THE NEOPLANE

Introducing the Neoplane

The Euclidean plane is a grossly uneven structure. It has the quality of vertical flatness; that is, it is flat when embedded in R³, but it is not flat in itself. For example, we can specify an origin in a Euclidean plane. Then every other point has a radial connection to the origin; this eschews any sense of flatness or orthogonality right from the start. Away with the origin. Away with the axes also, for the same reason.

All of these wild historical eruptions in the Euclidean plane mitigate against any sense of flatness, at least, against perfect flatness. Away with the rational numbers, the algebraic numbers and the tame transcendentals like p and e. They all distort the planarity.

What we have left, eventually, is a very flat subset of the Euclidean plane, to be called the 'almost neoplane'. The neoplane is a very thin subsection of the Euclidean plane, at least in two classifications. it differs from the Euclidean plane by a point set of measure zero.

What curvatures are requisite for being a plane? This is not entirely obvious; there are rotational flows possible in a plane, plus all sorts of twisty curves.

Now we will call the form in which all the components of all possible forms of curvature or higher analogues of curvature are set equal to zero or null, the 'neoplane'.

To what extent can the neoplane be modeled by the Euclidean plane? Suppose we consider how the Euclidean plane can be packed with curves. The curves can be winding any which way, gathered in streams, crossing and recrossing. there are many ways to pack the Euclidean plane with curves and lines. For example, we could add a line through every rational y in an (x,y) plane without changing the planar measure. All possible curve packings are represented in the Euclidean plane.

(Perhaps I should explain what I mean by curve packing. The Euclidean plane can be packed in many ways. One might consider all lines parallel to the x-axis in an (x,y) plane - or the y axis. One might also consider a packing consisting of all circles of real radius with the origin as center - or with any other point as center. These are just two of the simplest possibilities.)

To approach the neoplane imagine straightening all of the packing lines, or excluding curved packing lines. They could still be of random or any lengths and directions.

Alternative approaches to the neoplane include considering all packing lines either radial - but this implies an origin- or to consider all packing lines parallel - but this seems to imply an axis or at least a preferential direction. These are still not quite the neoplane.

The neoplane may be just beyond our models.

The neoplane does not fit into our classification of Euclidean packings. If you have a neoplane and change one line you certainly have a change of planar measure zero, but you no longer have the neoplane.

There may be different neoplanes.

It would appear that the neoplane is preserved under point alterations of measure zero, at least if the points are not distinguished. Then the neoplane would be stable under at least countable point transformations.

Thus the class of neoplanes is at least a countable infinity size subset of the Euclidean packings.

Is the class of neoplanes included in a generic class? The generic packing class is at least a dimension larger than the class of neoplanes, so it has enough room to hold the neoplane class.

Alternatively, the neoplane class may be distributed among the various generic classes, which would mean that the neoplane could be reached from any member of the class to which it was distributed, by a generic preserving transformation. Or, the neoplane class might include elements outside of all the generic classes. (It is not intrinsic to genericity to cover.)

The neoplanes outside of the generic classes could not be reached by a genericity preserving transformation from any element of a generic class. (For generic, you may take, if you like, within a set of measure zero.)

What does the packing class structure of the Euclidean plane look like?

it is a class of generic classes of z = 0f(x,y) preserved under finite two measure. it includes the class of neoplanes, preserved under one measure. It has all of these properties stacked together, not just linearly. We might imagine something like the neoplane in the center; surrounded by all of the planes which are generic transformations of the neoplane, then the other generic classes filling up the plane.

The axiom of choice does not hold in the neoplane. Since we cannot have an origin, we cannot distinguish a particular point, which could then be used as an origin.

Any closed sets in the neoplane are nowhere dense in the neoplane. Suppose we had a convergent sequence of nondistinguishable points in these closed sets. If such a set was dense in the neoplane, then the limit point would be included and thus distinguished.

The lack of distinguishable points in the neoplane gives us a clue for the embedding of the neoplane in the Euclidean plane. All rational points are excluded, as distinguishable, from the neoplane, as are also all algebraic numbers (What better specification of curvature than an algebraic form?) and all specified, or distinguishable, transcendental numbers. Thus the neoplane can consist only of the nondistinguishable transcendental points of R².

