horizontal parabolas

[Andrew Dinn]

Stuart Moulthrop writes:

> Yes yes yes....  maybe the Zone is constituted entirely of said parabolas,
> opening right and left?  A-and maybe they _overlap_, creating _interference
> patterns_...

> Anyway, Ryan, it's an idea worth pursuing.  Hope to see it in _Pynchon
> Notes_ some day.

I have been playing since the Warwick conference with a conceit which
encompasses parabolas and other related geometrical figures. Since I
probably will not be able to spend more time on it and apropos of the
current discussion here it is, with the usual apologies if it is
premature.

I think Steven Weisenburger's book points this out (although it is
relatively straightforward to prove): the movement of the rocket is
not correctly modelled as forward movement through a downward
gravitational force field but rather as motion under a central force,
the resultant trajectory being most readily solved as an ellipse.
However, as the discussion in Jules Verne's `Journey to the Moon'
points out there are three possible solutions to motion under a
central force, corresponding to the three conic sections, the
parabola, the ellipse and the hyperbola, the circle being a (very)
special case of the elliptical.

It is pretty evident that Pynchon's work manages (at the same time?)
to be elliptical, hyperbolic and parabolic. One might cite `V.' as the
paradigm of ellipsis (that's V period, damn it!) and `Gravity's
Rainbow' as *the* exemplar of C20th hyperbolic. How can these three
figures be united by some square-circling magic?  Consider how the
three terms come to acquire their literary connotation from their
origin as conic sections i.e. as the figures made by the intersection
of a cone and a plane.

Imagine if you will a circular cone balanced on its tip which rests at
the origin of a set of axes and which expands infinitely upwards
centered around the vertical axis. Pair it with another cone which
extends infinitely downwards the two noses touching at the zero
point. In fact the two cones are really one and the same surface,
rushing downwards from infinity from all sides in towards the origin
passing through the zero point and beyond expanding outwards to
negative infinity. In cross section they look like a giant V balanced
precariously but perfectly atop a giant A.

Now imagine if you will a plane which cuts through the V cone normal
to the vertical axis, parallel to the two horizontal axes. It
intersects with the V cone in a circle whose centre lies on the
vertical axis itself, a perfect figure with a single focus at its
centre, equidistant from all points in its locus.

Next imagine a giant hand entering the picture stage left, middle
finger extending to tilt one side of the plane upwards. The circle of
intersection stretches down towards the zero on the right side and
away from the zero on the left, producing an ellipse. The single focus
of the circle splits producing two foci one either side of the
vertical axis. The total distance from focus to a point on the ellipse
to the other focus is constant, but points on the path lie some nearer
to one some nearer to the other focus. So, the ellipse manages to
compress two foci within the scope of one figure, which feat of
encapsulation fails however to give independent measure of either. The
foci must be derived from motion around the figure itself.

Anchor the plane where it intersects the cone on the lower edge of the
ellipse and let the finger continue its upward displacement until the
angle of the plane approaches the angle of the cone itself. The rising
edge rushes ever upwards as does the upper focus. When the plane
becomes parallel to the line of the cone then the figure made by the
intersection sweeps dwonwards from infinity around the curve of the
cone and back upwards again to infinity. This section of the cone is a
parabola. It only has one focus proper near the base of the parabola
the limit point for the ellipse's lower focus but the disappearing
upper focus of the ellipse is implicitly present at infinity. The
parabolic figure is open ended so it does not contain this implied
focus but it is there as a shadow of its original for those who look
to the figure's origins.

The parabolic figure represents a transition point, a perfect
incidence of the plane and the cone. Let the finger push the plane
just a little further and not only does it intersect the V cone but it
also cuts the A cone. This section is the hyperbola, made of two
separate segments, one above and one below the zero point. The
parabolic focus at infinity returns at negative infinity and ascends
towards the zero as the plane tilts further. Let the finger continue
until the plane is vertical. It then cuts the V and A cones in two
equal figures, each with its own focus, neither related to the other
except through some distorted transition via the geometry of the
infinite. The original figure has been stretched so far it has snapped
with a visible discontinuity. It has reached so high it has ended up
subverting its own ground from underneath, kicking itself from
behind. An ass-backwards figure if ever there was one.

So, we have travelled from the elliptical V section with multiple
meanings crammed into one figure to the parabolic V section where the
figure contains a literal meaning but hints at an alternative meaning
approached but not arrived at by the figure itself. Then the finger
has shown us the A and V sections, a figure and its anti-figure, a
meaning and its opposed anti-meaning. Is the hand finished or is there
a final motion? Let the finger pull the vertical plane back towards
the vertical axis through the origin. As it does so the hyperbolic
arcs sharpen and steepen, their foci shift downwards until, when the
plane coincides with the axis of the A and V cones, the arcs
degenerate into lines meeting at the origin in a crossroads, the foci
converge again into a single point at the zero. The final section is a
degenerate form, distributed infinitely along its axes, not containing
but pin-pointing, crossing through, radiating outwards from its single
focus, a perfectly matched, opposed A and V.

Oh, I think I promised to square the circle earlier. Well, I know it's
impossible but you might note that the ellipse and the other other two
figures can be regarded as various different instances of the
`Quadratic Form'. You remember Quadratic? Quadratic Slothrop? He was
mentioned early on in the Slothrop genealogy section along with his
father and grandfather - not by name, of course, but he's there all
right.


Andrew Dinn
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there is no map / and a compass / wouldn't help at all