ENDPOINTS AND TRANSVERSALITY
Consider the end of a curve. Look closely at it in your mind's eye. Now imagine the end of a mathematical line. Where is it? What is it?

Suppose we attach a clockwise gene winding to the endpoint of a curve, to go along with the anticlockwise winding of a continuum point. Negative gene flow is like integration. This is appropriate to the finiteness of existence. We can thus make a distinction between actuality and existence. Existence is mathematical, but actuality also has stochastic gene tails, possibly both positive and negative. Existence has the modality of necessity, while actuality adds the modality of possibility. Call the transexistential corresponding to the gene tails 'ammetry'. Actuality = existence = ammetry.

Recall the meaning of 'distinguished' points or numbers. A number point is distinguished if it has been mentioned, such as pi or 5.46 or the square root of two. Consider the number 1.123248965374859673264869; this number is distinguished by my typing it, but it was in all likeliness 'nondistinguished' before that.

The nondistinguished points have their genes in ammetry. (Genes were defined for functions rather than for numbers or points, but any number-point 'a' can be made into a point function by f(x) = a or into an interval function by f(X) = a. If to any distinguished point we add an ammetric gene tail, this is how distinguished points fit into the continuum (of mainly nondistinguished points). This is obvious for distinguished transcendentals. But whenever we use '1' in any actual sense, we do not mean just exactly one, but very near to one:

1.000...0a0...0b0...

where the ellipses may have any length up to seminfinity.

Let us look at the subpoint relational structure between an endpoint and other points. An endpoint has a relation to all the other points in the curve of which it is an endpoint. It has a different kind of relation to all other points not in the curve, such as those belonging to a plane in which the curve may be embedded.

Suppose we mark the relation of connectivity by a connecting line as an inter subpoint relations, diagrammed as
Then, using subpoint geometry we can form many different kinds of connectivity, such as
We can also form more complex interconnectivities:
So we might consider the end point as having countably many connectivity connections to each of the aleph-1 (assuming the continuum hypothesis) points of its curve and countable disconnectivity relations to each exterior point. Now the possibility of new geometry arises when the connectivity and disconnectivity relations between points, either pairwise or in larger sets, are mixed. As we see from the last illustration, a point may be connected and disconnected to other points in many ways. Then there is the possibility of considering the set of all connectivity, all disconnectivity, or all of both sorts of relations between and among sets of points. If A is a set of points and B is the set of all possible connectivity relations among the points of A; then we could call the set union of A and B the 'thick geometry' of A.

The two dimensional representation of the relational structure is dense within its area of coverage of the two manifold continuum in which it is embedded; yet after any finite, seminfinite or countable number of steps has areal measure zero (the abstract diagram, not the physical drawing) and it is transversal for all of the intersections, except at the points themselves, to be of two lines only, in the same sense and directly following from the fact that any countable set of curves in the plane intersect transversally in two point crossings only. Call this the property of having 'stacking number' 2. We can measure degrees of nontransversality by specifying different stacking numbers for different geometries.

What happens when a line transversally intersects a non parallel surface in three space? There is one point which is distinguished by the intersection. It is the only point with relational connectivity with all the points both of the line and the surface.

Draw a curve that starts along the line and bends at the intersection point to a continuations a curve in the surface. What is the line' gene structure at the intersection point? Transversally the bend is a finite angle and its gene structure is g1 and a gene tail.

Exercise: Construct partial gene theory (as in partial differentiation) and apply to surfaces and general manifolds