This is a countable exclusion. But a countable set of points in the plane has both linear and planar measure zero. This embedding would thus allow the neoplane to have linear and planar measure of one, relative to the Euclidean plane.

If we look at the neoplane and then add a countable number of distinguishable points, in the proper arrangement, we can reconstruct the Euclidean plane. As we have seen above, each of the distinguishable points causes a distortion or curvature - as in around an origin - transforming the neoplane out of itself. A particular countable collection of these transformations can give rise to the Euclidean plane.

The Euclidean plane is thus a countable number of steps less flat than the neoplane.

Specified lines are also absent from the neoplane. For example, no hyperbolic curves have asymptotic limits as lines in the neoplane.

The construction of the neoplane by removing lines rather than points, from the Euclidean plane, is much more drastic, removing, as it does, a set of infinite linear measure, although still of zero planar measure.

The neoplane contains no curved packings and also contains no curved lines. Since we have taken out the rational and algebraic number-points from the neoplane, the neoplane is too disconnected to contain any finite length of curve which could be expressed by an algebraic form.

Let us consider the subpoint structures associated with the neoplane and the Euclidean plane.

Each of the countable number of specified Euclidean points must have a relation to each of the others. This network of points and relations is absent from the neoplane. Also, in the Euclidean plane with origin, there is a relations between the origin and each of the other points, also absent from the neoplane.

What sorts of relations would the points of the neoplane have? The only relation nameable, with no implication of curvature, is a simple relation between the points, which might be marked as

which, being real as a mark on the page (or screen, fjs2000), is an open set in any Euclidean plane passing through it, so is potentially qualified, if it has a countable number of holes, for being an open structure in the neoplane. Since real marks do have a countable number of holes (look at an ink mark under a magnifying glass), they are in this way fitting models for being in the neoplane. The line of the relation above should also be Euclideanly straight. Also, let the Euclidean length of a written mark be the length of the longest continuous (aside, in this case, from the ink holes or pixels) intersection of the mark with any straight Euclidean line passing through it.

Since the neoplane is a plane, i.e. two dimensional, we can look for it to have two (nondistinguishable) orthogonal dimensions, connecting three points (the 'same' three points, which were they distinguishable, as would determine the Euclidean plane). So it seems as if we are left, for the relational structure of the neoplane, with a square grid of points and relations, such as

Since this grid, having planar measure zero, cannot cover the neoplane, we find that the neoplane must assembles a large number of square point meshes.

We might consider the neoplane as consisting of an infinite set of (Euclideanly overlapping) nonintersecting square grids of nondistinguishable points.

Euclidean space may be reached from the neoplane by inserting a countable number of points into the collection of neoplane points, and all the required subpoint relations to fulfill the Euclidean subpoint relational structure.

The only twist we have presumed in the neoplane so far is orthogonality. We could give up orthogonality of the neoplane by giving up square gridding. We are left, then, since we cannot properly specify any other form of relation, with the neoplane as a subset of the Euclidean plane point and relation structure consisting of nondistinguishable relations between nondistinguishable points.

Applications of the Neoplane

We might consider using neoplanes. They are topologically more congruent to physical reality, which has at least a countable number of holes, than is Euclidean space, which does not immediately have such holes. And recall that the neoplane is equal to the Euclidean plane in planar measure.

Suppose we have a two manifold for which we ordinarily take the derivative in terms of Euclidean planes. Consider instead taking the derivative in terms of the neoplane. If the neoplane is considered a pointwise subset of the Euclidean plane, then it must have the same exterior directionality as the Euclidean plane. Suppose, however, that we try to take the derivative directly in terms of the neoplane. Call the direction of the normal to either a neoplane or a Euclidean plane, in R³, the normality. If we look at a neoplane on its own, without respect for its abstraction from some specific Euclidean plane, it is not immediately clear what normality the neoplane will have, if any, for our common device of attaching a normal vector to a surface will not be available in the neoplane since that would distinguish the point of attachment of the normal to the neoplane, which would no longer be a neoplane.

If we do not have a specific normality, then what can a tangent or derivative of a neoplane to a, generally curved, manifold mean? It would seem most relevant in terms of the line packing of the surface, particularly if there was a connection between the line packing of the surface and its curvature, which seems natural.

