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On Real and Imaginary (2)

Cones
Conics(1)
Conics(2)
Conics(3)

On Real and Imaginary (1)

Projective Geometry is concerned with incidence—with meetings

.

Two real lines in a real plane must meet, at some real, locatable place, so that meeting is real, and with it, they share a real point.

All the elements in this case
- the plane, the lines and the point -
are real because they are locatable.

Strictly speaking, planes and lines never have locations, as location (place, somewhere) is uniquely the quality of a point.
They are, however, findable by the locatable virtue of the (real) points with which they are incident.

  tworegion

Conics, being closed, divide the plane into two regions; both are enclosed by the conic.

nocut

This makes it possible for a line to be at the same instant entirely in one of the two regions and not at all in the other,

and this means that a line and a conic need not intersect.

REAL POINTS

iscut

If a Real Line and a Real Conic in a Real Plane intersect, they share two, locatable points—the so-called real points.

These points exist, and it is in principle always possible (though it might not be practical) to visit such points.

IMAGINARY POINTS

notcut

If a Real Line and a Real Conic do not intersect, they share two, non-locatable points—the so-called imaginary points.

These points exist, but it is in principle never possible to visit such points.

 

But real and imaginary elements have a
relational context in common

Please enable Java for an interactive construction (with Cinderella).

Real lines can connect
points in a line to points in a conic,

whether or not
that line and that conic intersect
.

Experiment with the moveable elements, and
see the effect on the bold blue interval
of having the thick blue line cut, or not cut, the conic.

You should find that

 
  • there is an interval whether or not the conic is cut by the blue line.
  • if the blue line does cut the conic, the blue interval degenerates to a double-point—twice.
  • if the blue line does not cut the conic, the interval never degenerates.
 

This simple thing underlies metrication and measurement

 
  • The intersecting, "real" version leads on to
    point-to-point metrication along a line,
    and the quantification of linear distance.
  • The non-intersecting, "imaginary" version leads on to
    line-to-line metrication around a point,
    and the quantification of rotational distance.
 
Click here for a more sophisticated demonstration, embodying some of this “metrication”.

 

REAL
RELATIONSHIPS

If line and conic do intersect, all the ways in which these bridging lines can connect points is the ensemble of real relationships.

IMAGINARY
RELATIONSHIPS

If line and conic do not intersect, all the ways in which these bridging lines can connect points is the ensemble of imaginary relationships.

On Real and Imaginary (2)

It is often asked what Imaginary elements, "actually are."
There seems to be an ontological issue!

It is, admittedly, rather hard to fathom how a point, whose only property is location, can be “non-locatable”.

Simplicity is the key.


Though a point may stand alone,
a point of the kind now in question exists by virtue of a relationship.

Whether real or imaginary, it is
one thing,
found as an intersection
(i.e., a relationship) of
two things

(lines, or a line and a curve),
and is traceable/detectable only by virtue of that relationship.

It is probably obvious that there are relationships – as opposed to their members – (such as marriage, or friendship, or family) without whereabouts.


It may not be so obvious that all relationships are without whereabouts
—they are all ‘non-locatable’.

Formally-

A relationship
is between at least
two things,
while a place is just
one thing.

Two things simply cannot be a place.


If the above seems dubious, consider instead that, if a line is not a point, it is still not a point on addition of a second line, or of a curve. Moreover, neither is the added line, or curve, a point, nor can it become one.


(See http://mathworld.wolfram.com/ContinuityPrinciple.html for a very formal statement of the position!)

Some History

On the conics(3) page, we see the conic defined algebraically, this being done there, and historically, in response to the realisations —

  • that plane curvature cannot be one of the unary qualities defining a fundamental element of geometry, like the point, line and plane all are, but must instead be fashioned from a binary combination (point and line),
  • that there seemed to be nothing native to the axioms with which to direct this combination in an orderly fashion.

To meet the case (and nevertheless stick as closely as possible to the axioms), metrics and co-ordinates were derived from the projective ground up, and used to direct the combination through the ordered numbers of algebra, and first and second order functions.

Straight lines are first order. Conics are the second order item. Algebraically, this means the conics function has terms in x raised to the second power - that is, x2, while straight-line functions have terms in x to the first power only.

Graphically, being second order corresponds to being the kind of curve cut exactly twice by a straight line. Working algebraically, on the other hand, means taking a square root to find the co-ordinates of the cuts, and there are always two answers to this, one for each of the cuts. (All this of course assumes that numbers and points-in-curves invariably go one-to-one. They don't, not invariably, but ignore this—at least until the foot of this panel.)

And it was in this business of extracting square roots that the imaginary was originally detected, because when there is no intersection, the numbers under the square root sign turn negative—and you can't get a real answer extracting roots from a number in that state!

But that negative square is real (it is calculable), so the inference is that there must be a root – actually, two of them – of some sort (they must be in some way calculable, too!), for all that we can't write them down in cold blood, and, in the nature of the case, they must refer to points.

The lesson for us now is that, from this, at any rate, the imaginary is numeric, not geometric. We seem here to be concerned about the nature of certain kinds of number, not about certain sorts of line, or point, or plane.

It hereby becomes clear, if it had not been already clear, that the connection of numbers to elements of geometry is not immediate, or indeed elementary, and we are left with a curious apposition—

  • There are numbers for which, seemingly, there are no locatable points (imaginaries)
  • There are points for which, seemingly, there are no locatable numbers (irrationals)