otherwise identical relations are said to have property g.
8. Subpoints which may have hanging relations are said to have property h.

Generally finite subpoints of a given class are designated by s( , , , , , , , ) where the blanks are filled in with the operative properties. The commas are not necessary in the actual designations. We will also make two separate provisos. First that our common original subpoints s(abcdg) my simply be designated by the bare letter s. Second, that if multiplicity property g does not apply to all of the entities which the subpoint may contain, it may be applied to a particular entity class if used as a subscript for that entity designation; for example s(a_g bcd) will contain multiple representations of the primary loops but only individual examples of any other given relation, for example

would be an example of s(a_g bcd), but

would not.

A. Subpoints which contain at least a countable number of primary loops are of class A.

B. Subpoints which contain at least a countable number of b relations are of class B.

C. Subpoints which contain at least a countable number of c relations are of class C.

D. Subpoints which contain at least a countable number of d relations are of class D.

E. Subpoints which contain at least a countable number of e relations are of class E.

F. Subpoints which contain at least a countable number of F relations are of class F.

G. Subpoints which contain at least a countable number of hanging relations are of class G. (Note the different letter than the finite version; this is because we need H for the next class.)

H. Heraclitian subpoints are of class H.

K. Complete subpoints are of class K.

P. A Parmenidean subpoint is P.

S. A generally infinite subpoint is designated by S and, as in the finite case, a polycategorical subpoint may be designated by S( , , , , , , ), where the spaces between the commas may be filled by any of the letters from A through G. We may also use H( , , , , , , ) and P( , , , , , , ).

The whole subpoint may be called W.

What is the wholest, most all, possible or imagined subpoint relational thing? What properties will it have?

• All entities will be connected to themselves.
• All entities will be connected to every other entity.
• All sets of entities will have a mutual connection.
• All finite subsets will be included.
• All infinite subpoints will be included.
• All torn flocks with hanging relations will be included.
• The Parmenidean completion will be included.
• All Heraclitian subpoints will be included.
• The relations of sets of relations will be included.
• The whole will be included.
• The whole in its relation to itself will be included.
• The whole in its relations to its parts will be included.
• The whole with holes will be included.
• Completions with voids will be included.
• Unknown forms will be included.
• Even the relations of contradiction may be included.

We may introduce ternary and higher relational structures. Looking at the diagrams above we can see that there is an ambiguity between the ternary relation on the left and a pair of binary relations (triadic dualism). To consider the pair of binary relations as a ternary relation we need to introduce a three way connecting point A, which leads us toward a new designated point, indeed, seems to provide us with one.

In the quatenary relation shown on the right above we are forced by transversality to introduce a new point of necessary connection; perhaps we can illustrate this more clearly

The diagram at the left (the same as the right above) is quaternary, nontransversal and designates a new point at B. The diagram on the right is transversal but also binary and already accomplished in our binary subpoint theory.

We are led to the idea that multiple point structures are naturally associated with nontransversal quaternary and higher relational structures. Ternary relations are ambiguous.

We can form sequences that explicate this concept by introducing terms like

(n)k - j, j', j" etc.

which means that relation number k is connected to relations j, j', j", etc. We have a bit of a sign problem here since we have previously used the comma to separate new relational entities. Periods are effectively invisible on the most common internet browsers and semicolons can be similarly difficult to distinguish from commas. We could write that whenever we shift into multiple relations we replace the commas between new entities by semicolons, although that is awkward. This is not a mathematical or logical problem, but rather a symbological one. It shall be resolved. This is essentially a simplistic problem which can be resolved by a variety of grammatical symbols in the finite case. Infinitely, however, we may need to add another alternative. We could use parenthesis again, or subscripts, etc.

We have not yet touched upon the subject of directional relations.

It is easy to obtain directed sequences. We merely determine that (n)k-j means that relation curve n goes from relation curve k to relation curve j and not the reverse. In nondirected subpoints we have

(n) k-j = (n) j-k

while in directed subpoints we have

(n) k-j ~= (n) j-k

It would be nice to use an arrow to specify directed relations and especially useful when we have a mixed subpoint using both directed and nondirected relations, but again we have a problem in getting browsers to show the arrow.

Although directions may be mixed and more or less random it is also possible to find some generally coherent categories of directed subpoints. We might, for example, have all directions flowing outwards from the designated point, or the opposite, or, using arithmogeometry, have all directions circulating. We might define directions with reference to hanging in the void or to some arithmogeometric depth. We might form directionality in reference to the finite/nonfinite divide. We might form basins of directionality (On I-80 crossing the Red Desert of Wyoming you pass a sign that indicates that you are crossing the continental divide and several miles later you pass another sign that indicates that you are crossing the continental divide, and then there is no other such sign along the highway.). We might have directions flowing outward from, or inward towards, the highest multiple connectivity points. We can also form more specific classes, such as in the relations of countable infinities, where we can use any sort of rational sequence (e.g. alternating) to determine directionality, or for that matter, in, say a B sequence, we could let every prime (n) relation curve be outward from the beginning point and all the others be inward, except that we might alternatively specify that the even (n) relation curves will be nondirectional. There is quite a large space of possibilities here, some of which might be mathematically, physically or metaphysically interesting, or not.

joestaley@mindalive.org

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Copyright(c) Joe Staley 1998