If you’ve been following the lessons pretty well so far, perhaps I’ve opened up a little space in your mind where you at least entertain the notion that two parallel lines might logically and consistently be thought to meet “at a point at infinity”. We’ve worked through the Kepler’s Propellor exercise, and the Harmonic Point exercises. All of this was intended to give your imagination a bit of a stretch . . . in order to be more open to further stretching to come!
I showed you how it’s clear that Euclid also struggled with the apparent simplicity of parallel lines being anything but. This planted perhaps the longest-lived “easter egg” in human history ever, hidden in plain sight (it’s pretty sure that Euclid didn’t know he was planting it, though!). The truth began to be unearthed starting in the 1400’s, but remained not fully recognized all the way until the 1800’s, when a mathematician called Janos Bolyai is credited with putting Projective Geometry on rigorous mathematical footing that could no longer be denied. But before we get to him, it’s worthwhile to look at a little more historical development.
As I have shared, Euclid’s geometry lasted a loooooong time, essentially unchanged. Students that have taken geometry in schools for the last however-many-years-compulsory-education-has-been-a-thing-in-the-West have learned Euclid’s geometry, and because no one understood that there was any other kind, it has just been called “geometry”. But, as I'll try to show you, Projective Geometry is best understood as a kind of “Ur-geometry,” the most general geometry that includes and transcends all others. Euclidean geometry, it turns out, is one specialized sub-class of Projective Geometry, achieved by placing certain points and lines “at infinity,” . . . and then forgetting or ignoring that they are there! There are in fact other sub-geometries that can be developed under the umbrella of the Mother Geometry of PG that are just as valid as Euclid's, but lead to different results. . . But we are not ready to go there yet!
Projective Geometry was developed in fits and starts, just like any new field of knowledge. However, the place where it really began to take off was not in mathematics, but in art. Things really started to shift in human collective consciousness as the artists of the Renaissance increasingly perfected methods to depict scenes with accurate optical perspective. If you’ve ever learned about how to draw in perspective, you know that there are “vanishing lines,” and “vanishing points”. One draws or paints lines that converge on the canvas, but which are supposed to be understood to be parallel in the scene. Those lines “recede into the distance” in the painting, collecting into a vanishing point. Below are a couple pictures taken from a slide show made by my friend Jeremy, who is Yoda to my Skywalker when it comes to Projective Geometry.
(Jeremy recently taught a course at Schumacher College, and in preparation for that made this very cool introductory video which he called “Archetypal Geometry”. In it, Jeremy makes a quadrilateral Net, but then adds another perspective point to turn it into a perspective 3-Dimensional cube. Check it out!)
If you study these two pictures, you’ll see that the artists are showing you how perspective pictures come about by creating lines that come together at the eye of the observer. In the first picture, you see a man (with a sweet toga, I feel compelled to add) holding strings to one eye as he sights along it. The standing screen bears a perspective image of the cube sitting on the ground:
The second picture is a woodcut from Albrecht Durer. You see a taut string held by the person on the left, running from the edge of a lute lying on the table, through an aperture with a screen on hinges. The person on the right is marking what we’d think of as the x and y coordinates of the location of the string. Then the string will be set down, the hinged screen closed, and another dot will be marked at the measured coordinates. When the dots are all connected, they will form the correct shape of the lute when viewed from that precise perspective. The “perspective point” is where the string is attached to a ring in the wall and the string is held taut by a hanging weight. All pretty cool, yes? What’s really cool is that these two illustrations are showing how artists depicted things in perspective . . . and the drawings themselves are also drawn in perspective!
So, there is something about our sense of sight that is very tied to Projective Geometry. When we take seriously that objects appear to shrink as they get further away from us, we can see that vanishing points and lines are correct from the standpoint of sight. In fact, when things shrink in our visual field, that is a cue that causes us to assume they are getting farther away from us.
Below are two perspective drawings “in process”, showing the lines and points that the artist must set up before starting to draw the scene. Spend a little time studying these, and look for the places where vanishing points and lines have been set up to give the appearance of depth.
We are poised now to try another “mind stretch, and again, I want to thank Jeremy for these images.
Recall that very first Projective Net that we made, with equidistant points:
We now understand that this Net is exactly the same process that artists set up to depict visual scenes that recede to the “infinite horizon”. This means that when we look at that horizon line, we are seeing a “line at Infinity,” right in our view.
Now, engaging your flexible imagination, what if we take that horizon line that we can see right now and actually move it out to infinity where we won't see it any more. Alternatively you can imagine that we are changing our angle of view, perhaps rising above the Net and gazing down as we move further away from the horizon line. What would we see and how would it be changing?
Here is the Net with the diagonals removed:
…And now we shift our perspective:
Can you see what is happening as the horizon line moves further out? The grid becomes more regular, the quadrilaterals are looking more and more similar. And now, can you predict how it will look when that horizon line has gone all the way to infinity and stopped there? . . .
We now have a regular, Euclidean grid, complete with measurement and even right angles!
The four harmonic points that we have been working with are all on that horizon line and therefore are all at infinity!
Wow. What we have discovered here is rather epic! First, that Euclidean geometry is only valid when all of the perspective points are placed at infinity, and then disregarded. Secondly, we have to conclude something new about the nature of a plane in projective geometry. We saw the horizon line move away from us “out to infinity” and now we see that the four points that are on that line are “shooting in from all directions”, creating a regular square with 45 degree diagonals.. This leads to the massively mind-bending conclusion that all of the points at infinity lie on a single line at infinity. . . and that you can reach that line at infinity by traveling in any direction in the plane!
Better stop here for now. See how you feel about all of the above and as always, let me know your reactions!