Keeping up my breakneck pace of introducing the ideas of Projective Geometry: At some point soon I will take a pause, so don’t worry, you’ll have time to catch up! I’m just using the available time while I’m teaching this course to get the PG conversation warmed up here on my Substack and lay some of the basic foundations as best I can, all the while keeping it nicely unpolished in Doing Whatever You Feel Like Doing, Gosh! style.
Last time we imagined the Diagonal points of the projective Net quadrilateral “mirroring” each other. As one point (D1) moved slowly to the left from S2 to S1 along a short line segment, the other (D2) moved to the right quickly, shot right “through infinity” and came out the other side!
If this seems hard to comprehend (in particular the shooting through infinity part) there is a simple drawing and imaginative exercise that could maybe help clarify what’s going on when I make a statement such as this, and when you try to make sense of saying that a point has “gone to infinity”.
This exercise is sometimes called Kepler’s Propellor, and it is named for the very same Johannes Kepler who is famous for many scientific discoveries in optics, astronomy and mathematics, among others. Perhaps most famously, Kepler worked out mathematically that the orbital paths of the planets around the Sun were elliptical rather than circular, thus improving upon the previous work of Copernicus who provided the mathematical proof of Galileo’s heliocentric solar system. Kepler was well aware of all of the geometry of his time. While Projective Geometry as such had not been put on solid theoretical footing yet (that came later with a mathematician in the 19th century named Janos Bolyai), Kepler was one (along with Gerard Desargues, more about his another time) who laid the foundation and seems to have been a person who intuited certain elements ahead of his time.
To do this exercise, simply draw a horizontal line. Let’s call it line b. Then somewhere above line b, place a point, call it point A. Then, imagine that there is a second line, a, which is drawn through point A. Line a meets line b in a point which we will call B.
Now, imagine that, as line b stays horizontal and stationary, line a is free to rotate slowly around point A. Let’s imagine that a rotates counterclockwise. Because point B is located where the two lines meet, point B will have to shift as line a rotates. Can you imagine how point B shifts? If you said it starts to slide to the right, you are correct!
(In my experience this movement is difficult for some people to picture. Sometimes I say that they can imagine that line a is a laser pointer and point B is the laser dot that is projected onto the flat floor as the pointer rotates slowly. Or, point A could be a lighthouse turning slowly and the light beam is falling upon the perfectly straight shoreline). Point B slides to the right, and, just like D2 in our previous drawings, the further B slides out to the right, the faster it goes. In fact, B speeds up very quickly indeed as the line rotates further. As the rotating line a approaches horizontal (and therefore parallel to line b), point B is going faster and faster, and shooting out “to infinity”, just like D2 did.
Then, as line a continues to rotate, it reaches and then passes the special position of being perfectly parallel to line b (we will come back to consider this special position closely); and so, just a tic later, suddenly point B which was far, far to the right, is now far, far to the left, moving back in toward our viewing page at a fast speed but now slowing down as it approaches its starting point.
The above drawing is meant to be an imaginative exercise that shows “snapshots” of the movement of line a and point B. Can you smoothly imagine the movement of point B as line a rotates at a slow constant speed? B is always moving to the right, with the exception of that special moment when B “pops” from being “all the way” to the right to being “all the way” to the left. The movement of B is constantly changing in speed. The slowest that B travels is when it’s directly under point A. The fastest is when the lines are parallel.
Now, to the heart of it: This exercise is intended to make it clear to you that you have a choice about that special position of parallel. There are two logical choices, both valid, but with very different outcomes.
You can choose (1) that at the horizontal parallel position, point B ceases to exist, and that a moment later, B reappears on the other side. In making this choice, you are introducing a discontinuity into the smooth movement of point B. You are saying that parallel lines do not have a meeting point. This choice could be thought of as the choice of Euclidean geometry, the usual geometry that most of us learned in school.
Alternatively, and this is the Projective Geometry choice, you can choose to say (2) that at the parallel position, B, the meeting of the two lines is still present, and is “at infinity” . . . and can be reached by going in either direction! Now, this may seem outlandish to the extreme at first. However, giving it some thought, one may be forced to conclude that there is no way to prove that one choice is more or less true than the other. This is because you have set up a paradox of sorts. You are thinking about two parallel lines separated by a finite distance, traveling out along an infinite plane for an infinite distance. This sets up a conflict between a couple different spatial intuitions. On the one hand, you might say it’s obvious that the two lines “never” meet. On the other hand, you might entertain the possibility that, since infinity is, after all, infinity, maybe it’s not so crazy to consider that they meet. After all, when one gazes along a pair of perfectly straight parallel lines (such as railroad tracks) as they recede into the distance, one sees the tracks appear to come together!
What we are talking about here is what are called “postulates” or “axioms” in mathematics. These are statements that are made without proof. We think of mathematics as the most rock solid of logical disciplines, but every mathematician knows that the only way to start doing math is to state some simple assumptions that are so obvious that they can be taken as true without proof, upon which you will build your mathematics. I will help to put this in context in another post soon! In the meantime, keep that propellor whirring in your mind and point B shooting through infinity and coming back from the other side!
Photo by Garrick Sangil on Unsplash