Implicit Geometry
On previous pages of this “Imaginary Circling Measure” set, there is reference to “overlap” of the points belonging to successive cycles of an imaginary circling measure, and to the fact that it can be exact, or inexact. If it is exact, then successive passes or cycles put points always in the same places. If it is inexact, they will be put in progressively (or, indeed, regressively) different places per cycle.
But on none of these pages does it say how to pre-dispose the determining elements of the transformation to produce a specified overlap.
This is because it cannot be done explicitly.
The situation can remind us of the dilemma of much-fabled Superman: he needed - in the original comic strip, if not in the movies - a supply of kryptonite in order to get to planet Krypton to secure a supply of kryptonite.
We need already to have the layout of all our elements doing just what we want to be able to discover which layout will do just what we want!
It is the geometric equivalent of an implicit equation, such as Kepler's equation, which implicitly expresses E, the eccentric anomaly of an elliptical orbit in terms of e, the eccentricity of the orbit, and M, the mean anomaly—
M = E - e.sin(E). ^{[*]}
Try as we will, we will never be able to change the subject of this equation so as to isolate E on the left side of it (which would make E “explicit”), and as a result, we cannot determine what value E must have to allow the equation to supply whatever we have chosen for the target value of M.
Similarly,
because the configuration of determining elements of our transformation is implicit—meaning that, because they are not independent, and interact, we cannot isolate and set the elements one-by-one in terms of all the others—it is impossible exactly to anticipate which disposition of transforming-line, conic or directrices will ensure that a measure recycles after exactly n steps, where n is a finite integer (except in the case of n = 2, which belongs to the well-known transformation called, “involution”).
We see that there can be implicit geometric configuration, too.
However, just as we can “solve” an implicit equation arbitrarily-nearly, using guesswork followed by some sort of error-driven iteration, we can in a similar fashion serially adjust the disposition of the elements of our geometric configuration so as to approach the layout for a measure repeating after a pre-specified n steps.
Of course, if we have a suitable applet, such as the one providing the interactive construction below, and here, we can do it by hand, eye and mind - though not perfectly accurately - and this is tantamount to “measuring” things into place, since we are adjusting things according to how far they are seen (guaged) to be, after each adjustment, from where we want them to be.
The Graphical ‘Solution’ The construction on the left has been sequentially set up to show eight points and seven intervals between them as an imaginary circling measure on the transforming line, as mediated by the conic, and by the directrices D_{1} and D_{2} on the conic. An eigth interval appears to lie between the first and last point and include the point at ∞, and, though it is an interval, it is not one in our series. The order of present business is, in fact, to ‘eliminate’ this interval, by degenerating its bounding points into one. For easy location, the first and last points in the set have been labelled. Drag white elements to have these dashed lines co-incide in a point on the transforming line as nearly as can be managed, dividing the entire line into just eight intervals, albeit approximately. Hint - first move the unlabelled white point on the conic to lie on the other arc between the directrices. |
That we do this at all shows that we are confident that the sought-for configuration of elements actually exists, for all that we are denied the means to work out exactly what it is, or to find it with perfect precision. Doing it means that we are sure it is there to be sought!
As an aside, in all this lies why we cannot directly trisect a sector or a segment, or indeed, divide it equally by any integer not a power of two (2).
* It will be appreciated that the choice of Kepler's equation as an example of an implicit equation was no accident.
Mass in orbit shows physics at its most certain, because very nearly only Mass, Gravity, Space and Time need be considered when attempting to account for what happens. Few physical phenomena manifest with such splendid absence of complication as planets in orbit round their primaries. The mathematics are correspondingly spare, and come very close to being the most elementary and fundamental mathematics available. So it should seem that we have here the most demonstrable and most secure connection between physical and mathematical things.
But the “controlling” equation is an implicit one, which therefore cannot be solved explicitly and exactly. This must bring at least the precision of cause and effect into question, if we are inclined to believe that an orbit is maintained as continued solutions of an equation - because, to say it yet again, solutions to implicit equations can't be found exactly, or indeed instantaneously, let alone in advance. (Are we not asked to accept that a cause must precede - that is, come in advance of - its effect?)
If we protest that no reference of any kind to any equation is made by the mass, while that mass is busy tracing its orbit around another mass, then the connection between mathematical and physical things must become less definite and clear than at first it seemed to be. Indeed, there may be no connection at all. If there in fact is one, what is it?