Exercise: Construct a mathematical theory connecting line packings of two manifolds with their curvatures.

If we removed the distinguishable points from a curved surface, what remains has the same set of points as the neoplane, up to a 1-1 nondistinguishable mapping.

We cannot select a particular point at which to fix the tangency contact between a surface and a neoplane. it becomes something of a question what we can mean by such a tangency contact. S surface and a neoplane can either share no points in common, or a nondistinguishable set of points.

If the surface and the neoplane share a set of points relative measure (relative measure, say, to the infinitesimal Euclidean measure they mutually traverse) one, then they are very close to being parallel and flat, except on a set of measure zero.

The measure of the intersection of the surface with the neoplane might be used to give a scalar measure of curvature.

Note that if the tangency between the surface and the neoplane were looked at from a Euclidean point of view, it would appear to be highly nontransversal. But in the neoplane system itself, it is transversal for a surface and its tangent neoplane to meet in nondistinguishable sets of points which may be of any measure from zero to one.

Suppose we look at our tangency neoplane from the point of view of the superposed square grid approximation. If some of these square grids may be mapped isomorphically onto identical grids in the curved surface, it would give us a more geometrically explicit picture of nonzero intersection. If all the grids were identical, the surface would conform to the neoplane up to the square grid approximation, and could be taken as flat, where the measure of the intersection of the surface with the neoplane would be one. Generally we would suppose some portion of the grids to be shared, that portion being the relative measure of the tangency.

The above shows that a generally curved manifold contains, in any pointwise nondistinguishable area, subgrids of the neoplane, if it is of finite curvature.

The isomorphism of the pairs of grids is a grid in which each point, in addition to its grid relations, contains a self relation. This is the tangent entity:

Since a curved surface may intersect a neoplane transversally with finite measure, how large in lateral extent is the intersection of the surface and the neoplane? Consider the following. It would be smaller than any distinguishable term in a series dominated by a series with measure zero limit, but it would still be greater than measure zero.

The essence of this situation is nondistinguishability.

If we consider an isomorphism of square grids, wherein the isomorphism is represented by a self relation at each point, then add a second self relation to each point, we arrive at:

This structure might be interpreted as a triple isomorphism between three grids, say manifold, neoplane, and Euclidean plane. Call this the transparent tangency, or 'transparencey'.

Except for the distinguished entities in a surface, we can just as well take the surface as consisting of grids in the neoplane. It is not necessary to imagine these grids as broken off in a curved surface. We may instead imagine each grid as unbounded in extent, and the curving of the surface as arising from a selection from the total set of grids available in the neoplane, plus the distinguished points and their induced relations with themselves and the grid points.

A model of the pattern of distinguished and nondistinguished points on a surface is the night sky. All the stars, galaxies, etc. in the sky occupy distinguishable, and if already discovered, distinguished points. The darkness between is the nondistinguished set. As new objects are discovered, new points, or sets of points, are correspondingly distinguished.

Consider a nondistinguishable cusp on a surface. This is not unreasonable. From general relativity we know of the possibility of black hole fields which would not be locally distinguishable from exterior gravitational fields. This might be described in subpoint relational language as the endpoint of a fold.

Could we find subpoint structures for curvatures? This allows us to ask if we can use neomathematics to understand the nature of curvature. We have analyzed the neoplane as one special example. Those curved surfaces which are crumpled out of Euclidean manifolds contain smoothly curving sections, folds and cusps, all transversally, with other nontransversal possibilities, such as planes and symmetries.

One thing to look at is the interaction of the neoplane tangency with folds and cusps.

We recall that any smooth curved surface contains a scattering of neoplane grids everywhere but at the distinguishable points. As we bend a surface towards a fold we have two surfaces approaching each other. They must either meet at a tangential fold or at a finite angles fold. If they meet tangentially at the fold then their neoplane grids merge to become locally the neoplane.
Reversed, this contains the idea of a neoplane grid splitting. It might be diagrammed like the following:

There remains the question of what kind of structures we might find by initiating the consideration with a subpoint diagram and adding the grid structure to indicate a surface.

Joe Staley
joestaley@mindalive.org

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