Involution and Conjugacy
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We find that some linear measures are confined to a segment of their line, and cannot escape from it, no matter how many steps are taken. The animation on the right shows a measure confined to the segment not containing the point at infinity.

The end-points of the segment, X and Y, mark the limits of stepping, and so, one might say, serve as numerical infinities (some call them "infinitudes"), for an infinite number of steps would seem to be required to reach either limit from any start.

Note, however, that stepping cannot continue along the line starting from either X or Y, as, in the case of X, the stepping point must remain colinear with U and D, and, in the case of Y, the stepping point must remain copunctual with Y.

Note also the corollary that the stepping point cannot actually reach X or Y, from any start, as it is never quite colinear with U and D, or copunctual with Y, and no number of steps, however large that number, can make it so.  For to take a step at all, the stepping point and U and D cannot be colinear, and it cannot be copunctual with Y.

We can do this stepping in two ways, or senses. Our steps may track right, or they may track left.

If we go right —> from X, we soon encounter Y. If we go left <— from X, we must trace the line via its point at infinity before encountering Y.

Please do note that changing from one sense to the other does not involve a change of direction. Such a change would involve rotation of the line, but here there is no such rotation. Direction and Sense are different things, often confused.

A Sense has an Opposite.
A Direction has not.


So, any measure started to the left of X in the segment containing the point at infinity and always stepping <— left, must be expected to step via infinity to Y—and can never leave that segment.

Passage through infinity affords no release!

It would thereby seem that the measures in the two segments can be independent of each other, but if we wish that a step in one of the segments should be matched by an exactly equivalent step in the other, then we need a measure that steps in both segments.

Now, it turns out that there is a construction which does step in both - but not at the same time. With it, the steps jump from one segment to the other.

They alternate.

Alternating (Oscillating) Measure uses Both Segments

Note the "Winding Quadrilateral"!

Save for the alternation, the measure behaves with respect to the end-points as other measures do. We see the steps expand from one end-point then converge on the other.

The animation reveals a winding quadrilateral being drawn by the stepping process, in this instance clearly "shrinking" towards Y.


We need only have the intermediate line, i, pass between the directing points on m—that is, through the segment, UD, of line m not containing the end-point, X.

If step N falls in S1, then step N+1 falls in S2, step N+2 falls in S1, and so on.

Thus even-numbered steps fall in one segment, and odd-numbered steps in the other.

(We could say that the point escapes its segment at every step!)

If we could "freeze" the shrinking (so that the quadrilateral only winds), then the steps would alternate between fixed positions in the segments.

Each step would send the measure back to the segment, and position in that segment, from which it had just come.

It turns out that we can freeze it, and the animation on the right indicates how such a frozen quadrilateral is made, and then shows the effect of it.

This can be thought of as the geometric equivalent of repeated multiplication by -1 (minus one).

Any operation that,
when applied twice to an object,
recovers that object,
is an Involution


We see the numbers increasing as before, odd on one segment, even on the other, but their positions do not alter.

We may say that
Involution finds Exactly Corresponding Points of segments.

Such Corresponding Points
are said to be
"Conjugate Pairs "

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