- a single point, and
- a plane (π) in common.
- then they must either be co-incident, or meet in a single point.
Infinity in its Proper Place.
Projective Geometry works with just three primitive qualities (‘primitive’ in the sense of ‘prime’—i.e., ‘first’):• The Point: this is just “place”, “location”, “somewhere”.
• The Line: this is just “extension”, “straightness”.
• The Plane: this is just “spread”, “flatness”.
None of these qualities can be made “more” or “less” what it is. For example, a point cannot be made more of a place than it is, or less.
Thus these three qualities are just that—qualities.
Accordingly, they are not quantified, or quantifiable.
It follows that a line is not a sum of contiguous points, because a sum is a quantity, not a quality. Also, a point has no size—of any value whatsoever, not even zero. A line may extend through places, but does not consist of those places.
It also follows
The latter of these two statements will probably be regarded by most as almost laughably self-evident, while the former will probably be considered self-evidently ridiculous. But the statements are both true, each being simply each other's complement. We have a first hint here of something fundamental called, “Duality”, and also a hint that common-sense and intuition are not quite the reliable guides we may have supposed them to be.
The Axioms of Incidence list the ways in which the three qualities interact. In particular—
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If two straight lines (such as 1 & 2, on the left) meet, they have both
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Axioms 1 and 2, besides being dual, brook no exceptions.
By them, even “parallel” lines (perhaps such as those marked by ? above left), having a plane in common, quite simply must have a point in common—for, if they do not, axiom number 3 in the list above applies, and they must be skew.
However, Projective Geometry has no axiomatic means of “knowing” whether or not lines are parallel, since that is a matter of measurement, of size, of units and of quantity. None of these is ‘sponsored’ by the axioms. The whereabouts of the meeting point of parallels (said to be the ‘infinite’) is likewise a quantitative issue of measurement.
Projective Geometry does not know what parallelity and infinity are.
But, if, in the intrinsically-measureless projective-geometric context, we come nevertheless to believe that we have discovered a valid way to measure—which always means defining, assigning and counting units—and use this way to give numerical meaning to the word, “constant”,
and then go on to describe parallels as having “constant” separation,
we blunder quite fundamentally.
By that rotational (angular) reckoning,
lines are parallel iff ("if and only if") the rotational distance between them is zero.
That is, lines which, by some measure-system, have zero rotational separation, are parallel in that measure-system.
Judging by the heat that discussions on this topic tend to produce, all this is, for many, counter-intuitive. In order to cope, some mathematicians propose that parallel, straight lines meet in a single, “ideal” point at infinity. Here the word, “ideal”, is perhaps being used in much the same way as it is in, “There would be no poverty in an ideal world.” That is, we can think, or conceive of, such an ideal state of affairs, but we do not expect to observe it—or, observing it, consider that it must be illusory. Some insist that the “vanishing points” of perspective firmly belong in this illusory category. But an ideal, by definition, does not exist, except in the mind—for, were it to exist out of mind, it would be operational (and observable) fact, not necessarily in any mind. These latter two properties, namely,
As is detailed in the panel alongside, |
The diagram below is from H.S.M. Coxeter, “The Real Projective Plane”, 2nd Edition, Cambridge University Press, 1961, page 3. The shadow of a circle is projected from point L on to a table top. The line from L through A in the circle is supposedly parallel to the table top, Euclidean Geometers would insist that there is no shadow. The famous “fifth postulate” (in John Playfair's version) states that no two distinct parallel lines can be drawn through a single point, which implies they do not meet, for the reason that such a meeting, if it occurs, must occur in just such a point. But Projective Geometry says axiomatically that any pair of lines on a plane must meet, else they are not on the same plane and are skew. The ‘ideal-pointists’ attempt to reconcile these apparent incompatibilities. They do this by displacing the meeting point of parallels to numeric infinity—that is, to a position which is distant by an infinite number of unit distances. This precisely is the “ideal point at infinity” mentioned on the left, and the geometry arising from the use of this device is called “Affine Geometry ”. | |
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Now, as I read Mr. Coxeter's diagram, there are certainly shadows for points of the circle other than A, and these shadows are themselves points, but, for the sake of argument, assume that point A is indeed the exception, and has no shadow. There is then, presumably, a gap in the shadow. A gap left by a "missing" point is just a place. But a point (and, for that matter, the shadow of a point) is, "just a place". Ergo, the gap is a point—so A has a shadow, and is not in fact exceptional! The lesson here is that points, being places, cannot be missing! To see this more simply and directly, just suppose that some points have by some unspecified means actually been “removed”. Then ask, "What remains where these points were?" |
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Because, in affine geometry, lengths seem independent of the orientation of the lines to which they pertain,
