A spiral space-filling curve as a natural continuum
Ross A. Finlayson
space from the origin.
Here basically the idea is to get around to the definitions of
geometry from other definitions, towards that then the idea is to get
some minimal list of axioms, towards defining a geometry in terms of
the point and space instead of points and lines.
There is a notion that a spiral space-filling curve starts from the
origin, then the curve encircles the origin (spirally), filling shells
(like a shell) out towards the unit hyperball (specifically the unit
disc to begin and then to generalize) and R^N (RxR to begin). The
idea is to describe this space-filling curve, in terms of its standard
and nonstandard properties in terms of classical and modern
definitions of curves, then to examine its properties.
Imagine then that there is a single points at the origin and a point
next to it. Here the idea of the definition of this curve is more
along the lines of the smooth infinitesimal analysis than the standard
"infinitesimal" analysis that is about finite asymptotes. Then, the
curve proceeds where from point to point, it is always a left turn
upwards a dimension. Here in restriction to the disc, imagine then
that there is thus formed a firts circle about the origin, then a
second, and etcetera until there is reached the circle that bounds the
unit disc, its boundary from the space outside the unit disc.
Then, in considering the points of these circles, which have
beginnings and ends, connecting the beginnings of the circles forms a
basis vector or unit line segment. Here this geometry starts with a
general curve made of points before it gets to defining lines. There
is a consideration then of what algorithms on the points of the
circles are needed to define then the line segment of a second basis
vector orthornormal to the first. Here first it seems there would be
indicated the ray opposite that of the unit basis vector, where the
algorithm used is to start at the beginning and end of a circle and
for each of infinitely many points process until they're the same
points, then that is a point opposite the origin. For each of these
circles of the unit disc the points opposite their beginnings and
endings comprise the ray of points opposite the first basis vector
(where that it's a basis in a space is yet to be defined). Then
similarly between the nearest and furthest points of the circles there
are those points of the unit line segments of a second pair of line
segments, where the line segment between the beginning and point
opposite the origin of the circles (or 2-spheres) of a spiral space-
filling curve of the unit disc (or 2-ball) is identified with, for
example e_2, a second basis vector of RxR.
Perhaps then an idea is to build this kind of framework up in a
discrete setting, otherwise there are definitions of circles before
lines. There would be a lot of cases where properties of some
considered algorithm would be along the lines of infinite passage, so
it's simple to model them in the discrete, but then they don't
necessarily have properties of the non-continuous.
Then, going along further with this notion, there are particular
structural features of the space-filling curve, in terms of reductions
or repetitions of as above some of these simple algorithms, which then
describe well-known structural features of a Euclidean geometry whose
defined terms are lines and points (and correspondingly the continuum
of real numbers and spaces structured in terms of real numbers). It
doesn't seem clear that there would be simpler representations of well-
known and intuitively available geometrical features like angle vis-a-
vis the intersections of these structures, but then as well there
would be structures more directly defineable in this manner, and as
exercise in pure mathematics it's beautiful. These are structural
features of a natural continuum.
Here I've been freely using canonical definitions of Euclidean
geometry, but the goal is to refine the definition of "these" "points"
to the extent that they're usable as mathematical primitives in a
nonstandard geometry with applications. Considering then some of the
properties of these points defined by this spiral space-filling curve
as a representation of a natural continuum: points define space, so
two points define a space.
1. Points are defined in a total order from the origin. There's an
origin.
2. There's a first point that isn't the origin. For each point along
the order there is another. (Those aren't necessarily the same
statement.)
3. Subsequent points along the order are as close to the origin as
they can be, but no closer than previous points along the order.
(Here then "close" is underdefined, it is assuming a metric in a
space.)
4. There are only so many points that can be equally close to the
origin, same as all the other distances. Then beyond those there are
just as many (in a particular manner of order in their description) at
the "next" distance, where those distances are defined in terms of the
previous dimensions' least distances.
Here there is much more to consider with regards to this
considerations of the formation of a monotonic substrate of the
natural continuum in the space-filling spiral curve.
Ross
Ross A. Finlayson
wrote:
Then here with these properties about the evolution of points along
the curve that define points in some space, there is a consideration
about why there would be circles instead of squares or any other n-
gon, where there is an expectation that there would be the formation
of a regular polygon of sorts with the circles as infinitely-"sided"
polygon. (While in the two-dimensional case there are regular n-gons
for each integer n > 2, in three dimensions and higher there aren't,
although the numbers of sides of regular polyhedrons correspond to
things like 4, 6, 8 in packing then 12, 20, 60, 100, 120, ..., in
polydimensional considerations).
A part of this consideration is that of the equality of the circles in
terms of that the rays from the origin go through each of the circles,
that a ray indicated as being a given number of points from the
beginning of an inner circle is a given number of points from the
beginning of an outer circle. A problem with that is in terms of re-
centering the origin, about the density of the points. The density of
the points should be uniform, given any neighborhood of a point, where
it is immediately adjacent to a variety of points (in this non-
standard continuum that is to correspond exactly with useful real
numbers.) Here this follows on to how many points there are in the
innermost circle, of the disc for the 2-D case, and as to what those
values could be for the higher cases. (The 1-dimensional case is a
line segment that is the unit interval of real numbers, here with a
nonstandard definition of the real numbers along the lines of R-bar-
umlaut with iota values in the polydimensional as a contiguous
sequence of points and a nonstandard completed infinity.) Basically
in consideration of how many points neighbor a point on the disc,
there are at least three. (On the 1-d line there are two.) There are
some cases where it is sufficient that there are three, others where
it is not sufficient that there are less than infinitely many, in the
innermost circle from considering the arbitrary point as the origin.
At the edge of the disc there's even a consideration that there is
just the one or two neighbor points. This is about a concept that as
the rays disperse the points are less dense, yet there still is to be
the support of uniform density of points throughout the space. Then
it would be to an extent about the perspective of algorithms. There's
a proliferation of cases. (Consider in analogy the path integral, vis-
a-vis, packing in the discrete.) In the polydimensional then the goal
is to determine what transcendent properties are synthesized from
these small primitives, and as well, from the large primitive or
space.
I'll write some more on this later, where basically there are
extensions of properties of the natural/unit equivalency function in
the context of the polydimensional, again towards defining a geometry
in terms of points and space.
Ross