Forewarn: This is just a bit of aimless amateur rambling. Harmless fun. Nothing profound. Hope you like it.
I was reading this paper “Pappus Theorem, 9 proofs and 3 variations” by Jurgen Richter-Gerber from “Perspectives on Projective Geometry” (thanks Christos Papavasiliou for telling me the author name in the comments!!)
Don’t understand all of it but one bit is fascinating.
So many theorems of plane geometry are true when projected in some way, so that our vision of relationships between lines and points can teach us relationships between circles and ellipses and lines, and on and on.
But what about Relativity? If space time itself is curved, then every plane geometry theorem becomes a theorem about curved geometry as well. Every theorem about lines in a plane becomes a theorem about lines in curved space.
If “an observer”, that famous character of Relativity texts, was observing a teacher talk about the beautiful patterns discovered by Pappus or Pythagoras or whoever, but this teacher was opposite some kind of space-time distortion bubble such as a black hole, then this observer would not view someone describing straight lines, in a plane. She would instead observe a teacher describing weird curves and arcs… and yet the theorems would be as true as ever.
We already see simpler examples of this in life, after all we do plane geometry on a 3 dimensional surface, perhaps a blackboard in a room, which in space is oriented differently depending on where we sit in class, and so the shapes are slightly off depending on our perspective. But we can understand the ideas anyways, the square of the legs of a triangle and so forth.
And when we describe 3 dimensional objects, well, perhaps we draw them on a blackboard and again, we have projected the dimensions down but still retain the underlying truths, such as perhaps Gauss’ formula about edges and faces and whatnot. But even if we have a 3d model of a polyhedron, a block of wood or plastic, when we view it visually it is in fact two dimensional. Human vision is still dealing with two dimensions, two retinas are simply two 2-dimensional surfaces, arrays of rods and cones.
And then in the Projective study of Geometry, they prove many fascinating theorems, such as making a statement about the relationship between lines and points, and then saying that the same thing holds true if you replace the word “point” with “line” and vice versa. Some famous examples are Pappus theorem about lines drawn between points on two other lines, themselves will intersect in three new points that are on a third line. But the dual is true (And I am too unknowledgable to state it properly here myself). Then there is Pascals theorem for six-sided polygons inside a circle, and a “Dual” of Brianchon’s theorem.
Some people say that Pascals’ theorem is like an enhanced version of Pappus.
And then there is “inversion”, where you take points inside a circle and ‘invert’ them through the circle and create patterns outside of it. Suddenly circles can become lines and vice versa. Then there is the projective sphere! Reimann and Hyperbolic projection Mobius transforms and all that! For example these youtube videos:
So then. . . .
If we went from Pappus talking about lines 2000 years ago, to Pascal talking about circles and modern people talking about cubics…
… then what are we actually talking about? What is the fundamental object we are discussing? If it is a line, then how come we can take it out and replace it so easily with circles or points? But if circles or points are the objects, why can we replace them so easily with lines? If you think of it from the observer watching the proofs through a black hole, it’s not even simple conics and such anymore, its warped arcs and stuff out of Interstellar…
Is it the relationship between these objects itself that is the more fundamental thing than the objects themselves?
It reminds me of something that has been in the back of my mind for years. Two equations
The first is from Norma Wildberger’s Rational Geometry. Q = Quadrance, or distance squared. It describes the relationship between the squares of the sides of a triangle with no area. Or…. you could say the 3 different possible Quadrances between three different collinear points.
The second is Descarte’s circle theorem. B= Bend (also called Curvature), which is .B4 can either be the big red circle or the tiny Cyan circle. (It’s a long story.)
You know the funny bit? If you replace one of those Bends with 0, then you can imagine your radius is infinite and you have a straight line. Which means you are describing three circles and a straight line, all tangent, which is exactly the configuration of Ford’s Circles.
Let’s have a look at those equations again, with B4 as 0.
In other words, every Ford circle triplet corresponds to a single triangle of no area. Or… quadrances between three collinear points.
You see, from one perspective, we are talking about lines and points and circles and radii and curvature, and patterns and relationships between them. But from a different perspective, we are talking about a single pattern, a single type of relationships, and the various ways it is expressed by objects. In a way, the relationship itself could be seen as more ‘fundamental’ or ‘primitive’ than the objects themselves.
This is an echo from Wildberger’s “Chromogeometry” in which he takes several of his theorems involving ordinary “Quadrance” (square of distance) and points out that they also apply if you consider Quadrance as any of these three:
blue (ordinary) quadrance =
red quadrance =
green quadrance =
So then… if there are these patterns and relationships that are out there, but we only discover them by looking at different objects in different projections and contexts…
… I wonder what we might discover by examining some of these relationships from “Relativistic projection”?