seem also to be preserved. But, for this to “work”—that is, to be consistent with the axioms of point, line and plane—we must have a plane bearing all the points at which sets of mutually parallel lines meet. This is the so-called Ideal Plane at Infinity. Recalling that Infinity is unknown to Projective Geometry, the ‘Infinity’ referred to here must depend entirely on non-geometric properties, namely, the numerical properties of the Measure System imported into, and grafted on to, the geometric context. Physical bodies ( especially solid ones) are widely supposed to preserve properties of measure just as above: the units engraved on a ruler, for example, are expected to retain their sizes whatever the orientation of that ruler. If such bodies do preserve measure in this way, then we must expect that for them the ideal plane at infinity will have particular significance. For a non-abstract application of all this, see below, under "Club Quarters Hotel". |
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Thus, if we are persuaded that we actually have a pair of
parallel lines, such as we might suppose straight railway lines or the verges of a straight road to be, it must follow that their
(very obvious and observable) convergence to a meeting-point (the so-called 'vanishing' point of perspective) tells us where infinity is, at least in respect of these lines. The point where they meet cannot be a fiction, or some sort of abstract (i.e., unreal), 'ideal' state, because as noted already, the lines either meet, and have both a plane and a point in common, or they are skew. In other words, they cannot both meet and fail to meet. Their meeting must be an actual place, to which we
could - in
principle, if not in practice - travel.
The key realisation here is that the practical limits on travel to infinity are not geometric. Rather, they are physical. So it is probable that it is to physics, not geometry, that we must turn for the true properties of infinity. We reiterate that pure projective geometry simply does not know what infinity is, and emphasise that any element that may seem, for whatever reason, actually to be at infinity will be treated by the geometry just like any other element of its kind. For the pure geometry, "infinity" is a redundant and, in fact, meaningless notion. |
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Consider the blue quadrangle (or dually, the quadrilateral) depicted on the left. It is formed by four lines in the same plane. By axiom 2, every line meets every other in just one point, giving us six such points in all. In particular, lines a and b meet in point X, Now, if line a is held actually to be parallel to line b, then X is actually - not 'ideally' - at infinity. Similarly, if line c is held actually to be parallel to line d, then Y too is at infinity. In which case, the entire line l joining X to Y must lie at infinity - and what we have here is a parallelogram in perspective, for which X and Y are the so-called "vanishing points", and line l is the so-called "horizon". This could be a diagram traced from a photograph of a sheet of blue foolscap floating in the air above our heads—or of a skylight in the Club Quarters Hotel, Trafalgar Square, London, UK! The opposite sides of the sheet, or of the skylight, certainly do not look parallel, but yet - somehow they do, and, somehow, we "know" they are. So here we literally see lines believed (mistakenly, as we now know) to remain at a constant distance from each other actually meeting! So it seems that all we need do to resolve the paradox of the parallels is look askance at them! |
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Next consider the cuboid-looking item on the right. It is just the diagram above, once again, with the addition of a second plane of four lines apparently "below", or "behind", the first, with these lines defined by the points of projection of the four corners of the blue sheet onto the second plane by the orange lines through point Z. Now, if the orange lines are parallel, then point Z is at infinity, and if the green lines are parallel, and the black also, then points X, Y and Z form a triangle that defines: a
plane We have a parallelepiped
in perspective. |
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Interactive version. White points are draggable. |
But I need to emphasise that the only way that you, gentle viewer, know that these are parallelograms/parallelepipeds,
etc., is because I, the drawer of the diagrams, have told you so! They
can just as well not be para-whatevers;
the diagrams proffer no real clues as to their status in this respect.
They themselves do not "know" what they are besides being a collection
of planes, lines and points obeying the laws of incidence. |
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| That being so, and noting that when we look upon our world, we never see its objects but in perspective, we may ask ourselves how we ever come to the conclusion that a bit of pipe, say, or a plank, retains constant dimensions (length, etc.) no matter how we brandish it, because that is most certainly not what we see. |  |
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But we do conclude so, appearances notwithstanding. And we might say, "Well, if you want to be sure, you can always measure them!", but this recommendation has an obvious flaw—the thing (yardstick, ruler) with which we measure is subject to the same apparent variations as the thing being measured by it. The intervals on the ruler visibly do not maintain a constant size. Is our firm belief that they actually have a constant size then merely an article of blind faith? This would seem to be a good moment to have that proper look at measurement! |
