How Do You Define “Parallel”

It amused me yesterday to discover that to ambitious mathematicians the word parallel need not mean what the dictionary says. Webster’s says, “extending in the same direction, everywhere equidistant, and not meeting.”

The crux of the definitional problem arises from Euclid’s Fifth Postulate. It says:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Not what you might call an elegant postulate. To make it plain we encounter it usually with an illustration, such as you see to the left. The sum of the two angles is greater than 180°, therefore the lines converge. Now what underlies this postulate are some assumptions. One is the nature of the plane on which the lines live: flat and two dimensional. Another  is that straight on such a plane is not curved in any way. A third is embedded in Euclid’s Second Postulate, namely that any line segment can be extended continuously—no limits.

The Scottish mathematician John Playfair (1748-1819) formulated a simpler version. Here it is:

Given a line and a point not on it, at most one parallel to the given line can be drawn through the point.

That postulate is also usually illustrated, as shown. The point Playfair mentions is the P, the dotted line the only parallel line. Playfair obviously understood that a parallel line means a line always at the same distance from another line to which it is parallel.

Non-Euclidean geometries all rely on two assumptions. One is that line segments may be limited in length or that the plane on which they persist may be curved. Here for instance is an illustration of how the word “parallel” is abused (in my opinion). I show a diagram and quote the explanation for it—it concerns the Beltrami-Klein model of non-Euclidean geometry. The link to the site is here:

The Beltrami-Klein model considers the region strictly inside a circle as a model of a plane. Note that the region does not include points on the circle itself. Lines are chords connecting points on the circle with the endpoints excluded. (For lines should belong to the plane.) Line AC and BD pass through point P and both are parallel to AB. Furthermore, all the lines between AC and BD (inside the angles APD and BPC) are also parallel to AB. It's very easy to verify that the first four Euclid’s postulates hold but there is [sic] infinitely many lines through a given point and parallel to a given line.

Note that here “parallel” obviously means “lines that do not intersect.” And that possibility exists only because the nature of the plane has come to be defined in a limited way. Using that definition, of course, the very illustration used above to make Euclid’s Fifth Postulate plain shows “parallel” lines if we restrict the plane to the illustration’s white area.

Too much of modern science relies on such trickeries to be inventive of brand new concepts, among them my favorite bête noire, spacetime.